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Physics
Dr. Jay Maron

Units    Index of equations

Aerodynamic drag    Pendulum    Thermodynamics    Balance    Spin II

Units

One-second ticks

The fundamental units are the meter, second, and kilogram, and all other units are derived from these.

```Quantity                                        Units

Length                                          Meter
Time                                            Second
Mass                                            Kilogram
Velocity        =  Length  / Time               Meter / second
Acceleration    =  Velocity/ Time               Meter / second2
Momentum        =  Mass    * Velocity           Kilogram meter / second
Force           =  Mass    * Acceleration       Kilogram meter / second2    = Newtons
Energy          =  Force   * Length             Kilogram meter2 / second2   = Joules
=  ½ *Mass * Velocity2
Area            =  Length2                      Meter2
Volume          =  Length3                      Meter3
Pressure        =  Force   / Area               Kilogram / meter / second2  = Pascals = Joules/meter3
Density         =  Mass    / Volume             Kilogram / meter3
Frequency       =  1       / Time               Second-1                    = Hertz
Angular momentum=  Mass * Velocity * Length     Kilogram meter2 / second    = Joule seconds
```

Electric charge is also a fundamental unit and it is measured in Coulombs, but it won't appear in the laws of mechanics.

Index of equations
```Linear variables             Spin variables

Position    = X              Angle             = $\theta$ =  X / R
Velocity    = V              Angular velocity  = $\omega$ =  V / R
Acceleration= A              Angular accel.    = $\alpha$ =  A / R
Mass        = M              Moment of inertia = I =  M * R2
Force       = F              Torque            = $\Gamma$ =  F * R
Momentum    = Q              Angular momentum  = L =  Q * R
Energy      = E
Time        = T

Force        Momentum            Energy          Centripetal    Constant   Constant
acceleration   velocity   acceleration
Linear motion:  F = M A     Q = M V = F T     E = .5 M V2 = F X    A = V2 / R     X = V T    V = A T
Spin:           $\Gamma$ = I $\alpha$     L = I $\omega$ = Q R     E = .5 I $\omega 2$ = $\Gamma$ $\theta$    A = $\omega 2$ R       $\theta$ = $\omega$ T    $\alpha$ = $\omega$ T

Energy = .5 M V2 + .5 I $\omega 2$ + M g Height
Fgrav  = - M g = - G M0 M / R2
```

Motion in 1D

Constant velocity

If an object starts at X=0 and moves with constant velocity,

```Time              =  T             (seconds)
Velocity          =  V             (meters/second)
Distance traveled =  X  =  V T     (meters)
```

Constant acceleration

In the previous case the acceleration is 0.

If an object starts at rest with X=0 and V=0 and moves with constant acceleration,

```Time              =  T             (seconds)
Acceleration      =  A             (meters/second2)
Final velocity    =  V  =  A T     (meters/second)
Average velocity  =  Va = .5 V     (meters/second)
Distance traveled =  X  =  Va T  =  .5 V T  =  .5 A T2        (meters)
```
All of these equations contain the variable T. We can solve for T to obtain an equation in terms of (X, A, V).
```V2 = 2 A X
```
Sim:
Position, velocity, and acceleration
Equations for constant acceleration

There are four variables (X, V, A, T) and four equations, and each equation contains three of the variables.
At T=0, X=0 and V=0.

```
Equations                       Variables in       Variable not
the equation      in the equation

V = A T                             V  A  T             X

X = .5 A T2                      X     A  T             V

X = .5 V T                       X  V     T             A

V2 = 2 A X                       X  V  A                T
```

The figure shows the position of a ball at regular time intervals and the green arrow shows the direction of the acceleration.

```Top row        Zero acceleration (constant velocity)
Second row     Positive acceleration
Third row      Negative acceleration (deceleration)
Fourth row     Free-fall in gravity
```
In the language of calculus,
```Time         =  T
Position     =  X  =         =  $\int$ V dT
Velocity     =  V  =  ∂X/∂T  =  $\int$ A dT
Acceleration =  A  =  ∂V/∂T
```
Examples of position, velocity, and acceleration.

Sim:    Position, velocity, and acceleration #2

Force

```Mass         =  M
Acceleration =  A
Force        =  F  =  M A         (Newton's law)
```

Gravitational acceleration

For an object falling in gravity, the acceleration doesn't depend on mass and the acceleration is the same everywhere on the surface of the Earth.

```Mass                 =  M
Gravity constant     =  g  =  9.8 m/s2
Gravity acceleration =  A
Gravity force        =  F  =  M g       (Law of gravity)
=  M A       (Newton's law)
```
Cancelling the "M's", the acceleration experienced by the object is
```A = g
```
If gravity is the only force involved, then all objects experience the same gravitational acceleration.

We can distinguish between gravitational mass and inertial mass.

```Mgrav    =  Gravitational mass
Minertial =  Inertial mass
F        =  Mgrav    g            Gravitational mass causes gravitational force
F        =  Minertial A            Inertial mass governs the response to force
```
For all known forms of matter,
```Mgrav  =  Minertial
```

Falling

For this example we set g=10 m/s2 and assume there is no air drag. If an object starts at rest and falls under gravity, the distance fallen is

``` Time   Velocity   Average   Distance   Acceleration
(s)     (m/s)     velocity   fallen      (m/s2)
(m/s)      (m)

0        0         0         0           10
1       10         5         5           10
2       20        10        20           10
3       30        15        45           10
4       40        20        80           10

Distance fallen  =  ½ * Acceleration * Time2
```

Pounds
```g            =    9.8 m/s2
PoundAsMass  =  .4535 kg       =  Pound interpreted as mass
PoundAsForce =  4.448 Newtons  =  Pound interpreted as force
=  The force exerted by .4535 kg in Earth's gravity
=  .4535 kg  *  9.8 m/s2

PoundAsForce =  PoundAsMass * g
```

Momentum

```Mass      =  M
Velocity  =  V
Momentum  =  Q  =  M V
```

Impulse

Suppose an object undergoes a constant force for time T.

```Impulse  =  F T  =  M A T  =  M V  =  Momentum
```

Equal and opposite forces

Suppose 2 objects both start with X=0, V=0, and T=0, and that they exert a constant repelling force on each other.

Object 1 accelerates to the right (positive force) and object 2 accelerates to the left (negative force).

```            Mass   Acceleration     Force     Velocity after time T     Momentum after time T

Object 1     M1        A1         F1 = M1 A1      V1 = A1 T              Q1 = M1 V1 = F1 T
Object 2     M2        A2         F2 = M2 A2      V2 = A2 T              Q2 = M2 V2 = F2 T
```
The forces and momenta are equal and opposite.
```F2 = -F1
Q1 = -Q1

Total momentum  =  Q1 + Q2  =  0
```
The total momentum is constant in time. This is the principle of "conservation of momentum".

Conservation of momentum is equivalent to the fact that forces are equal and opposite.
Equal and opposite forces imply conservation of momentum.
Conservation of momentum implies equal and opposite forces.

Kinetic energy

Suppose an object starts from rest at X=0 and experiences a constant force.

```X  =  Distance traveled
F  =  Force

F X  =  M A X  =  .5 M V2                       (using V2 = 2 A X)
```
We can define a kinetic energy, which is equal to the force times the distance.
```Kinetic energy  =  F X  =  .5 M V2
```
Newton's law implies conservation of energy. For example, suppose an object starts at rest at height X and falls in Earth's gravity until it reaches the ground.
```Initial height                      =  X
Mass                                =  M
Gravity constant                    =  g  =  -9.8 meters/second2
Velocity upon reaching ground       =  V
Time to reach the ground            =  T
Kinetic energy upon reaching ground =  Ek =  .5 M V2
Gravity energy when released        =  Eg =   M A X
Total energy                        =  Et =   Ek + Eg

Eg = Ek     because    .5 M V2 = M A X
```
The gravitational energy at the start of the fall is converted to kinetic energy at the end of the fall so that the total energy is constant.
```
[d/dT] Et =  [∂/∂T] Eg  +  [∂/∂T] Ek
=   -M A V    +  M A V
=  0
```

Gravity

For a test mass experiencing Earth gravity,

```Mass of Earth      =  M                 =  5.972e24 kg
Radius of Earth    =  R                 =      6371 km
Gravity constant   =  G                 = 6.67⋅10-11 Newton m2/kg2
Test mass          =  m
Force on test mass =  F  =  G M m / R2  =  g m
Acceleration       =  g  =  G M   / R2  =  9.8 m/s2
```

Collisions

The simpliest case of a collision is two billiard balls colliding head-on and with equal speeds.

We henceforth use dimensionless units.

```Mass      =  M
Time      =  T
Velocity  =  V
Position  =  X  =  X T
Momentum  =  Q  =  M V
Energy    =  E  = .5 M V2

Mass   Initial    Final     Initial     Final     Initial  Final
velocity  velocity   Momentum   momentum   energy   energy
Ball 1     1       +1        -1         +1         -1        1/2      1/2
Ball 2     1       -1        +1         -1         +1        1/2      1/2
Total      2      n/a       n/a          0          0         1        1
```
Momentum is always conserved. In this example the total initial momentum is equal to the total final momentum.

Energy is either conserved or some energy is lost to heat. In this example energy is conserved.

If energy is lost to heat then the rebound velocity is less than the initial velocity. If the rebound velocity is "K" then

```          Mass   Initial    Final     Initial     Final     Initial  Final
velocity  velocity   Momentum   momentum   energy   energy
Ball 1     1       +1        -K         +1         -K         .5     .5 K2
Ball 2     1       -1        +K         -1         +K         .5     .5 K2
Total      2      n/a       n/a          0          0        1          K2
```

We can define a collision coefficient "K" as

```K2  =  Collision coefficient  =  FinalEnergy / InitialEnergy
```
If you know the value of K for a collision then you can solve it by writing down down equations for momentum and energy conservation.
Solving a collision

The easiest case is if you are in the frame of the center of mass so that the total momentum is zero.

```            Mass   Initial    Final     Initial     Final      Initial       Final
velocity  velocity   Momentum   momentum    energy        energy
Object 1     M1      V1i       V1f        M1 V1i     M1 V1f     .5 M1 V1i2    .5 M1 V1f2
Object 2     M2      V2i       V2f        M2 V2i     M2 V2f     .5 M2 V2i2    .5 M2 V2f2

Total momentum  =  M1 V1i  + M2 V2i
=  M1 V1f  + M2 V2f
=  0
```
If energy is conserved then
```V1f = -V1i
V2f = -V2i
```
If energy is not conserved then
```V1f = -V1i K
V2f = -V2i K
```
where K is the collision coefficient.
Power
```Power  =  Energy / Time  =  Force * Distance / Time  =  Force * Velocity
```
Suppose you climb a flight of stairs.
```Height of a flight of stairs              =  X          =  4 meters
Typical time to climb a flight of stairs  =  T          =  2 seconds
Mass of a typical human                   =  M          =  75 kg
Gravitational energy gained               =  E  =  MgX  =  3000 Joules
Power delivered in climbing the stairs    =  P  =  E/T  =  1500 Watts
```

Frames of reference

Galilean transformation

Black: no drag.    Blue: Stokes drag.    Green: Newtonian drag
Earthquake defense

Often a problem can be simplified with a strategic choice of reference frame. For example, for an object in free fall the center of mass follows a parabola regardless of its angular momentum.

The motion of an object can be described as the motion of the center of mass plus an angular momentum vector. In the above figure the red dot is the center of mass.

Practice being in the frame of the sword, or your opponent, or the center of mass between you and your opponent.

There is a sequence of reference frames: The Earth, the tip of the spine (the atlas vertebra), the tip of the index finger, and the tip of the sword. They should be programmed hierarchically in this order. For example, you should be able to move the atlas vertebra while preserving the reference frame of the Earth (maintain balnance), you should be able to move the index finger while preserving the frame of the atlas vertebra, etc.

Center of mass

If two objects are placed on a seesaw, the center of mass is the position of the fulcrum.

```Distance from the left ball to the fulcrum  =  a   =   1
Distance from the right ball to the fulcrum =  b   =  20
Mass of the left ball                       =  M1  = 100
Mass of the right ball                      =  M2  =   5

M1 a = M2 b
```

Initial position and velocity

The equations of constant acceleration usually assume an initial velocity and position of 0. If not, then

```Time             =  T
Initial position =  Xi
Final position   =  X
Initial velocity =  Vi
Final velocity   =  V
Acceleration     =  A

X(T) = Xi + Vi T + 1/2 A T2
V(T) = Vi + A T
A(T) = A
```
If Xi = Vi = 0 then the equations reduce to
```X(T) = .5 A T2
V(T) = A T
A(T) = A
```
For example, suppose a car starts at X=5, has an initial speed of 20 meters/second, and decelerates uniformly at a rate of 10 meters/second2.
```X(T) = 5 + 20 T - 5 T2
```

Velocity of mass

The "center of mass" (COM) and "velocity of mass" (VOM) are defined as

```                 Mass   Position  Velocity
Object 1         M1      X1         V1
Object 2         M2      X2         V2
Center of mass   Mc      Xc         Vc

Total mass       =  Mc  =      M1 +    M2
Center of mass   =  Xc  =  (X1 M1 + X2 M2) / Mc
Velocity of mass =  Vc  =  (V1 M1 + V2 M2) / Mc
```
If object 1 and 2 are balanced on a seesaw then the center of mass is at the fulcrum.
```(Xc - X1) M1  =  (X2 - Xc) M2
```
If two objects exert a force on each other then the trajectories are
```X1  =  X1i  +  V1i T  +  .5 A1 T2
X2  =  X2i  +  V2i T  +  .5 A2 T2
Xc  =  Xc   +  Vc  T
```
The center of mass moves with constant velocity.

Hooke's law

```Displacement of the spring =  X
Spring constant            =  K
Force on the spring        =  F  =  K X        (Hooke's law)
```

Energy of a spring

For a spring, the force is proportional to displacement.

A spring contains compression energy if compressed and tension energy if stretched.

```Energy of the spring  =  E  =  $\int$ F dX  =  $\int$ K X dX  =  .5 K X2
```

Motion in 2D

Punt

Suppose a football is kicked vertically upward with a velocity of 20 m/s.

```Initial vertical velocity =  V                =  20 m/s
Maximum height reached    =  X  =  .5 V2 / g  =  20 m
Time to reach max height  =  T  =  V/g        =   2 s
```
The "Hang time" is the total time the football spends in the air. The upward and downward parts of the trajectory each take 2 seconds, for a hang time of 4 seconds. The downward trajectory is the mirror image of the upward trajectory.

We could write the Y trajectory as

```Y(T)  =  20 T - 5 T2
Vy(T) =  20 - 10 T
A(T)  = -10
```

2D parabolic trajectory

Suppose a punt is kicked with a horizontal velocity of 20 m/s and a vertical velocity of 20 m/s.

The horizontal motion corresponds to constant velocity and the vertical motion corresponds to constant acceleration.

```Time                  =  T
Horizontal velocity   =  Vx =  20 m/s          (constant)
Vertical velocity     =  Vy =  20 - 10 T
Football X coordinate =  X  =  Vx T
Football Y coordinate =  Y  =  20 T - 5 T2
```
We can eliminate T from the trajectories and express Y in terms of X.
```Y  =  X - X2 / 80
```
The ball follows a parabolic trajectory. In general, the trajectory of any object moving under gravity is a parabola.

The ball hits the ground when X=80 and Y=0.

Baseball

Fastball
Curveball

Suppose a ball is pitched horizontally, with no initial vertical velocity. Suppose a second ball is dropped with zero initial velocity from the same release point as the pitch. Both balls hit the ground at the same time.

```Distance from the pitcher's plate to home plate             =  18.0 m     (measurement)
Distance from the pitcher's plate to the ball release point =   2.0 m     (estimate)
Distance between the ball release point and home plate      =  16.0 m     (estimate)
Height of the pitcher's mound                               =   .25 m     (measurement)
Height of the ball release point above the pitcher's mound  =  1.25 m     (estimate)
Height of the ball release point above the field            =  1.50 m     (estimate)
Speed of a typical fastball                                 =    43 m/s   (96 mph)
Ball travel time from the release point to home plate       =  .372 s
Vertical distance the ball drops before reaching home plate =   .69 m  =  .5 g T2
```

Magnus force

Topspin

Topspin allows for faster shots
Topspin enhances the bounce

A spinning ball curves, a fact that was first observed by Newton while playing tennis. This is called the "Magnus force".

A ball with topspin curves downward and a ball with backspin curves upward.

A fastball is thrown with maximum speed, which puts backspin on the ball, giving it an upward force. This frce works against gravity and makes the ball easier to hit. A curveball has topspin, which works with gravity and makes the ball harder to hit. The topsin comes with a sacrifice in speed.

```                      mph
Fastball              95
Split finger fastball 90
Curveball             85
Knuckleball           60
```
A split finger fastball is thrown with wide fingers to decrease the backspin. A knuckleball is thrown with minimal spin so that it curves in multiple directions on its way to the plate. A knuckleball curves because of airflow around the seams.

Split finger fastball
Split finger fastball
Knuckleball

Circular acceleration

```Radius of the circle    =  R
Velocity                =  V
Centripetal acceleraton =  A  =  V2 / R
```

Artificial gravity on a spaceship

If artificial gravity is generated by spinning a spaceship, then according to en.wikipedia.org/wiki/Artificial_gravity, the spin period has to be at least 30 seconds for the inhabitants to not get dizzy.

```Spin period                           =  T  =  2$\pi$ R / V    =   30 s
Centripetal acceleration              =  A  =  V2 / R      =   10 m/s2
Spin radius of the spaceship          =  R  =  T2 A / (2$\pi$)2 = 228 meters
Tangential velocity of the spaceship  =  V  =  (A R)1/2     =  48 m/s
```

Circular gravitational orbit

Geosynchronous orbit

Suppose an satellite is on a circular orbit around a central object.

```Graviational constant                     =  G
Mass of central object                    =  M
Mass of satellite                         =  m
Distance of satellite from central object =  R
Velocity of satellite                     =  V
Gravity force                             =  F  =  G M m / R2
Gravity potential energy                  =  E  = -G M m / R  =  $\int$ F ∂R
```
The velocity of a circular orbit is obtained by setting gravitational force equal to centripetal force.
```G M m / R  =  m V2 / R
```
The escape velocity is obtained by setting gravitational energy equal to kinetic energy.
```G M m / R  =  1/2 m V2

Circular orbit velocity  =    (G M / R)$1/2$
Escape velocity          = $\surd 2 \left(G M / R\right)1/2Escape velocity =\surd 2 * Circular orbit velocity Escape velocity Circular orbit velocity \left(km/s\right) \left(km/s\right) Earth 11.2 7.9 Mars 5.0 3.6 Moon 2.4 1.7$```

Gravitational energy

For an object on a circular orbit,

```Gravitational energy  =  -2 * Kinetic energy
```
The relationship between the kinetic and gravitational energy doesn't depend on R. If a satellite inspirals toward a central object, the gain in kinetic energy is always half the loss in gravitational energy.

The total energy is negative.

```Total energy     =  Gravitational energy  +  Kinetic energy
=  ½ * Gravitational energy
= -½ G M m / R

Angular momentum =  m V R
=  m (G M R)1/2
```
As R decreases, both energy and angular momentum decrease. In order for a satellite to inspiral it has to give energy and angular momentum to another object.
Gravity for a uniform-density sphere
```Density                 =  D
Volume                  =  Υ  =  (4/3) π R3
Mass                    =  M  =  D Υ
Acceleration at surface =  A  =  G M / R2     =  (4/3) π G D R
Orbit speed at surface  =  V  = (G M / R)1/2  = [(4/3) π G D]½ R
Orbit time at surface   =  T  =  2 π R / V    = [(16/3) π3 G D]½
```
Acceleration is proportional to R
```
g/cm2   (Earth=1)  m/s2

Earth    5.52    1.00      9.8
Venus    5.20     .95      8.87
Uranus   1.27    3.97      8.69
Mars     3.95     .53      3.71
Mercury  5.60     .38      3.7
Moon     3.35     .27      1.62
Titan    1.88     .40      1.35
Ceres    2.08     .074      .27
```

Friction

```Fcontact  =  Contact force between the object and a surface (usually gravity)
Ffriction =  Maximum friction force transverse to the surface of contact.
C        =  Coefficient of friction, usually with a magnitude of ~ 1.0.

Ffriction  =  C Fcontact
```
The larger the contact force the larger the maximum friction force.
```      Coefficient of friction
Ice           .05
Tires        1
```
When two surfaces first come together there is an instant of large surface force, which allows for a large friction force.
Agassi returning a Sampras serve. At T=0:07 Agassi's feet hit the ground simultaneous with when he reads the serve.

Maximum drag racing acceleration
```Mass                                             =  M
Contact force between the car and the road       =  Fcontact   =  M g
Maximum friction force that the road can provide =  Ffriction  =  C Fcontact
Maximum acceleration that friction can provide   =  A  =  Ffriction / M
=  C Fcontact / M
=  C g M / M
=  C g
```
This clip shows the magnitude and direction of the acceleration while a Formula-1 car navigates a racetrack.
Formula-1 lap

Villeneuve vs. Arnoux At 0:49 Arnoux breaks before he hits the turn.

Maximum cornering acceleration

For maximum cornering acceleration, the same equations apply as for the maximum drag racing acceleration. It doesn't matter in which direction the acceleration is.

```Maximum cornering acceleration  =  C g
```

Friction on a ramp

Suppose an object with mass m rests on a ramp inclined by an angle theta. The gravitational force on the object is

```F = m g
```
The force between the object and the surface is equal to the component of the gravitational force perpendicular to the surface.
```Fcontact = Fgrav * cos($\theta$)
```
The force of gravity parallel to the ramp surface is
```Framp = Fgrav sin($\theta$)
```
Th maximum friction force that the ramp can exert is
```Ffriction = C Fcontact
```
This is balanced by the gravitational force along the ramp
```Ffriction = Framp

Fgrav sin($\theta$) = C Fgrav cos($\theta$)

C = tan($\theta$)
```
This is a handy way to measure the coefficient of friction. Tilt the ramp until the object slides and measure the angle.
Spin

```
$\theta$  =  Angle in radians   (dimensionless)
X  =  Arc distance around the circle in meters (the red line in the figure)
R  =  Radius of the circle in meters

X  =  $\theta$ R

```

Pi is defined as the ratio of the circumference to the diameter.

```Full circle  =  360 degrees  = 2 $\pi$ radians

1 radian  =  57.3 degrees
1 degree  = .0175 radians
```

Polar coordinates

```Radius  =  R
Angle   =  $\theta$
X coordinate  =  X  =  R cos($\theta$)
Y coordinate  =  Y  =  R sin($\theta$)
```

Spin and angular velocity

```T  =  Time in seconds
$\theta$  =  Angle  (dimensionless)
$\omega$  =  Angular velocity in radians/second = 1.75 in the animated figure
```

Wikipedia: circular motion

Spin frequency
```$\omega$  =  Angular frequency
F  =  Spin frequency in Hertz or 1/second

$\omega$  =  2 Pi F

1 Hertz  =  1 revolution/second  =  2$\pi$ radians/second
```

Frequency and period

```T  =  Period in seconds
F  =  Frequency in Hertz or 1/second

F T = 1
```

Speed of a record

If an object is at the edge of a record then the position is the arc length around the circumference.

```Time  =  T
Radius  =  R                   =         .15  meters             (for a vinyl record)
Angular velocity               =  $\omega$   = 3.49  radians/second     (= 33.33 revolutions per minute)
Velocity of the outer edge  V  =  $\omega$ R =  .523 meters
```

Rolling ball

Suppose a billiard ball rolls across a table with a speed of 2 m/s.

```Ball velocity                 =  V           =    2 meters/second
Ball radius                   =  R           =  .03 meters
Angular frequency             =  $\omega$  =  V/R   =   67 radians/second
Ball spin frequency in Hertz  =  F  =  $\omega$/(2$\pi$)= 10.6 Hertz
```
A point on the edge of the ball is moving at Velocity=0 when it is in contact with the ground and it is moving at Velocity=2V when it is at the opposite point from the ground.
Spinning ball
```Ball velocity   =  V
Edge velocity   =  Vedge
Spin parameter  =  Z  =  Vedge / V  =  2 π R F / V
Spin frequency  =  F  =  Z V / (2 π R)
```
For a rolling ball, Z=1. For curveballs thrown in the air, Z is typically less than 1. If Z=.5 then the following table shows typical speeds and spin rates for various balls.
```          Radius   Speed   Spin
(mm)   (m/s)   (1/s)

Ping pong    20      20     80
Golf         21.5    80    296
Tennis       33.5    50    119
Baseball     37.2    40     86
Soccer      110      40     29
```

Spin variables

For a mass moving around a circle, we can describe the motion in terms of either a position or an angle.

```R  =  Radius of the circle
X  =  Position of the mass on the circumference of the circle
$\theta$  =  Angle pointing to X

X  = $\theta$ R
```
Every linear quantity has a corresponding spin quantity, and every linear equation has a crresponding spin equation. By changing variables from X to $\theta$ we can translate a linear equation into a spin equation.
```
Linear variables             Spin variables

Position    = X              Angle             = $\theta$ =  X / R
Velocity    = V              Angular velocity  = $\omega$ =  V / R
Acceleration= A              Angular accel.    = $\alpha$ =  A / R
Mass        = M              Moment of inertia = I =  M * R2
Force       = F              Torque            = $\Gamma$ =  F * R
Momentum    = Q              Angular momentum  = L =  Q * R
Energy      = E
Time        = T

Force        Momentum             Energy          Centripetal    Constant   Constant
acceleration   velocity   acceleration
Linear motion:  F = M A     Q = M V = F T      E = .5 M V2 = F X    A = V2 / R     X = V T    V = A T
Spin:           $\Gamma$ = I $\alpha$     L = I $\omega$ = F T R    E = .5 I $\omega 2$ = $\Gamma$ $\theta$    A = $\omega 2$ R       $\theta$ = $\omega$ T    $\alpha$ = $\omega$ T

Derivation:

Force            Momentum           Energy                 Energy         Constant           Constant
velocity           acceleration
F   = M A        Q   = M V          E = .5 M V2            E = F X        X   = V T          V   = A T
F R = M A R      Q R = M V R        E = .5 M R2 (V/R)2     E = F R X/R    X/R = (V/R) T      V/R = (A/R) T
$\Gamma$   = M R2 A/R   L   = M R2 V/R     E = .5 I $\omega 2$            E = $\Gamma$ $\omega$        $\theta$   = $\omega$ T          $\omega$  =  $\alpha$ T
$\Gamma$   = I $\alpha$        L   = I $\omega$
```

Angular velocity and angular acceleration
```Angle             = $\theta$  =  X / R
Angular velocity  = $\omega$  =  V / R
Angular accel.    = $\alpha$  =  A / R

Constant velocity:        X = V T
Constant spin:            $\theta$ = $\omega$ T

Constant acceleration:    V = A T
Constant angular accel.:  $\omega$ = $\alpha$ T
```

Torque, moment of inertia, and angular acceleration

The equivalent of Newton's law for spin is

```Newton's law for linear motion:    F = M A
Newton's law for circular motion:  $\Gamma$ = I $\alpha$ = F R

Momentum:                          Q = M V
Angular momentum:                  L = I $\omega$ = M V R

Energy:                            E = .5 M V2
Angular energy:                    L =  .5 I $\omega 2$
```
Mometum, angular momentum, force, and torque

Angular momentum is conserved. In the following figure, the angular momentum is constant and V*R is constant.

Angular momentum vector

The direction of the angular momentum vector is given by the right hand rule. The Earth's angular momentum points to the north pole and it is constant throughout the orbit.

Right hand rule
Two perpendicular spin axes

The angular momentum vector is the cross product of the position and momentum vectors. Denoting the cross product by "x" and the angle between R and Q by Theta,

```L  =  R x Q
= |R| |Q| sin(Theta)
```
Vector cross product

Projection

Moment of inertia

The moment of inertia of an object depends on its mass and how far the mass is from the axis of rotation.

Point mass: I = M R2

Ring: I = M R2
Solid disk: I = .5 M R2

Spherical shell: I = (2/3) M R2
Solid sphere: I = .4 M R2

Pole: I = M L2 / 12
Sword: I = M L2 / 3

```              I / (M R2)

Point mass        1        Tetherball
Ring              1        Hula hoop
Solid disk       1/2       Pizza
Spherical shell  2/3       Tennis ball, soccer ball
Solid sphere     2/5       Billiard ball
Pole             1/12      Grasped at the center
Sword            1/3       Grasped at the end
```
Pizzeria Port'Alba, the first pizzeria

Spherical cow

If the object has a simple shape then we can calculate the moment of inertia using the formulae above. If the shape is not simple then we often assume it is a solid sphere. For example, the moment of inertia of Chuck Norris as a solid sphere is

```M  =  Mass of Chuck               =  100 kg
R  =  Radius of Chuck             =  .25 meters                      (estimate)
I  =  Moment of inertia of Chuck  =  .4 * 100 * .252  =  2.5 kg m2
```

Energy forms
```Kinetic energy        =  .5 M V2

Gravitational energy  =  M g Height               (In a constant gravitational field)

Gravitational energy  =  - G M1 M2  / R

Rotational energy     =  .5 I $\omega 2$

Energy of matter      =  M C2                     C = Speed of light = 3.00e8 m/s
```

Conservation

Conservation arises from multiplying F=MA by various quantities.

```Force                 = Mass * Acceleration

Force * Time          = Momentum

Force * Distance      = Energy

Force * Radius        = Torque

Force * Time * Radius = Angular Momentum
```

Energy of a rolling ball

For a rolling ball,

```Velocity           =  V
Mass               =  M
Angular frequency  =  $\omega$  =  V/R
Moment of inertia  =  I  =  .4 M R2
Kinetic energy     =  Ek  =  .5 M V2
Spin energy        =  Es  =  .5 I $\omega 2$  =  .2 M V2
Total energy       =  Et  =  Ek + Es  =  1.4 Ek

Spin energy  / Kinetic energy  =  Es / Ek  =  .4
Total energy / Kinetic energy  =  Es / Ek  = 1.4
```

Bowling

Suppose a bowling ball is launched so that it initially slides along the floor with zero spin. The friction force decreases the ball velocity. It also exerts a torque on the ball that increases the spin angular velocity.

```Ball initial velocity      =  V0       =  10 m/s
Mass of the ball           =  M           =  7.26 kg              (=16 pounds, which is the maximum)
Radius of the ball         =  R
Friction coefficient       =  C
Gravitational force        =  Fg =  M g
Friction force             =  Ff =  C M g              (directed opposite to the ball's velocity)
Ball acceleration          =  A  =  -Ff / M  =  -C g   (The friction force decelerates the ball)
Torque on ball             =  $\Gamma$  =  R C M g
Moment of inertia          =  I  = .4 M R2
Angular acceleration       =  $\alpha$  =  $\Gamma$ / I  =  2.5 C g / R

Time sliding               =  T
Rolling angular velocity   =  $\omega$  =  $\alpha$ T  =  2.5 C g T / R
Ball velocity when rolling =  V  =  Vo + A T
=  Vo - C g T
=  Vo - .4 C g $\omega$ R / C / g
=  Vo - .4 $\omega$ R
=  Vo - .4 V
```
Solving for V,
```V  =  V0 / 1.4  =  7.1 m/s
```
The time it takes to being rolling smoothly is
```T  =  (2/7) V0 / C / g  =  .29 / C
```
For surfaces with C=1, it usually takes less than half a second for a ball to begin rolling smoothly.

A bowling lane is covered in a layer of oil and has a friction coefficient of C=.08, giving T=3.6, giving the ball plenty of time to slide before it starts rolling. This allows one to use sidespin. While the ball is still sliding, sidespin can deliver a sideways force. Once the ball starts rolling the sidespin is lost.

Orientation

If there are no torques on an object then the angular momentum is conserved.

In free fall you can't change your angular momentum but you can change your orientation.

In the absence of external torques, a rigid object can't change its orientation axis and a deformable object can. Cats change their orientation axis by generating internal torques and by varying their moment of inertia.

A cat can change its orientation using either a 2-axis strategy or a 3-axis strategy. Each can work indepenently and the cat uses a combination of both. The 3-axis strategy is depicted in the figure above an the 2-axis strategy is as follows:

A can can right itself with the following 2-step procedure. The first step is to compact the arms and extend the legs, turning the upper torso one direction and the legs the opposite direction. Because the upper torso has a smaller moment of inertia it rotates farther than the legs. The second step is to extend the arms and compact the legs and perform an opposite set of rotatons as step 1. The sum of steps 1 and step 2 produces a net change in orientation.

The more deformable you are the more precise internal torques you can generate.

Bruce Lee: "Be like water"

Fumio, from the film "Fist of Legend": "if you learn to be fluid, to adapt, you will be unbeatable."

Paul Atreides, from the film "Dune" during the duel with Feyd-Rautha Harkonnen: "I will bend like a reed in the wind."

Most of the elements of the breathing cycle and axis cycle are determined by conservation of momentum.

Airborne with non-zero angular momentum

The cat's first move is to maximize its moment of inertia to slow down its rotation.

3D moment of inertia

Sphere
Prolate spheroid
Oblate spheroid
Triaxial spheroid

Jupiter's spin makes it oblate.

For a spinning 3D object,

```Torque:                3D vector
Angular acceleration:  3D vector
Moment of inertia:     3x3 matrix

Torque  =  MomentOfInertia * AngularAcceleration
```
If the axis of rotation passes through the object's center of mass then the moment of inertia matrix has a tridiagonal form.

Bicycle

A typical set of parameters for a racing bike is

```Velocity        =  V          =   20 m/s       (World record=22.9 m/s)
Power           =  P          = 2560 Watts     (Typical power required to move at 20 m/s, measured experimentally)
Force on ground =  F  =  P/V  =  128 Newtons
```

We assume a high gear, with 53 teeth on the front gear and 11 teeth on the rear gear.

```Number of links in the front gear      =  Nf  =  53
Number of links in the rear gear       =  Nr  =  11
Length of one link of a bicycle chain  =  L          =  .0127 m =  .5 inches
Radius of the front gear               =  Rf  =  Nf L / (2 π)   =  .107  m
Radius of the rear gear                =  Rr  =  Nr L / (2 π)   =  .0222 m

```
Torque balance:
```Ground force * Wheel radius  =  Chain force * Rear gear radius
Pedal force  * Pedal radius  =  Chain force * Front gear radius

Chain force  =  Ground force * Wheel radius / Rear gear radius
=  128 * .311 / .0222
=  1793 Newtons

Pedal force  =  Ground force * Wheel radius / Pedal radius * Front gear radius / Rear gear radius
=  Ground force * Wheel radius / Pedal radius * Front gear teeth  / Rear gear teeth
=  128 * .311 / .17 * 53 / 11
=  1128 Newtons

Radius  Force  Torque  Gear
(m)    (N)    (Nm)   teeth
Pedal crank  .170   1128   191.9     -
Front gear   .107   1793   191.9    53
Rear gear    .0222  1793    39.8    11
Rear wheel   .311    128    39.8     -

Wheel frequency =  Velocity / (Radius * 2Pi)
=  20 / (.311 * 2$\pi$)
=  10.2 Hertz
Pedal frequency =  Wheel frequency * Rear gear teeth / Front gear teeth
=  10.2 * 53 / 11
=  2.12 Hertz
=  127 revolutions per minute
```
Humans can pedal effectively in the range from 60 rpm to 120 rpm. Gears allow one to choose the pedal frequency. There is also a maximum pedal force of around 1200 Newtons.

When going fast the goal of gears is to slow down the pedals.

When one is climbing a hill the goal of gears is to speed up the pedals so that you don't have to use as much force on the pedals.

```Pedal period                   * Rear gear teeth   =  Wheel period                   * Front gear teeth
Pedal radius / Pedal velocity  * Front gear teeth  =  Wheel radius / Wheel velocity  * Front gear teeth

Pedal force  =  Power / Pedal velocity
=  Power / Wheel velocity * Wheel radius / Pedal radius * Front gear teeth / Rear gear teeth
=  Power / Wheel velocity * .311 / .17 * Front gear teeth / Rear gear teeth
=  Power / Wheel velocity * 1.83 * (Front gear teeth / Rear gear teeth)
=  Power / Wheel velocity * 1.83 * Gear ratio

Gear ratio   =  Front gear teeth / Rear gear teeth
```
For a given power and wheel velocity, the pedal force can be adjusted by adjusting the gear ratio.

Suppose a bike is going uphill at large power and low velocity.

```Power            =  1000 Watts
Velocity         =  3 m/s
Front gear teeth =  34              (Typical for the lowest gear)
Rear gear teeth  =  24              (Typical for the lowest gear)

Pedal force  =  Power / Wheel velocity * 1.83 * Front gear teeth / Rear gear teeth
=  1000 / 3 * 1.83 * 34 / 24
=  864 Newtons
=  88 kg equivalent force
```
This is a practical force. If you used the high gear,
```Pedal force  =  Power / Wheel velocity * 1.83 * Front gear teeth / Rear gear teeth
=  1000 / 3 * 1.83 * 53 / 11
=  2939 Newtons
=  300 kg equivalent force
```
This force is impractically high.
Pressure
```Surface area =  A
Force        =  F
Pressure     =  P  =  F / A     (Pascals or Newtons/meter2 or Joules/meter3)
```

Atmospheric pressure

```Mass of the Earth's atmosphere  =  M              =  5.15e18 kg
Surface area of the Earth       =  A              =  5.10e14 m^2
Gravitational constant          =  g              =  9.8 m/s^2
Pressure of Earth's atmosphere  =  P  =  M g / A  =  101000 Pascals
=  15 pounds/inch2
=  1 Bar
```
One bar is defined as the Earth's mean atmospheric pressure at sea level

```              Height   Pressure   Density
(km)     (Bar)     (kg/m3)

Sea level         0      1.00     1.225
Denver            1.6     .82     1.05        One mile
Everest           8.8     .31      .48
Airbus A380      13.1     .16      .26
F-22 Raptor      19.8     .056      .091
SR-71 Blackbird  25.9     .022      .034
Space station   400       .000009   .000016
```
Earth
Titan
Veuns
Mars

Properties of atmospheres:

```       Density   Pressure    S       Gravity
(kg/m2)    (Bar)   (tons/m2)   (m/s2)

Venus    67       92.1      1050      8.87
Titan     5.3      1.46      109      1.35
Earth     1.22     1          10.3    9.78
Mars       .020     .0063       .54   3.71

Mass of atmosphere above one meter2 of surface  =  M  =  10.3 tons for the Earth

P  =  M g
```
You don't need a pressure suit on Titan. You can use the kind of gear arctic scuba divers use. Also, the gravity is so weak and the atmosphere is so thick that human-powered flight is easy. Titan will be a good place for the X games.
Temperature

```                       Kelvin   Celsius   Fahrenheit
Absolute zero            0      -273.2     -459.7
Water freezing point   273.2       0         32
Room temperature       294        21         70
Water boiling point    373.2     100        212

Kelvin
Absolute zero               0
Helium boiling point        4.2
Hydrogen boiling point     20.3
Pluto                      44
Nitrogen boiling point     77.4
Oxygen boiling point       90.2
Hottest superconductor    135          Mercury barium calcium copper oxide
Mars                      210
H2O melting point         273.15         0 Celcius = 32 Fahrenheit
Room temperature          293           20 Celcius = 68 Fahrenheit
H2O boiling point         373.15       100 Celcius = 212 Fahrenheit
Venus                     740
Wood fire                1170
Iron melting point       1811
Bunsen burner            1830
Tungsten melting point   3683          Highest melting point among metals
Earth's core             5650          Inner-core boundary
Sun's surface            5780
Solar core               13.6 million
Helium-4 fusion           200 million
Carbon-12 fusion          230 million
```

Ideal gas law

Molecules in a gas
Brownian motion

```P   =  Pressure
T   =  Temperature
Vol =  Volume
E   =  Kinetic energy of gas molecules within the volume
e   =  Kinetic energy per volume of gas molecules in Joules/meter3
=  E / Vol
Mol =  Number of moles of gas molecules in the volume
```
Ideal gas law:
```P  =  2/3 e                   Form used in physics

P Vol  =  8.3 Mol T           Form used in chemistry
```
Pressure has units of energy density, where the energy corresponds to kinetic energy of gas molecules.
History

Boyle's law
Charles' law

```1660  Boyle law          P Vol     = Constant          at fixed T
1802  Charles law        T Vol     = Constant          at fixed P
1802  Gay-Lussac law     T P       = Constant          at fixed Vol
1811  Avogadro law       Vol / N   = Constant          at fixed T and P
1834  Clapeyron law      P Vol / T = Constant          combined ideal gas law
```

Water pressure

```Distance below the surface                      =  X
Density of water                                =  D   =  1000 kg/m3
Mass of water above 1 meter2 of surface        =  M   =  D X
Force from the water above 1 meter2 of surface =  F   =  D X g
Pressure at depth X relative to the surface     =  P   =  D X g
```
At a depth of 10 meters,
```P  =  1000 * 10 * 10
=  100000 Pascals
=  1 Bar
```

Deflategate

In the 2014 AFC championship football game the Patriot's footballs were found to be underinflated. The Patriots claimed this was because the balls were inflated at warm temperature and used at cold temperature.

"Gauge pressure" is the pressure difference between the inside and outside of the football. The rules state that the gauge pressure for a football should be between 12.5 and 13.5 psi.

When the ball is moved from warm temperature to cold temperature, the external atmospheric pressure doesn't change and the pressure inside the football decreases.

The game was played at 4 Celsius and the gauge pressure was measured to be 11 psi. If we assume the footballs were inflated at warm temperature at 12.5 psi then we can solve for the inflation temperature.

```Atmospheric pressure                          =  Patm  =  15 psi
Pressure of the cold football during the game =  Pcold =  Patm + 11 psi   = 15 + 11.0 psi = 26.0 psi
Pressure of the warm football at inflation    =  Pwarm =  Patm + 12.5 psi = 15 + 12.5 psi = 27.5 psi
Temperature of the cold football              =  Tcold =  277 Kelvin  =  4 Celsius
Temperature of the warm football              =  Twarm
```
The number of gas molecules inside the football is constant and we assume that the volume of the football doesn't change. Using the ideal gas law for constant volume and molecule number,
```Pressure  =  Constant * Temperature

Pcold  =  Constant * Tcold
Pwarm  =  Constant * Twarm

Twarm  =  Tcold * Pwarm / Pcold
=  277 * 27.5 / 26
=  293 Kelvin  =  20 Celsius
```

Sound speed

Gas pressure arises from kinetic energy of gas molecules, and the average kinetic energy per molecule is proportional to the temperature.

For air at sea level and room temperature,

```P    =  Pressure                        =  101325 Pascals
D    =  Density                         =  1.22 kg/m3
$\gamma$     =  Adiabatic constant             =  7/5 for air
Vtherm=  Thermal speed                   =  544 meters/second
Vsound=  Sound speed                     =  343 meters/second at 20 Celsius
=  ($\gamma$ P / D)1/2
=  ($\gamma$/3)1/2 Vtherm
=  .63 Vtherm
M    =  Average mass of an air molecule =  4.78e-26 kg
n    =  Number of molecules per volume  =  2.55e25 meter-3
=  D / M
k    =  Boltzmann constant              =  1.38e-23 Joules/Kelvin
T    =  Gas temperature                 =  293 Kelvin   (Room temperature, or 20 Celcius)
E    =  Ave kinetic energy per molecule =  5.96e-21 Joules
=  .5 M Vtherm2
=  1.5 k T
=  e / n
e    =  Kinetic energy per volume       =  152000 Joules/meter3
=  1.5 P
Mol  =  Moles of molecules in 1 meter3  =  42.34
=  n / 6.022e23
Avo  =  Avogadro number                 =  6.022e23 molecules
H    =  Heat capacity                   =  1004 Joules/kg/Kelvin  (calculated in the thermodynamics section)
```
The characteristic thermal speed of a gas molecule is defined in terms of the mean energy per molecule. The Boltzmann constant relates the average kinetic energy to the temperature.
```E  =  .5 M Vtherm2  =  1.5 k T
```
The ideal gas law can be written as:
```P  =  2/3 e                 (Physics form)
=  8.3 Mol T / Vol       (Chemistry form)
```
Writing the pressure as an energy density allows one to connect pressure with molecular kinetic energy.

The following table estimates the average mass per air molecule. We have neglected the argon molecules.

```Atmosphere oxygen fraction   =  .21
Atmosphere nitrogen fraction =  .78
Atmosphere argon fraction    =  .01
Mass of a nitrogen molecule  =  28 Atomic mass units
Mass of an oxygen molecule   =  32 Atomic mass units
Mass of one Atomic mass unit =  1.66e-27 kg
Average molecule mass        =  Oxygen fraction * Oxygen mass  +  Nitrogen fraction * Nitrogen mass
=  28.8 Atomic mass units
=  4.78e-26 kg
```

Heat capacity
```Ice heat capacity                           =    2110 J/kg/K      At -10 Celsius
Water heat capacity                         =    4200 J/kg/K      At  20 Celsius
Steam heat capacity                         =    2080 J/kg/K      At 100 Celsius
Air heat capacity                           =    1004 J/kg/K
Melting energy of water at 0 Celsius        = 2501000 J/kg
Vaporization energy of water at 100 Celsius = 2257000 J/kg

Energy required to raise the temperature of 1 kg of H2O from -40 Celsius to 140 Celsius
=  Energy to raise the temperature of ice from -40 C to 0 C
+  Energy to turn ice to water (at 0 C)
+  Energy to raise the temperature of water from 0 C to 100 C
+  Energy to turn the water from a liquid to steam (at 100 C)
+  Energy to raise the temperature of steam from 100 C to 140 C
=  2110 * 40  +  2501000  +  4200 * 100  +  2257000  +  1004 * 40
=  5302560 Joules
```

Buoyancy

Archimedes' principle

Archimedes was commissioned by the king to develop a method to measure the volume of an irregular object, such as a crown. The king wanted to measure the crown's density to determine if it was made of pure gold.

In the animation above, the crown and the cylinder have equal masses and densities and they displace equal volumes of water. This is "Archimedes' principle". A submerged mass displaces an equal mass of water.

Inventions of Archimedes:
Concept of a limit
Water pump
Defense

Buoyancy

For a ship floating in water,

```Gravitational acceleration           =  g       =     9.8 m/s2
Density of water                     =  Dwater   =  1000 kg/m3
Mass of a ship                       =  Mship
Mass of water displaced by the ship  =  Mwater   =  Mship             (Archimedes' principle)
Gravity force on ship                =  Fgrav    =  Mship g
Buoyancy force on ship               =  Fbuoy    =  Mwater g
Volume of water displaced by the ship=  Υwater   =  Mwater / Dwater

Mgrav  =  Fbuoy

Mship  =  Mwater
```
The mass of water displaced is the same for a floating and a sunk ship.
Icebergs

For ice floating in water,

```Gravitational acceleration           =  g       =     9.8 m/s2
Density of ice                       =  Dice     =   920   kg/m3
Density of water                     =  Dwater   =  1000   kg/m3
Volume of the iceberg                =  Υice
Volume of water displaced by iceberg =  Υwater
Mass of the iceberg                  =  Mice     =  Dice   Υice
Mass of water displaced by iceberg   =  Mwater   =  Dwater Υwater
Gravity force on the iceberg         =  Fgrav    =  Mice   g
Buoyant force on the iceberg         =  Fbuoy    =  Mwater g
Fraction of iceberg above surface    =  f  =  (Υice - Υwater) / Υice

Fgrav  =  Fbuoy            (Principle of bouyancy)

Mice   =  Mwater           (Principle of Archimedes)

f  =  (Dice - Dwater) / Dice
=  (Mice/Υice - Mwater/Υwater) / (Mice/Υice)
=  1 - Υice / Υwater
=  .08
```

Helium balloons

We estimate the number of helium balloons required to lift a person.

```Gravitational acceleration           =  g       =   9.8 m/s2
Density of air                       =  Dair    =  1.22 kg/m3
Density of helium                    =  Dhelium  =  .179 kg/m3
Radius of one balloon                =  Rballoon =    .2 m
Volume of one balloon                =  Υballoon =  .0335 m3
Volume of helium in all balloons     =  Υhelium  =     83 m3
Mass of helium in one balloon        =  Mballoon = .00600 kg  =  Dhelium Υballoon
Mass of helium in all balloons       =  Mhelium  =   14.6 kg  =  Dhelium Υhelium
Mass of air displaced by the balloons=  Mair    =   101.3 kg  =  Dair Υhelium
Mass of payload                      =  Mpayload =     80 kg                           (Typical person)
Gravity force on payload & balloons  =  Fgrav   =     927 N   =  (Mpayload + Mhelium) g
Buoyant force on a helium balloon    =  Fbuoy   =     927 N   =  Fair                  (Principle of buoyancy)
Number of balloons required          =  Z      =    2477     =   Υhelium / Υballoon

Fbuoy  =  Fair               (Principle of buoyancy)

Fgrav  =  Fbuoy              (Balance of gravity and buoyancy)

(Mpayload + Mhelium) g  =  Mair g

(Mpayload + Dhelium Υhelium) g  =  Dair Υhelium g

Υhelium  =  Mpayload / (Dair - Dhelium)  =  1.04 Mpayload  =  83 m3
```
The volume of helium doesn't depend on gravity.
Density

Helium is more expensive and more dense than hydrogen, but it is not flamable.

```               kg/m3

Hydrogen           .0899
Helium             .179
Air (hot)         1.12          320 Kelvin
Air (room temp)   1.22          293 Kelvin
Ice             920
Water          1000
```

Hot air balloon

For a hot air balloon, the volume is fixed and the pressures on the inside and outside are equal For example,

```Inside temperature  =  Tin  =  320 Kelvin
Outside temperature =  Tout =  293 Kelvin
Inside density      =  Din  = 1.12 kg/m3
Outside density     =  Dout = 1.22 kg/m3

Din Tin  =  Dout Tout
```

Jacques Charles made the first hot air balloon flight in 1783.

History of physics
```1585  Simon Stevin introduces decimal numbers to Europe.
(For example, writing 1/8 as 0.125)

1586  Simon Stevin drops objects of varying mass from a church tower to demonstrate
that they accelerate uniformly.

1604  Galileo publishes a mathematical description of acceleration.

1614  Logarithms invented by John Napier, making possible precise calculations
of equal tuning ratios.  Stevin's calculations were mathematically sound but
the frequencies couldn't be calculated with precision until logarithms were
developed.

1637  Cartesian geometry published by Fermat and Descartes.
This was the crucial development that triggered an explosion of mathematics
and opened the way for the calculus.

1676  Leibniz defines kinetic energy and notes that it is conserved in many
mechanical processes

1684  Leibniz publishes the calculus

1687  Newton publishes the Principia Mathematica, which contained t hecalculus,
the laws of motion (F=MA), and a proof that planets orbit as ellipses.

1776  Smeaton publishes a paper on experiments related to power, work, momentum,
and kinetic energy, supporting the principle of conservation of energy

1798  Thompson performs measurements of the frictional heat generated in
boring cannons and develops the idea that heat is a form of kinetic energy

1802  Gay-Lussac publishes Charles's law.
For a gas at constant pressure, Temperature * Volume = Constant

1819  Dulong and Petit find that the heat capacity of a crystal is proportional to the
number of atoms

1824  Carnot analyzes the efficiency of steam engines; he develops the notion of a
reversible process and, in postulating that no such thing exists in nature,
lays the foundation for the second law of thermodynamics, and initiating the
science of thermodynamics

1831  Melloni demonstrates that infrared radiation can be reflected, refracted,
and polarised in the same way as light

1834  Clapeyron combines Boyle's Law, Charles's Law, and Gay-Lussac's Law to
produce a Combined Gas Law.
Pressure * Volume  =  Constant * Temperature

1842  Mayer calculates the equivalence between heat and kinetic energy
```

Waves

Wave equation

Wavelength
Wave speed

Frequency and period

The properties of a wave are

```Frequency  =  F  (seconds-1
Wavelength =  W  (meters)
Wavespeed  =  V  (meters/second)
Period     =  T  (seconds)  =  The time it takes for one wavelength to pass by
```
Wave equations:
```F W = V

F T = 1
```

Trains

A train is like a wave.

```Length of a train car =  W  =  10 meters         (The wavelength)
Speed of the train    =  V  =  20 meters/second  (The wavespeed)
Frequency             =  F  =  2 Hertz           (Number of train cars passing by per second)
Period                =  T  =  .5 seconds        (the time it takes for one train car to pass by)
```

Speed of sound in air

Your ear senses changes in pressure as a wave passes by

```Speed of sound at sea level    =  V  =  340 meters/second
Frequency of a violin A string =  F  =  440 Hertz
Wavelength of a sound wave     =  W  =  .77 meters  =  V/F
Wave period                    =  T  =  .0023 seconds
```

Speed of a wave on a string

A wave on a string moves at constant speed and reflects at the boundaries.

For a violin A-string,

```Frequency                           =  F  =  440 Hertz
Length                              =  L  =  .32 meters
Time for one round trip of the wave =  T  =  .0023 s  =  2 L / V  =  1/F
Speed of the wave on the string     =  V  =  688 m/s  =  F / (2L)
```

Octave

If two notes are played at the same time then we hear the sum of the waveforms.

If two notes are played such that the frequency of the high note is twice that of the low note then this is an octave. The wavelength of the high note is half that of the low note.

```Color       Frequency       Wavelength

Orange      220 Hertz           1
Red         440 Hertz          1/2
```
Because the red and orange waves match up after a distance of 1 the blue note is periodic. This makes it easy for your ear to process.

Orange = 220 Hertz          Red = 440 Hertz   (octave)          Blue = Orange + Red

If we double both frequencies then it also sounds like an octave. The shape of the blue wave is preserved.

Orange = 440 Hertz          Red = 880 Hertz   (octave)          Blue = Orange + Red

```Color       Frequency       Wavelength

Orange      440 Hertz          1/2
Red         880 Hertz          1/4
```
When two simultaneous pitches are played our ear is sensitive to the frequency ratio. For both of the above octaves the ratio of the high frequency to the low frequency is 2.
```440 / 220  =  2
880 / 440  =  2
```
If we are talking about frequency ratios and not absolute frequencies then for simplicity we can set the lower frequency to 1.
```Frequency   Normalized frequency

220         1
440         2
880         4
```

Gallery of intervals

Octave

Orange = 1 Hertz          Red = 2 Hertz   (The note "A")          Blue = Orange + Red

Perfect fifth

Orange = 1 Hertz          Red = 3/2 Hertz    (the note "E")

Beat frequencies: consequences of playing out of tune

If two notes are out of tune they produce dissonant beat frequencies.

```F1 = Frequency of note #1
F2 = Frequency of note #2
Fb = Beat frequency
```
If F1 and F2 are played together the beat frequency is
```Fb = F2 - F1
```
For the beats to not be noticeable, Fb has to be less than one Hertz. On the E string there is little margin for error. Vibrato is often used to cover up the beat frequencies.

Examples of beat frequencies

Orange = A          Red = A

Orange = A          Red = 1.03 A

Orange = A          Red = 1.06 A

Orange = A          Red = 1.09 A

The more out of tune the note, the more pronounced the beat frequencies. In the first figure, the notes are in tune and no beat frequencies are produced.

If you play an octave out of tune you also get beat frequencies.

An octave played in tune
Orange = A          Red = 2 A

An octave played out of tune
Orange = A          Red = 2.1 A

Instruments

Stringed instruments

A violin, viola, cello, and double bass
String quartet
Orchestra

Violin and viola
Cello
Bass
Guitar
Electric guitar

Strings on a violin

Strings on a viola or cello

Violin fingering
Strings on a guitar

Violins, violas, and cellos are tuned in fifths. String basses, guitars, and bass guitars are tuned in fourths. Pianos are tuned with equal tuning.

```             Hertz
Violin E      660      =  440*1.5
Violin A      440
Violin D      293      =  440/1.5
Violin G      196      =  440/1.5^2

Viola  A      440      Same as a violin A
Viola  D      293
Viola  G      196
Viola  C      130

Cello  A      220      One octave below a viola A
Cello  D      147
Cello  G       98
Cello  C       65

String bass G  98      =  55 * 1.5^2
String bass D  73      =  55 * 1.5
String bass A  55      3 octaves below a violin A
String bass E  41      =  55 / 1.5

Guitar E      326
Guitar B      244
Guitar G      196
Guitar D      147
Guitar A      110      2 octaves below a violin A
Guitar E       82
```
When an orchestra tunes, the concertmaster plays an A and then everyone tunes their A strings. Then the other strings are tuned in fifths starting from the A.

A bass guitar is tuned like a string bass.

The viola is the largest instrument for which one can comfortably play an octave, for example by playing a D on the C-string with the first finger and a D on the G-string with the fourth finger. Cellists have to shift to reach the D on the G-string.

Wind and brass instruments

Flute
Oboe
Clarinet
Bassoon

Trumpet
French horn
Trombone
Tuba

In a reed instrument, a puff of air enters the pipe, which closes the reed because of the Bernoulli effect. A pressure pulse travels to the other and and back and when it returns it opens the reed, allowing another puff of air to enter the pipe and repeat the cycle.

Piano

Range of instruments

Green dots indicate the frequencies of open strings.

An orchestral bass and a bass guitar have the same string tunings.

The range of organs is variable and typically extends beyond the piano in both the high and low directions.

Overtones

Linearity

If a wave is linear then it propagates without distortion.

Wave interference

If a wave is linear then waves add linearly and oppositely-traveling waves pass through each other without distortion.

If two waves are added they can interfere constructively or destructively, depending on the phase between them.

Two speakers

If a speaker system has 2 speakers you can easily sense the interference by moving around the room. There will be loud spots and quiet spots.

The more speakers, the less noticeable the interference.

Noise-cancelling headphones use the speakers to generate sound that cancels incoming sound.

Standing waves

Two waves traveling in opposite directions create a standing wave.

Reflection

Whan a wave on a string encounters an endpoint it reflects with the waveform preserved and the amplitude reversed.

Overtones of a string

Standing waves on a string
Standing waves on a string
Notes in the overtone series

Notes in the overtone series

When an string is played it creates a set of standing waves.

```L  =  Length of a string
V  =  Speed of a wave on the string
N  =  An integer in the set {1, 2, 3, 4, ...}
W  =  Wavelength of an overtone
=  2 L / N
F  =  Frequency of the overtone
=  V/W
=  V N / (2L)

N = 1  corresponds to the fundamental tone
N = 2  is one octave above the fundamental
N = 3  is one octave plus one fifth above the fundamental.
```
Audio: overtones

For example, the overtones of an A-string with a frequency of 440 Hertz are

```Overtone  Frequency   Note

1         440       A
2         880       A
3        1320       E
4        1760       A
5        2200       C#
6        2640       E
7        3080       G
8        3520       A
```

Overtones of a half-open pipe

Overtones of a half-open pipe
Airflow for the fundamental mode

In the left frame the pipe is open at the left and closed at the right. In the right frame the pipe is reversed, with the left end closed and the right end open. Both are "half-open pipes".

An oboe and a clarinet are half-open pipes.

```
L  =  Length of the pipe
~  .6 meters for an oboe
V  =  Speed of sound
N  =  An odd integer having values of {1, 3, 5, 7, ...}
W  =  Wavelength of the overtone
=  4 L / N
F  =  Frequency of the overtone
=  V / W
=  V N / (4L)

```
The overtones have N = {1, 3, 5, 7, etc}

A cantilever has the same overtones as a half-open pipe.

N=1 mode
N=3 mode

Overtones of an open pipe

Overtones of an open pipe
Airflow for the fundamental mode

A flute and a bassoon are pipes that are open at both ends and the overtones are plotted in the figure above. In this case the overtones have twice the frequency as those for a half-open pipe.

```L  =  Length of the pipe
V  =  Speed of sound
N  =  An odd integer having values of {1, 3, 5, 7, ...}
W  =  Wavelength of the overtone
=  2 L / N
F  =  Frequency of the overtone
=  V / W
=  V N / (2L)
```

Overtones of a pipe that is closed at both ends

Airflow for the fundamental mode
Airflow for the N=2 mode
A string is like a closed pipe

A string has the same overtones as a closed pipe.

A closed pipe doesn't produce much sound. There are no instruments that are closed pipes. A muted wind or bass instrument can be like a closed pipe.

Modes 1 through 5 for a closed pipe.

Mode 1
Mode 2
Mode 3
Mode 4
Mode 5

Overtones for various instruments

Overtones

An instrument of length L has overtones with frequency

```Frequency  =  Z * Wavespeed / (2 * Length)
```
Z corresponds to the white numbers in the figure above.

An oboe is a half-open pipe (open at one end), a flute is an open pipe (open at both ends), and a string behaves like a pipe that is closed at both ends.

If a violin, an oboe, and a flute are all playing a note with 440 Hertz then the overtones are

```
Violin      440, 2*440, 3*440, 4*440, ...
Oboe        440, 3*440, 5*440, 7*440, ...
Flute       440, 3*440, 5*440, 7*440, ...

```

Drum modes

1
2.295
3.598

1.593
2.917
4.230

2.136
3.500
4.832

The fundamental mode is at the upper left. The number underneath each mode is the frequency relative to the fundamental mode. The frequencies are not integer ratios.

In general, overtones of a 1D resonator are integer multiples of the fundamental frequency and overtones of a 2D resonator are not.

Wikipedia: Virations of a circular membrane

In 1787 Chladni published observations of resonances of vibrating plates. He used a violin bow to generate a frequency tuned to a resonance of the plate and the sand collects wherever the vibration amplitude is zero.

Modes of a vibrating plate

Chladni modes of a guitar

Vocal modes

A "formant" is a vocal resonance. Vowels can be identified by their characteristic mode frequencies.

Whispering gallery

The whispering gallery in St. Paul's Cathedral has the same modes as a circular drum.

Whispering gallery waves were discovered by Lord Rayleigh in 1878 while he was in St. Paul's Cathedral.

St. Paul's Cathedral
St. Paul's Cathedral
U.S. Capitol
A mode in a circular chamber
Grand Central Station

The interior of a football is a spherical resonator.

Normal modes

Overtones are ubiquitous in vibrating systems. They are usually referred to as "normal modes".

Guitar overtones

Guitar overtones in relation to the positions of the frets

Table of fret values for each overtone

Guitar tuning

You can increase the pitch by pulling the string sideways. This increases the string tension, which increases the wavespeed and hence the frequency.

If you are playing a note on a guitar using a fret, you can change the frequency of the note by bending the string behind the fret.

```Tension  =  Tension of a string
D        =  Mass per meter of the string
V        =  Speed of a wave on the string
=  (Tension/D)^(1/2)
L        =  Length of the string
T        =  Wave period of a string (seconds)
=  2 L / V
F        =  Frequency of a string
=  1/T
=  V / (2L)
```

Plucked string

The vibration of the string depends on where it is plucked. Plucking the string close to the bridge enhances the overtones relative to the fundamental frequency.

Plucked at the center of string
Plucked at the edge of the string

A bow produces a sequence of plucks at the fundamental frequency of the string.

Reeds

As a sound waves travels back and forth along the clarinet it forces the reed to vibrate with the same frequency.

In a brass instrument your lips take the function of a reed.

Bernoulli principle

In the figure, as the flow constricts it speeds up and drops in pressure.

```P  =  Pressure
V  =  Fluid velocity
H  =  Height
g  =  Gravity  =  9.8 meters/second^2
D  =  Fluid density
```
The bernoulli principle was published in 1738. For a steady flow, the value of "B" is constant along the flow.
```B  =  P  +  .5 D V^2  +  D g H
```
If the flow speeds up the pressure goes down and vice versa.

A wing slows the air underneath it, inreasing the pressure and generating lift. In the right panel, air on the top of the wing is at increased speed and reduced pressure, causing condensation of water vapor.

Angle of attack
Lift as a function of angle of attack
Turbofan

Lift incrases with wing angle, unless the angle is large enough for the airflowto stall.

A turbofan compresses the incoming airflow so that it can be combusted with fuel.

In a reed instrument, a puff of air enters the pipe, which closes the reed because of the Bernoulli effect. A pressure pulse travels to the other and and back and when it returns it opens the reed, allowing another puff of air to enter the pipe and repeat the cycle.

Vocal chords

The vocal tract is around 17 cm long. For a half-open pipe this corresponds to a resonant frequency of

```Resonant frequency  =  WaveSpeed / (4 * Length)
=  340 / (4*.17)
=  500 Hertz
```
One has little control over the length of the vocal pipe but one can change the shape, which is how vowels are formed. Each of the two vocal chords functions like a string under tension. Changes in muscle tension change the frequency of the vibration.

Male vocal chords tend to be longer than female vocal chords, giving males a lower pitch. Male vocal chords range from 1.75 to 2.5 cm and female vocal chords range from 1.25 to 1.75 cm.

When air passes through the vocal chords the Bernoulli effect closes them. Further air pressure reopens the vocal chords and the cycle repeats.

The airflow has a triangle-shaped waveform, which because of its sharp edges generates abundant overtones.

Waves: sine, square, triangle, sawtooth
Creating a triangle wave from harmonics
Creating a sawtooth wave from harmonics

```                     Lung pressure (Pascals)

Passive exhalation          100
Singing                    1000
Fortissimo singing         4000
```
Atmospheric pressure is 101000 Pascals.

For a lung volume of 2 liters, 4000 Pascals corresponds to an energy of 8 Joules.

Singers, wind, and brass musicians train to deliver a continuous stable exhalation. String musicians train locking their ribcage in preparation for delivering a sharp impulse.

Nyquist frequency

Sampling a wave at the Nyquist frequency

Suppose a microphone samples a wave at fixed time intervals. The white curve is the wave and the orange dots are the microphone samplings.

```
F    =  Wave frequency
Fmic =  Sampling frequency of the microphone
Fny  =  Nyquist frequency
=  Minimum frequency to detect a wave of frequency F
=  2 F

```
In the above figure the sampling frequency is equal to the Nyquist frequency, or Fmic = 2 F. This is the minimum sampling frequency required to detect the wave.

This figure shows sampling for Fmic/F = {1, 2, 4, 8, 16}. In the left panel the wave and samplings are depicted and in the right panel only the samplings are depicted.

The top row corresponds to Fmic=F, and the wave cannot be detected at this sampling frequency.

The second row corresponds to Fmic=2F, which is the Nyquist frequency. This frequency is high enough to detect the wave but accuracy is poor.

For each successive row the value of Fmic/F is increased by a factor of 2. The larger the value of Fmic/F, the more accurately the wave can be detected.

Human hearing has a frequency limit of 20000 Hertz, which corresponds to a Nyquist frequency of 40000 Hertz. If you want to sample the highest frequencies accurately then you need a frequency of at least 80000 Hertz.

Overtones can generate high-frequency content in a recording, which is why the sampling frequency needs to be high.

Oscillators

Wikipedia:     Harmonic oscillator     Q factor     Resonance     Resonance

Hooke's law for a spring

A force can stretch or compresses a spring.

A spring oscillates at a frequency determined by K and M.

Frequency = Squareroot(K/M) / (2 Pi)

```
T     =  Time
X     =  Displacement of the spring when a force is applied
K     =  Spring constant
M     =  Mass of the object attached to the spring
Force =  Force on the spring
=  - K X      (Hooke's law)

```
Solving the differential equation:
```Force  =  M * Acceleration

- K X  =  M * X''

```
This equation has the solution
```X  =  sin(2 Pi F T)
```
where
```F = SquareRoot(K/M) / (2 Pi)

```
Wikipedia: Hooke's law

Damping

Damped spring

Vibrations of a damped string

After a string is plucked the amplitude of the oscillations decreases with time. The larger the damping the faster the amplitude decays.

```T    =  Time for one oscillation of the string
Tdamp=  Characteristic timescale for vibrations to damp
q    =  "Quality" parameter of the string
=  Characteristic number of oscillations required for the string to damp
=  Tdamp / T
```
In the above figure,
```q = Tdamp / T = 4
```
The smaller the damping the larger the value of q. For most instruments, q > 100.

Damping of a string for various values of q

The above figure uses the equation for a damped vibrating string.

```t    =  Time
X(t) =  Position of the string as a function of time
T    =  Time for the string to undergo one oscillation if there is no damping
q    =  Quality parameter, defined below
Typically  q>>1
F    =  Frequency of the string if there is no damping
=  1/T
Fd   =  Frequency of string oscillations if there is damping
=  F Z
Z    =  [1 - 1/(4 Pi^2 q^2)]^(1/2)
~  1  if  q>>1

```
A damped vibrating string follows a function of the form: (derived in the appendix)
```X  =  exp(-t/(Tq)) * cos(Zt/T)
```
The consine part generates the oscillations and the exponential part reflects the decay of the amplitude as a function of time.

For large q, the oscillations have a timescale of T and the damping has a timescale of T*q. This can be used to measure the value of q.

```q = (Timescale for damping)  /  (Time of one oscillation)
```
For example, you can record the waveform of a vibrating string and measure the oscillation period and the decay rate.

Resonance

If you shake a spring at the same frequency as the oscillation frequency then a large amplitude can result. Similarly, a swing can gain a large amplitude from small impulses if the impulses are timed with the swing period.

Suppose a violin A-string is tuned to 440 Hertz and a synthesizer produces a frequency that is close to 440 Hertz. If the synthesizer is close enough to 440 Hertz then the A-string rings, and if the synthesizer is far from 440 Hertz then the string doesn't ring.

This is a plot of the strength of the resonance as a function of the synthesizer frequency. The synthesizer frequency corresponds to the horizontal axis and the violin string has a frequency of 440 Hertz. The vertical axis corresponds to the strength of the vibration of the A-string.

A resonance has a characteristic width. The synthesizer frequency has to be within this width to excite the resonance. In the above plot the width of the resonance is around 3 Hertz.

```
F      =  Frequency of the resonator
=  440 Hertz
f      =  Frequency of the synthesizer
Fwidth =  Characteristic frequency width for resonance

If  |f-F|  <  Fwidth          then the resonator vibrates
If  |f-F|  >  Fwidth          then the resonator doesn't vibrate
```
Resonance simulation at phet.colorado.edu

Wind can make a string vibrate (The von Karman vortex).

The Tacoma Narrows bridge collapse was caused by wind exciting resonances in the bridge.

Resonant strength

The larger the value of q, the stronger the resonance. The following plot shows resonance curves for various values of q.

If q>>1 then

```
Amplitude of the resonance  =  Constant * q

```

You can break a wine glass by singing at the same pitch as the glass's resonanant frequency. The more "ringy" the glass the stronger the resonance and the easier it is to break.

Resonant width

The width of the resonance decreases with q. In the following plot the peak amplitude of the resonance curve has been set equal to 1 for each curve. As q increases the width of the resonance decreases.

```T      =  Time for one oscillation of the string
Td     =  Characteristic timescale for vibrations to damp
q      =  Characteristic number of oscillations required for the string to damp
=  Td / T
F      =  Frequency of the resonator
=  1/T
f      =  Frequency of the synthesizer
Fwidth =  Characteristic frequency width for resonance  (derived in appendix)
=  F / (2 Pi q)

If  |f-F|  <  Fwidth          then the resonator vibrates
If  |f-F|  >  Fwidth          then the resonator doesn't vibrate

```
If q>>1 then
```
Width of the resonance  =  F / (2 Pi q)

```
Overtones can also excite a resonance. For example, if you play an "A" on the G-string of a violin then the A-string vibrates. The open A-string is one octave above the "A" on the G-string and this is one of the overtones of the G-string.

The strings on an electric guitar are less damped than the strings on an acoustic guitar. An acoustic guitar loses energy as it generates sound while an electric guitar is designed to minimize damping. The resonances on an electric guitar are stronger than for an acoustic guitar.

Coupled oscillators

Oscillators that are mechanically connected can transfer energy back and forth between them.

Uncertainty principle

Suppose you measure the frequency of a wave by counting the number of crests and dividing by the time.

```T  =  Time over which the measurement is made
N  =  Number of crests occurring in a time T
F  =  N/T
dF =  Uncertainty in the frequency measurement
=  1/T
```
Suppose the number of crests can only be measured with an uncertainty of +-1. The uncertainty in the frequency is dF = 1/T. The more time you have to observe a wave the more precisely you can measure the frequency.

The equation for the uncertainty in a frequency measurement is

```dF T  >=  1
```

Gases

Ideal gas law

Molecules in a gas
Brownian motion

```Pressure                          =  P             (Pascals or Newtons/meter2 or Joules/meter3)
Temperature                       =  T             (Kelvin)
Volume                            =  Vol           (meters3)
Total gas kinetic energy          =  E             (Joules)
Kinetic energy per volume         =  e  =  E/Vol   (Joules/meter3)
Number of gas molecules           =  N
Mass of a gas molecule            =  M
Gas molecules per volume          =  n  =   N / Vol
Gas density                       =  D  = N M / Vol
Avogadro number                   =  Avo=  6.022⋅1023  moles-1
Moles of gas molecules            =  Mol=  N / Avo
Boltzmann constant                =  k  =  1.38⋅10-23 Joules/Kelvin
Gas constant                      =  R  =  k Avo  =  8.31 Joules/Kelvin/mole
Gas molecule thermal speed        =  Vth
Mean kinetic energy / gas molecule=  ε  =  E / n  =  ½ M Vth2     (Definition of the mean thermal speed)
```
Gas pressure arises from the kinetic energy of gas molecules and has units of energy/volume.
The ideal gas law can be written in the following forms:
```P  =  2⁄3 e                    Form used in physics
=  R Mol T / Vol            Form used in chemistry
=  k N   T / Vol
=  1⁄3 N M Vth2/ Vol
=  1⁄3 D Vth2
=  k T D / M
```
Gas simulation at phet.colorado.edu
Derivation of the ideal gas law
History

Boyle's law
Charles' law

```1660  Boyle law          P Vol     = Constant          at fixed T
1802  Charles law        T Vol     = Constant          at fixed P
1802  Gay-Lussac law     T P       = Constant          at fixed Vol
1811  Avogadro law       Vol / N   = Constant          at fixed T and P
1834  Clapeyron law      P Vol / T = Constant          combined ideal gas law
```

Boltzmann constant

For a system in thermodynamic equilibrium each degree of freedom has a mean energy of ½ k T. This is the definition of temperature.

```Molecule mass                =  M
Thermal speed                =  Vth
Boltzmann constant           =  k  =  1.38⋅10-23 Joules/Kelvin
Molecule mean kinetic energy =  ε
```
A gas molecule moving in N dimensions has N degrees of freedom. In 3D the mean energy of a gas molecule is
```ε  =  3⁄2 k T  =  ½ M V2th
```

Speed of sound

The sound speed is proportional to the thermal speed of gas molecules. The thermal speed of a gas molecule is defined in terms of the mean energy per molecule.

```Adiabatic constant  =  γ
=  5/3 for monatomic molecules such as helium, neon, krypton, argon, and xenon
=  7/5 for diatomic molecules such as H2, O2, and N2
=  7/5 for air, which is 21% O2, 78% N2, and 1% Ar
≈  1.31 for a triatomic gas such as CO2
Pressure            =  P
Density             =  D
Sound speed         =  Vsound
Mean thermal speed  =  Vth
K.E. per molecule   =  ε  =  ½ M Vth2

V2sound  =  γ  P / D  =  1⁄3  γ  V2th
```
The sound speed depends on temperature and not on density or pressure.

For air, γ = 7/5 and

```Vsound  =  .68  Vth
```
These laws are derived in the appendix.

We can change the sound speed by using a gas with a different value of M.

```                   M in atomic mass units

Helium atom                4
Neon atom                 20
Nitrogen molecule         28
Oxygen molecule           32
Argon atom                40
Krypton atom              84
Xenon atom               131
```
A helium atom has a smaller mass than a nitrogen molecule and hence has a higher sound speed. This is why the pitch of your voice increases if you inhale helium. Inhaling xenon makes you sound like Darth Vader. Then you pass out because Xenon is an anaesthetic.

In a gas, some of the energy is in motion of the molecule and some is in rotations and vibrations. This determines the adiabatic constant.

Ethane
Molecule with thermal vibrations

History of the speed of sound
```1635  Gassendi measures the speed of sound to be 478 m/s with 25% error.
1660  Viviani and Borelli produce the first accurate measurement of the speed of
sound, giving a value of 350 m/s.
1660  Hooke's law published.  The force on a spring is proportional to the change
in length.
1662  Boyle discovers that for air at fixed temperature,
Pressure * Volume = Constant
1687  Newton publishes the Principia Mathematica, which contains the first analytic
calculation of the speed of sound.  The calculated value was 290 m/s.
```
Newton's calculation was correct if one assumes that a gas behaves like Boyle's law and Hooke's law.

The fact that Newton's calculation differed from the measured speed is due to the fact that air consists of diatomic molecules (nitrogen and oxygen). This was the first solid clue for the existence of atoms, and it also contained a clue for quantum mechanics.

In Newton's time it was not known that changing the volume of a gas changes its temperature, which modifies the relationship between density and pressure. This was discovered by Charles in 1802 (Charles' law).

Gas data
```       Melt   Boil  Solid    Liquid   Gas      Mass   Sound speed
(K)    (K)   density  density  density  (AMU)  at 20 C
g/cm3    g/cm3    g/cm3            (m/s)

He        .95   4.2            .125   .000179    4.00  1007
Ne      24.6   27.1           1.21    .000900   20.18
Ar      83.8   87.3           1.40    .00178    39.95   319
Kr     115.8  119.9           2.41    .00375    83.80   221
Xe     161.4  165.1           2.94    .00589   131.29   178
H2      14     20              .070   .000090    2.02  1270
N2      63     77              .81    .00125    28.01   349
O2      54     90             1.14    .00143    32.00   326
Air                                   .0013     29.2    344     79% N2, 21% O2, 1% Ar
H2O    273    373     .917    1.00    .00080    18.02
CO2    n/a    195    1.56      n/a    .00198    44.00   267
CH4     91    112              .42    .00070    16.04   446
CH5OH  159    352              .79    .00152    34.07           Alcohol
```
Gas density is for 0 Celsius and 1 Bar. Liquid density is for the boiling point, except for water, which is for 4 Celsius.

Carbon dioxide doesn't have a liquid state at standard temperature and pressure. It sublimes directly from a solid to a vapor.

Height of an atmosphere

```M  =  Mass of a gas molecule
V  =  Thermal speed
E  =  Mean energy of a gas molecule
=  1/2 M V^2
H  =  Characteristic height of an atmosphere
g  =  Gravitational acceleration
```
Suppose a molecule at the surface of the Earth is moving upward with speed V and suppose it doesn't collide with other air molecules. It will reach a height of
```M H g  =  1/2  M  V^2
```
This height H is the characteristic height of an atmosphere.
```Pressure of air at sea level      =  1   Bar
Pressure of air in Denver         = .85  Bar      One mile high
Pressure of air at Mount Everest  = 1/4  Bar      10 km high
```
The density of the atmosphere scales as
```Density ~ (Density At Sea Level) * exp(-E/E0)
```
where E is the gravitational potential energy of a gas molecule and E0 is the characteristic thermal energy given by
```E0 = M H g = 1/2 M V^2
```
Expressed in terms of altitude h,
```Density ~ Density At Sea Level * exp(-h/H)
```
For oxygen,
```E0  =  3/2 * Boltzmann_Constant * Temperature
```
E0 is the same for all molecules regardless of mass, and H depends on the molecule's mass. H scales as
```H  ~  Mass^-1
```

Atmospheric escape
```S = Escape speed
T = Temperature
B = Boltzmann constant
= 1.38e-23 Joules/Kelvin
g = Planet gravity at the surface

M = Mass of heavy molecule                    m = Mass of light molecule
V = Thermal speed of heavy molecule           v = Thermal speed of light molecule
E = Mean energy of heavy molecule             e = Mean energy of light molecule
H = Characteristic height of heavy molecule   h = Characteristic height of light molecule
= E / (M g)                                   = e / (m g)
Z = Energy of heavy molecule / escape energy  z = Energy of light molecule / escape energy
= .5 M V^2 / .5 M S^2                         = .5 m v^2 / .5 m S^2
= V^2 / S^2                                   = v^2 / S^2

For an ideal gas, all molecules have the same mean kinetic energy.

E     =     e      =  1.5 B T

.5 M V^2  =  .5 m v^2  =  1.5 B T
```
The light molecules tend to move faster than the heavy ones. This is why your voice increases in pitch when you breathe helium. Breathing a heavy gas such as Xenon makes you sound like Darth Vader.

For an object to have an atmosphere, the thermal energy must be much less than the escape energy.

```V^2 << S^2        <->        Z << 1

Escape  Atmos    Temp    H2     N2      Z        Z
speed   density  (K)    km/s   km/s    (H2)     (N2)
km/s    (kg/m^3)
Jupiter   59.5             112   1.18    .45   .00039   .000056
Saturn    35.5              84   1.02    .39   .00083   .00012
Neptune   23.5              55    .83    .31   .0012    .00018
Uranus    21.3              53    .81    .31   .0014    .00021
Earth     11.2     1.2     287   1.89    .71   .028     .0041
Venus     10.4    67       735   3.02   1.14   .084     .012
Mars       5.03     .020   210   1.61    .61   .103     .015
Titan      2.64    5.3      94   1.08    .41   .167     .024
Europa     2.02    0       102   1.12    .42   .31      .044
Moon       2.38    0       390   2.20    .83   .85      .12
Ceres       .51    0       168   1.44    .55  8.0      1.14
```
Even if an object has enough gravity to capture an atmosphere, it can still lose it to the solar wind. Also, the upper atmosphere tends to be hotter than at the surface, increasing the loss rate.

The threshold for capturing an atmosphere appears to be around Z = 1/25, or

```Thermal Speed  <  1/5 Escape speed
```

Heating by gravitational collapse

When an object collapses by gravity, its temperature increases such that

```Thermal speed of molecules  ~  Escape speed
```
In the gas simulation at phet.colorado.edu, you can move the wall and watch the gas change temperature.

For an ideal gas,

```3 * Boltzmann_Constant * Temperature  ~  MassOfMolecules * Escape_Speed^2
```
For the sun, what is the temperature of a proton moving at the escape speed? This sets the scale of the temperature of the core of the sun. The minimum temperature for hydrogen fusion is 4 million Kelvin.

The Earth's core is composed chiefly of iron. What is the temperature of an iron atom moving at the Earth's escape speed?

```      Escape speed (km/s)   Core composition
Sun        618.             Protons, electrons, helium
Earth       11.2            Iron
Mars         5.03           Iron
Moon         2.38           Iron
Ceres         .51           Iron
```

Derivation of the ideal gas law

We first derive the law for a 1D gas and then extend it to 3D.

Suppose a gas molecule bounces back and forth between two walls separated by a distance L.

```M  = Mass of molecule
V  = Speed of the molecule
L  = Space between the walls
```
With each collision, the momentum change = 2 M V

Time between collisions = 2 L / V

The average force on a wall is

```Force  =  Change in momentum  /  Time between collisions  =  M  V^2  /  L
```
Suppose a gas molecule is in a cube of volume L^3 and a molecule bounces back and forth between two opposite walls (never touching the other four walls). The pressure on these walls is
```Pressure  =  Force  /  Area
=  M  V^2  /  L^3
=  M  V^2  /  Volume

Pressure * Volume  =  M  V^2
```
This is the ideal gas law in one dimension. For a molecule moving in 3D,
```Velocity^2  = (Velocity in X direction)^2
+ (Velocity in Y direction)^2
+ (Velocity in Z direction)^2

Characteristic thermal speed in 3D  =  3  *  Characteristic thermal speed in 1D.
```
To produce the 3D ideal gas law, replace V^2 with 1/3 V^2 in the 1D equation.
```Pressure * Volume  =  1/3  M  V^2        Where V is the characteristic thermal speed of the gas
```
This is the pressure for a gas with one molecule. If there are n molecules,
```Pressure  Volume  =  n  1/3  M  V^2            Ideal gas law in 3D
```
If a gas consists of molecules with a mix of speeds, the thermal speed is defined as
```Kinetic dnergy density of gas molecules  =  E  =  (n / Volume) 1/2 M V^2
```
Using this, the ideal gas law can be written as
```Pressure  =  2/3  E
=  1/3  Density  V^2
=  8.3  Moles  Temperature  /  Volume
```
The last form comes from the law of thermodynamics:
```M V^2 = 3 B T
```

Virial theorem

A typical globular cluster consists of millions of stars. If you measure the total gravitational and kinetic energy of the stars, you will find that

```Total gravitational energy  =  -2 * Total kinetic energy
```
just like for a single satellite on a circular orbit.

Suppose a system consists of a set of objects interacting by a potential. If the system has reached a long-term equilibrium then the above statement about energies is true, no matter how chaotic the orbits of the objects. This is the "Virial theorem". It also applies if additional forces are involved. For example, the protons in the sun interact by both gravity and collisions and the virial theorem holds.

```Gravitational energy of the sun  =  -2 * Kinetic energy of protons in the sun
```

Newton's calculation for the speed of sound

Hooke's law for a spring
Wave in a continuum
Gas molecules

```
Because of Hooke's law, springs oscillate with a constant frequency.

X = Displacement of a spring
V = Velocity of the spring
A = Acceleration of the spring
F = Force on the spring
M = Spring mass
Q = Spring constant
q = (K/M)^(1/2)
t = time
T = Spring oscillation period
```
Hooke's law and Newton's law:
```F  =  - Q X  =  M A

A  =  - (Q/M) X  =  - q^2 X
```
This equation is solved with
```X  =      sin(q t)
V  =  q   cos(q t)
A  = -q^2 sin(q t)  =  - q^2 X
```
The oscillation period of the spring is
```T  =  2 Pi / q
=  2 Pi (M/Q)^(1/2)

```
According to Boyle's law, a gas functions like a spring and hence a gas oscillates like a spring. An oscillation in a gas is a sound wave.

For a gas,

```P   =  Pressure
dP  =  Change in pressure
Vol =  Volume
dVol=  Change in volume
```
If you change the volume of a gas according to Boyle's law,
```P Vol            =  Constant
P dVol + Vol dP  =  0

dP = - (P/Vol) dVol
```
The change in pressure is proportional to the change in volume. This is equivalent to Hooke's law, where pressure takes the role of force and the change in volume takes the role of displacement of the spring. This is the mechanism behind sound waves.

In Boyle's law, the change in volume is assumed to be slow so the gas has time to equilibrate temperature with its surroundings. In this case the temperature is constant as the volume changes and the change is "isothermal".

```P Vol = Constant
```
If the change in volume is fast then the walls do work on the molecules, changing their temperature. If there isn't enough time to equilibrate temperature with the surroundings then the change is "adiabatic". You can see this in action with the "Gas" simulation at phet.colorado.edu. Moving the wall changes the thermal speed of molecules and hence the temperature.

If a gas consists of pointlike particles then

```Vol =  Volume of the gas
Ek  =  Total kinetic energy of gas molecules within the volume
E   =  Total energy of gas molecules within the volume
=  Kinetic energy plus the energy from molecular rotation and vibration
dE  =  Change in energy as the volume changes
P   =  Pressure
dP  =  Change in pressure as the volume changes
D   =  Density
C   =  Speed of sound in the gas
d   =  Number of degrees of freedom of a gas molecule
=  3 for a monotomic gas such as Helium
=  5 for a diatomic gas such as nitrogen
G   =  Adiabatic constant
=  1 + 2/d
=  5/3 for a monatomic gas
=  7/5 for a diatomic gas
k   =  Boltzmann constant
T   =  Temperature
```
The ideal gas law is
```P Vol =  (2/3) Ek                    (Derived in www.jaymaron.com/gas/gas.html)
```
This law is equivalent to the formula that appears in chemistry.
```P Vol = Moles R T
```
For a gas in thermal equilibrium each degree of freedom has a mean energy of .5 k T. For a gas of pointlike particles (monotomic) there are three degrees of freedom, one each for motion in the X, Y, and Z direction. In this case d=3. The mean kinetic energy of each gas molecule is 3 * (.5 k T). The total mean energy of each gas molecule is also 3 * (.5 k T).

For a diatomic gas there are also two rotational degrees of freedom. In this case d=5.

In general,

```Ek  =  3 * (.5 k T)
E   =  d * (.5 k T)

Ek  =  (3/d) E
```
If you change the volume of a gas adiabatically, the walls change the kinetic and rotational energy of the gas molecules.
```dE  =  -P dVol
```
The ideal gas law in terms of E instead of Ek is
```P Vol =  (2/d) E

dP  =  (2/d) (dE/Vol - E dVol/Vol^2)
=  (2/d) [-P dVol/Vol - (d/2) P dVol/Vol]
= -(1+2/d) P dVol/Vol
= - G P dVol/Vol
```
This equation determines the speed of sound in a gas.
```C^2  =  G P / D
```
For air,
```P = 1.01e5 Newtons/meter^2
D = 1.2    kg/meter^3
```
Newton assumed G=1 from Boyle's law and calculated the speed of sound in air to be
```C  =  290 m/s
```
The correct value for air is G=7/5, which gives a sound speed of
```C = 343 m/s
```
which is in accord with the measurement.

For a gas, G can be measured by measuring the sound speed. The results are

```Helium     5/3    Monatomic molecule
Argon      5/3    Monatonic molecule
Air        7/5    4/5 Nitrogen and 1/5 Oxygen
Oxygen     7/5    Diatomic molecule
Nitrogen   7/5    Diatomic molecule
```
The fact that G is not equal to 1 was the first solid evidence for the existence of atoms and it also contained a clue for quantum mechanics. If a gas is a continuum (like Hooke's law) it has G=1 and if it consists of pointlike particles (monatonic) it has G=5/3. This explains helium and argon but not nitrogen and oxygen. Nitrogen and oxygen are diatomic molecules and their rotational degrees of freedom change Gamma.
```                             Kinetic degrees   Rotational degrees    Gamma
of freedom         of freedom
Monatonic gas                      3                  0               5/3
Diatomic gas  T < 1000 K           3                  2               7/5
Diatomic gas, T > 1000 K           3                  3               4/3
```
Quantum mechanics freezes out one of the rotation modes at low temperature. Without quantum mechanics, diatomic molecules would have Gamma=4/3 at room temperature.

The fact that Gamma=7/5 for air was a clue for the existence of both atoms, molecules, and quantum mechanics.

Dark energy

For dark energy,

```E  =  Energy
dE =  Change in energy
e  =  Energy density
Vol=  Volume
P  =  Pressure
```
The volume expands as the universe expands.

As a substance expands it does work on its surroundings according to its pressure.

```dE = - P dVol
```
For dark energy, the energy density "e" is constant in space and so
```dE = e dVol
```
Hence,
```P = - e
```
Dark energy has a negative pressure, which means that it behaves differently from a continuum and from particles.

Dark matter consists of pointlike particles but they rarely interact with other particles and so they exert no pressure.

Density

Size of atoms
Dot size corresponds to atom size.

For gases, the density at boiling point is used.   Size data

Density

Copper atoms stack like cannonballs. We can calculate the atom size by assuming the atoms are shaped like either cubes or spheres. For copper atoms,

```Density         = D              =       8900 kg/m3
Atomic mass unit= M0             = 1.661⋅10-27 kg
Atomic mass     = MA             =      63.55 Atomic mass units
Mass            = M   =  MA⋅M0    = 9.785⋅10-26 kg
Number density  = N   =  D / M   = 9.096⋅1028  atoms/m3
Cube volume     = Υcube=  1 / N   = 1.099⋅10-29 m3            Volume/atom if the atoms are cubes
Cube length     = L   =  Υ1/3cube = 2.22⋅10-10  m             Side length of the cube
Sphere fraction = f   =  π/(3√2) =     .7405                Fraction of volume occupied by spheres in a stack o spheres
Sphere volume   = Υsph=  Υcube f  = 8.14⋅10-30  m3 = 4⁄3πR3    Volume/atom if the atoms are spheres
Sphere radius   = R              = 1.25⋅10-10  m
```

Strength and density
```Dot size  =  Density
Color     =  Shear Modulus, an indicator of the element's strength.

Blue:    The element is a liquid at room temperature
Red:     Weak
White:   Strong
```
Shear data Density data
Chemistry

Bohr model of the atom

The "de Broglie wavelength" of a particle is

```Particle momentum    =  Q
Planck constant      =  h  =  6.62*10^-34 Joule seconds
Particle wavelength  =  W  =  h/Q             (de Broglie formula)
```
The Bohr hypothesis states that for an electron orbiting a proton, the number of electron wavelengths is an integer. This sets the characteristic size of a hydrogen atom.
```Orbit circumference  =  C  =  N W           where N is a positive integer

N   Orbital

1     S
2     P
3     D
4     F

Electron mass      =  m                =  9.11*10-31 kg
Electron velocity  =  V
Electron momentum  =  Q  = m V
Electron charge    =  e                =  -1.60*10-19 Coulombs
Coulomb constant   =  K                =  9.0*109  Newtons meters / Coulombs2
Electric force     =  Fe  =  K e2 / R2
Centripetal force  =  Fc  =  M V2 / R
Orbit radius       =  R  =  N h2 / (4 π2 K e2 m)  =  N * 5.29e-11 meters
Electron energy    =  E  =  - .5 K e2 / (R N2)    =  N-2 2.18e-18 Joules  =  N-2 13.6 electron Volts          (Ionization energy)
```
For an electron on a circular orbit,
```Fe = Fc
```

Wavefunctions

Shells

Electronegativity

Electromagnetism

Electric force

The fundamental unit of charge is the "Coulomb", and the electric force follows the same equations as the gravitational force.

Charges of the same sign repel and charges of opposite sign attract.

```Charge 1    Charge 2     Electric Force

+           +         Repel
-           -         Repel
+           -         Attract
-           +         Attract

Charge                   =  Q  (Coulombs)       1 Proton = 1.602e-19 Coulombs
Distance between charges =  R
Mass of the charges      =  M

Gravity constant         =  G  = 6.67e-11 Newton m2 / kg2
Electric constant        =  K  = 8.99e9 Newton m2 / Coulomb2

Gravity force            =  F  =  -G M1 M2 / R2  =  M2 g
Electric force           =  F  =  -K Q1 Q2 / R2  =  Q2 E

Gravity field from M1    =  g  =  G M1 / R2
Electric field from Q1   =  E  =  K Q1 / R2

Gravity voltage          =  H g               (H = Height, g = Gravitational acceleration)
Electric voltage         =  H E               (H = Distance parallel to the electric field)

Gravity energy           =  -G M1 M2 / R
Electric energy          =  -K Q1 Q2 / R
```

A charge generates an electric field. The electric field points away from positive charges and toward negative charges.

Electric current

A moving charge is an "electric current". In an electric circuit, a battery moves electrons through a wire.

```Charge            =  Q
Time              =  T
Electric current  =  I  =  Q / T   (Coulombs/second)
```
The current from a positive charge moving to the right is equivalent to that from a negative charge moving to the left.
Magnetic force

Parallel currents attract

Moving charges and currents exert forces on each other. Parallel currents attract and antiparallel currents repel.

```Charge                          =  Q
Velocity of the charges         =  V
Current                         =  I
Length of a wire                =  L
Distance between the charges    =  R
Electric force constant         =  Ke  =  8.988e9 N m2/C2
Magnetic force constant         =  Km  =  2e-7 = Ke/C2
Electric force between charges  =  Fe  =  Ke Q1 Q2 / R2
Magnetic force between charges  =  Fm  =  Km V2 Q1 Q2 / R2  =  (V2/C2) Fe
Magnetic force between currents =  Fm  =  Km I1 I2 Z / R
Magnetic force / Electric force =  V2 / C2
```
The magnetic force is always less than the electric force.
Magnetic field

A current generates a magnetic field
A magnetic field exerts a force on a current

The electric force can be interpreted as an electric field, and the magnetic force can be interpreted as a magnetic field. Both interpretations produce the same force.

```Radial distance                =  R                 (Distance perpendicular to the velocity of the charge)
Magnetic field from charge Q1  =  B  =  Km V Q1 / R2
Magnetic field from current I1 =  B  =  Km I1 / R
Magnetic force on charge Q2    =  Fm =  Q2 V B  =  Km V2 Q1 Q2 / R2
Magnetic force on current I2   =  Fm =  I2 Z B  =  Km I1 I2 Z / R
```

Right hand rule

The direction of the magnetic force on a positive charge is given by the right hand rule. The force on a negative charge is in the opposite direction (the left hand rule).

Positive and negative charge

A vertical magnetic field deflects positive charges rightward and negative charges leftward
A vertical field causes positive charges to circle clockwise and negative charges to circle counterclockwise.

We use the above symbols to depict vectors in the Z direction. The vector on the left points into the plane and the vector on the right points out of the plane.

Magnetic field generated by a magnet
Iron filings align with a magnetic field

Cross product

The direction of the force is the cross product "×" of V and B. The direction is given by the "right hand rule".

```Magnetic field              =  B
Magnetic force on a charge  =  F  =  Q V × B
Magnetic force on a current =  F  =  2e-7 I × B
```

Electricity and magnetism

Index of variables and equations
```
Quantity                         MKS units                       CGS units     Conversion factor

Mass                             M   kg                          gram            .001
Wire length                      Z   meter                       cm               .01
Radial distance from wire        R   meter                       cm               .01
Time                             T   second                      second             1
Force                            F   Newton                      dyne          100000
Charge                           Q   Coulomb                     Franklin   3.336e-10
Velocity of a charge             V   meter/second                cm/s             .01
Speed of light                   C   2.999e8 meter/second        cm/s             100
Energy                           E   Joule                       erg              e-7
Electric current                 I   Ampere = Coulomb/s          Franklin/s 3.336e-10
Electric potential               V   Volt                        Statvolt      299.79
Electric field                   E   Volt/meter                  StatVolt/cm    29979
Magnetic field                   B   Tesla                       Gauss          10000
Capacitance                      C   Farad                       cm          1.11e-12
Inductance                       L   Henry                       s2/cm  9e-11
Electric force constant          Ke  = 8.988e9 N m2/C2            Ke = 1 dyne cm2 / Franklin2
Magnetic force constant          Km  = 2e-7 = Ke/C2               Km = 1/C2
Vacuum permittivity              ε   = 8.854e-12 F/m =1/4/π/Ke
Vacuum permeability              μ   = 4 π e-7 Vs/A/m =2 π Km
Proton charge                    Qpro = 1.602e-19 Coulomb         Qpro= 4.803e-10 Franklin
Electric field from a charge     E   = Ke Q / R2                  E  = Q / R2
Electric force on a charge       F   = Q E                        F  = Q E
Electric force between charges   F   = Ke Q Q / R2                F  = Q Q / R2
Magnetic field of moving charge  B   = Km V Q / R2                B  = (V/C) Q / R2
Magnetic field around a wire     B   = Km I / R                   B  = (V/C) I / R
Magnetic force on a charge       F   = Q V B                      F  = (V/C) Q B
Magnetic force on a wire         F   = Km B  Z                    F  = I B z
Magnetic force between charges   F   = Km V2 Q1 Q2 / R2            F  = (V/C)2 Q Q / R2
Magnetic force between wires     F   = Km I1 I2 Z / R              F  = I1 I2 Z / R
Energy of a capacitor            E   = .5 C V2
Field energy per volume          Z   = (8 π Ke)-1 (E2 + B2/C2)      Z = .5 (E2 + B2/C2)
```

Maxwell's equations
```Speed of light                      C
Electric field                      E
Electric field, time derivative     Et
Magnetic field                      B
Magnetic field, time derivative     Bt
Charge                              Q
Charge density                      q
Current density                     J

MKS                           CGS

Ke=8.988e9                    Ke=1
Km=2e-7                       Km=2/C

∇˙E = 4 π Ke q                ∇˙E = 4 π q
∇˙B = 0                       ∇˙B = 0
∇×E = -Bt                     ∇×E = -Bt / C
∇×B = 2 π Km J + Et / C2       ∇×B = 4 π J / C + Et / C
```

Circuits

Battery

A battery moves charge upwards in voltage
A resistor dissipates energy as charges fall downwards in voltage

```Charge          =  Q           Coulombs
Voltage         =  V           Volts
Energy          =  E  =  VQ    Joules
Time            =  T           seconds
Current         =  I  =  Q/T   Amperes
Resistance      =  R  =  V/I   Ohms
Power           =  P  =  QV/T  Watts
=  IV
=  V2/R
=  I2R

Ohm's Law:  V = IR
```

Resistance

Superconductor
Resistor

In a superconductor, electrons move without interference.
In a resistor, electrons collide with atoms and lose energy.

```                 Resistance (Ohms)

Copper wire            .02          1 meter long and 1 mm in diameter
1 km power line        .03
AA battery             .1           Internal resistance
Light bulb          200
Human             10000
```

Capacitors
```Voltage          =  V             Volts
Capacitance      =  C             Farads
Total energy     =  E  =  ½ C V2  Joules
Effective        =  Ee =  ¼ C V2  Joules
```
Not all of the energy in a capacitor is harnessable because the voltage diminishes as the charge diminishes, hence the effective energy is less than the total energy.
Capacitance
```A   =  Plate area
Z   =  Plate spacing
Ke  =  Electric force constant  =  8.9876e9 N m2 / C2
Q   =  Max charge on the plate     (Coulombs)
Emax=  Max electric field       =  4 Pi Ke Q / A
V   =  Voltage between plates   =  E Z     =  4 Pi Ke Q Z / A
En  =  Energy                   =  .5 Q V  =  .5 A Z E2 / (4 π Ke)
e   =  Energy/Volume            =  E / A Z =  .5 E2 / (4 π Ke)
q   =  Charge/Volume            =  Q / A / Z
C   =  Capacitance              =  Q/V     =  (4 Pi Ke)-1 A/Z   (Farads)
c   =  Capacitance/Volume       =  C / A / Z =  (4 Pi Ke)-1 Emax2 / V2
Eair=  Max electric field in air=  3 MVolt/meter
k   =  Dielectric factor        =  Emax / Eair

Continuum                                                 Macroscopic

Energy/Volume  =  .5 E2  / (4 Pi Ke)           <->        Energy = .5 C V2
=  .5 q V                                         =  .5 Q V
c              =  (4 Pi Ke)-1 Emax2  / V2      <->        C      = (4 Pi Ke)-1 A / Z

```
A capacitor can be specified by two parameters:
*)   Maximum energy density or maximum electric field
*)   Voltage between the plates

The maximum electric field is equal to the max field for air times a dimensionless number characterizing the dielectric

```Eair =  Maximum electric field for air before electical breakdown
Emax =  Maximum electric field in the capacitor
=  Characteristic size of atoms
=  5.2918e-11 m
=  hbar2 / (ElectronMass*ElectronCharge2*Ke)
Ebohr=  Bohr electric field
=  Field generated by a proton at a distance of 1 Bohr radius
=  5.142e11 Volt/m
Maximum energy density  =  .5 * 8.854e-12 Emax2

Emax (MVolt/m)   Energy density
(Joule/kg)
Al electrolyte capacitor     15.0            1000
Supercapacitor               90.2           36000
Bohr limit               510000            1.2e12            Capacitor with a Bohr electric field
```

Inductance

A solenoid is a wire wound into a coil.

```N  =  Number of wire loops
Z  =  Length
A  =  Area
Mu =  Magnetic constant  =  4 π 10-7
I  =  Current
It =  Current change/time
F  =  Magnetic flux      =  N B A        (Tesla meter2)
Ft =  Flux change/time                   (Tesla meter2 / second)
B  =  Magnetic field     =  Mu N I / Z
V  =  Voltage            =  Ft =  L It  =  N A Bt  =  Mu N2 A It / Z
L  =  Inductance         =  Ft / It  =  Mu N2 A / Z      (Henrys)
E  =  Energy             =  .5 L I2
```
Hyperphysics: Inductor
Conductivity

```White: High conductivity
Red:   Low conductivity
```

Electric and thermal conductivity
```         Electric  Thermal  Density   Electric   C/Ct     Heat   Heat      Melt   \$/kg  Young  Tensile Poisson  Brinell
conduct   conduct            conduct/            cap    cap                                   number   hardness
(e7 A/V/m) (W/K/m)  (g/cm^3)  Density   (AK/VW)  (J/g/K) (J/cm^3K)  (K)         (GPa)  (GPa)             (GPa)

Silver      6.30   429      10.49       .60      147       .235   2.47     1235    590    83   .17      .37      .024
Copper      5.96   401       8.96       .67      147       .385   3.21     1358      6   130   .21      .34      .87
Gold        4.52   318      19.30       .234     142       .129   2.49     1337  24000    78   .124     .44      .24
Aluminum    3.50   237       2.70      1.30      148       .897   2.42      933      2    70   .05      .35      .245
Beryllium   2.5    200       1.85      1.35      125      1.825   3.38     1560    850   287   .448     .032     .6
Magnesium   2.3    156       1.74      1.32      147      1.023   1.78      923      3    45   .22      .29      .26
Iridium     2.12   147      22.56       .094     144       .131   2.96     2917  13000   528  1.32      .26     1.67
Rhodium     2.0    150      12.41       .161     133       .243   3.02     2237  13000   275   .95      .26     1.1
Tungsten    1.89   173      19.25       .098     137       .132   2.54     3695     50   441  1.51      .28     2.57
Molybdenum  1.87   138      10.28       .182     136       .251            2896     24   330   .55      .31     1.5
Cobalt      1.7    100       8.90       .170               .421            1768     30   209   .76      .31      .7
Zinc        1.69   116       7.14                          .388             693      2   108   .2       .25      .41
Nickel      1.4     90.9     8.91                          .444            1728     15
Ruthenium   1.25   117      12.45                                          2607   5600
Cadmium     1.25    96.6     8.65                                           594      2    50   .078     .30      .20
Osmium      1.23    87.6    22.59                          .130            3306  12000
Indium      1.19    81.8     7.31                                           430    750    11   .004     .45      .009
Iron        1.0     80.4     7.87                          .449            1811          211   .35      .29      .49
Palladium    .95    71.8                                                   1828
Tin          .83    66.8                                                    505     22    47   .20      .36      .005
Chromium     .79    93.9                                   .449            2180
Platinum     .95                                           .133            2041
Tantalum     .76                                           .140            3290
Gallium      .74                                                            303
Thorium      .68
Niobium      .55    53.7                                                   2750
Rhenium      .52                                           .137            3459
Vanadium     .5     30.7                                                   2183
Uranium      .35
Titanium     .25    21.9                                   .523            1941
Scandium     .18    15.8                                                   1814
Neodymium    .156                                                          1297
Mercury      .10     8.30                                  .140             234
Manganese    .062    7.81                                                  1519
Germanium    .00019                                                        1211

Dimond iso 10    40000
Diamond     e-16  2320                                     .509
Tube       10     3500                                                Carbon nanotube. Electric conductivity = e-16 laterally
Tube bulk          200                                                Carbon nanotubes in bulk
Graphene   10     5000
Graphite    2      400                                     .709       Natural graphite
Al Nitride  e-11   180
Brass       1.5    120
Steel               45                                                Carbon steel
Bronze       .65    40
Steel Cr     .15    20                                                Stainless steel (usually 10% chromium)
Quartz (C)          12                                                Crystalline quartz.  Thermal conductivity is anisotropic
Quartz (F)  e-16     2                                                Fused quartz
Granite              2.5
Marble               2.2
Ice                  2
Concrete             1.5
Limestone            1.3
Soil                 1
Glass       e-12      .85
Water       e-4       .6
Seawater    1         .6
Brick                 .5
Plastic               .5
Wood                  .2
Wood (dry)            .1
Plexiglass  e-14      .18
Rubber      e-13      .16
Snow                  .15
Paper                 .05
Plastic foam          .03
Air        5e-15      .025
Nitrogen              .025                                1.04
Oxygen                .025                                 .92
Silica aerogel        .01

Siemens:    Amperes^2 Seconds^3 / kg / meters^2     =   1 Ohm^-1
```
For most metals,
```Electric conductivity / Thermal conductivity  ~  140  J/g/K
```

Magnetic field magnitudes
```                                     Teslas

Field generated by brain             10-12
Wire carrying 1 Amp                  .00002     1 cm from the wire
Earth magnetic field                 .0000305   at the equator
Neodymium magnet                    1.4
Magnetic resonance imaging machine  8
Large Hadron Collider magnets       8.3
Field for frog levitation          16
Strongest electromagnet            32.2         without using superconductors
Strongest electromagnet            45           using superconductors
Neutron star                       1010
Magnetar neutron star              1014
```

Dielectric strength

The critical electric field for electric breakdown for the following materials is:

```
MVolt/meter
Air                3
Glass             12
Polystyrene       20
Rubber            20
Distilled water   68
Vacuum            30        Depends on electrode shape
Diamond         2000
```

Relative permittivity

Relative permittivity is the factor by which the electric field between charges is decreased relative to vacuum. Relative permittivity is dimensionless. Large permittivity is desirable for capacitors.

```             Relative permittivity
Vacuum            1                   (Exact)
Air               1.00059
Polyethylene      2.5
Sapphire         10
Concrete         4.5
Glass          ~ 6
Rubber           7
Diamond        ~ 8
Graphite       ~12
Silicon         11.7
Water (0 C)     88
Water (20 C)    80
Water (100 C)   55
TiO2         ~ 150
SrTiO3         310
BaSrTiO3       500
Ba TiO3     ~ 5000
CaCuTiO3    250000
```

Magnetic permeability

A ferromagnetic material amplifies a magnetic field by a factor called the "relative permeability".

```                Relative    Magnetic   Maximum    Critical
permeability  moment     frequency  temperature
(kHz)      (K)
Metglas 2714A    1000000                100               Rapidly-cooled metal
Iron              200000      2.2                 1043
Iron + nickel     100000                                  Mu-metal or permalloy
Cobalt + iron      18000
Nickel               600       .606                627
Cobalt               250      1.72                1388
Carbon steel         100
Neodymium magnet       1.05
Manganese              1.001
Air                    1.000
Superconductor         0
Dysprosium                   10.2                   88
EuO                           6.8                   69
Y3Fe5O12                      5.0                  560
MnBi                          3.52                 630
MnAs                          3.4                  318
NiO + Fe                      2.4                  858
CrO2                          2.03                 386
```

Effect of temperature on conductivity

Resistivity in 10^-9 Ohm Meters

```              293 K   300 K   500 K

Beryllium     35.6    37.6     99
Magnesium     43.9    45.1     78.6
Aluminum      26.5    27.33    49.9
Copper        16.78   17.25    30.9
Silver        15.87   16.29    28.7
```

Wire gauges
```Gauge  Diameter  Continuous  10 second  1 second  32 ms    Resistance
mm      current    current    current   current
Ampere     Ampere     Ampere    Ampere   mOhm/meter

0        8.3      125        1900      16000     91000       .32
2        6.5       95        1300      10200     57000       .51
4        5.2       70         946       6400     36000       .82
6        4.1       55         668       4000     23000      1.30
12        2.0       20         235       1000      5600      5.2
18        1.02      10          83        250      1400     21.0
24         .51       3.5        29         62       348     84
30         .255       .86       10         15        86    339
36         .127       .18        4         10        22   1361
40         .080                  1          1.5       8   3441
```

Nuclei

Proton = 2 up quarks + down quark
Helium atom
Neutron = 1 up quark + 2 down quarks
```
Particle   Charge    Mass

Proton       +1     1          Composed of 2 up quarks, 1 down quark,  and gluons
Neutron       0     1.0012     Composed of 1 up quark,  2 down quarks, and gluons
Electron     -1      .000544
Up quark    +2/3     .0024
Down quark  -1/3     .0048
Photon        0     0          Carries the electromagnetic force and binds electrons to the nucleus
Gluon         0     0          Carries the strong force and binds quarks, protons, and neutrons
```
Charge and mass are relative to the proton.

All of these particles are stable except for the neutron, which has a half life of 886 seconds.

```Proton charge  =  1.6022 Coulombs
Proton mass    =  1.673⋅10-27 kg
Electron mass  =  9.11⋅10-31 kg
Hydrogen mass  =  Proton mass + Electron mass  =  1.6739⋅10-27 kg
```

Isotopes

Isotopes of hydrogen

An element has a fixed number of protons and a variable number of neutrons. Each neutron number corresponds to a different isotope. Naturally-occuring elements tend to be a mix of isotopes.

```Isotope   Protons   Neutrons   Natural fraction

Hydrogen-1    1        0        .9998
Hydrogen-2    1        1        .0002

Helium-3      2        1        .000002
Helium-4      2        2        .999998

Lithium-6     3        3        .05
Lithium-7     3        4        .95

Beryllium-9   4        5        1

Boron-10      5        5        .20
Boron-11      5        6        .80

Carbon-12     6        6        .989
Carbon-13     6        7        .011
```
Teaching simulation for isotopes at phet.colorado.edu

Beta and gamma rays are harmful and alpha particles are harmless.
Beta decay

```Alpha particle  =  Helium nucleus  =  2 Protons and 2 Neutrons
Beta particle   =  Electron
Gamma ray       =  Photon

Alpha decay:   Uranium-235  ->  Thorium-231  +  Alpha
Beta decay:    Neutron      ->  Proton       +  Electron  +  Antineutrino     (From the point of view of nuclei)
Beta decay:    Down quark   ->  Up quark     +  Electron  +  Antineutrino     (From the point of view of quarks)
```
Beta decay is an example of the "weak force".

Teaching simulation for beta decay

Half life

For a radioactive material,

```Time                           =  T
Half life                      =  Th
Original mass                  =  M
Mass remaining after time "T"  =  m  =  M exp(-T/Th)
```

Suppose an element has a half life of 2 years.

```Time    Mass of element remaining (kg)

0            1
2           1/2
4           1/4
6           1/8
8           1/16
```

Weak force (beta decay)

The weak force can convert a neutron into a proton, ejecting a high-energy electron.

```From the point of view of nucleons:     Neutron     ->  Proton   + electron + antineutrino

From the point of view of quarks:       Down quark  ->  Up quark + electron + antineutrino
```
Teaching simulation for beta decay

Energy

The unit of energy used for atoms, nuclei, and particle is the "electron Volt", which is the energy gained by an electron upon descending a potential of 1 Volt.

```Electron Volt (eV)  =  1  eV  =  1.602e-19 Joules
Kilo electron Volt  =  1 keV  =  103 eV
Mega electron Volt  =  1 MeV  =  106 eV
Giga electron Vlt   =  1 GeV  =  109 eV
```

Fusion

Fusion of hydrogen into helium in the sun

```Proton + Proton  ->  Deuterium + Positron + Neutrino
```
Hydrogen fusion requires a temperature of at least 4 million Kelvin, which requires an object with at least 0.08 solar masses. This is the minimum mass to be a star. The reactions in the fusion of hydrogen to helium are:
```P    + P    -->  D    +  Positron + Neutrino +   .42 MeV
P    + D    -->  He3  +  Photon              +  5.49 MeV
He3  + He3  -->  He4  +  P   +  P            + 12.86 MeV
```

Helium fusion

As the core of a star star runs out of hydrogen it contracts and heats, and helium fusion begins when the temperature reaches 10 million Kelvin.

Heavy element fusion

A heavy star continues to fuse elements until it reaches Iron-56. Beyond this, fusion absorbs energy rather than releasing it, triggering a runaway core collapse that fuses elements up to Uranium. If the star explodes as a supernova then these elements are ejected into interstellar space.

Stars

```Star type    Mass   Luminosity    Color   Temp   Lifetime   Death      Remnant       Size of      Output
(solar   (solar             (Kelvin) (billions                           remnant
masses) luminosities)                 of years)

Brown Dwarf  <0.08                        1000  immortal
Red Dwarf     0.1         .0001   red     2000   1000      red giant   white dwarf  Earth-size
The Sun       1          1        white   5500     10      red giant   white dwarf  Earth-size    light elements
Blue star     10     10000        blue   10000      0.01   supernova   neutron star Manhattan     heavy elements
Blue giant    20    100000        blue   20000      0.01   supernova   black hole   Central Park  heavy elements
```
Fate of stars, with mass in solar masses:
```       Mass <   9   ->  End as red giants and then turn white dwarf.
9 <  Mass         ->  End as supernova
9 <  Mass <  20   ->  Remnant is a neutron star.
20 <  Mass         ->  Remnant is a black hole.
130 <  Mass < 250   ->  Pair-instability supernova (if the star has low metallicity)
250 <  Mass         ->  Photodisintegration supernova, producing a black hole and relativistic jets.
```

Fission

A neutron triggers the fission of Uranium-235 and plutonium-239, releasing energy and more neutrons.

Chain reaction

Fizzle

Fission releases neutrons that trigger more fission.

Chain reaction simulation

Critical mass

Two pieces of uranium, each with less than a critical mass, are brought together in a cannon barrel.

If the uranium is brought together too slowly, the bomb fizzles.

Plutonium fission

Plutonium is more difficult to detonate than uranium. Plutonium detonation requires a spherical implosion.

Nuclear isotopes relevant to fission energy

Abundance of elements in the sun, indicated by dot size

Blue elements are unstable with a half life much less than the age of the solar system.

The only elements heavier than Bismuth that can be found on the Earth are Thorium and Uranium, and these are the only elements that can be tapped for fission energy.

Natural Thorium is 100% Thorium-232
Natural Uranium is .72% Uranium-235 and 99.3% Uranium-238.
Plutonium doesn't exist in nature.

```
Protons  Neutrons  Halflife   Critical   Isotope
(10^6 yr)  mass (kg)  fraction

Thorium-232    90    142      14000          -       1.00     Absorbs neutron -> U-233
Uranium-233    92    141           .160     16        -       Fission chain reaction
Uranium-235    92    143        700         52        .0072   Fission chain reaction
Uranium-238    92    146       4500          -        .9927   Absorbs neutron -> Pu-239
Plutonium-238  94    144           .000088   -        -       Produces power from radioactive heat
Plutonium-239  94    145           .020     10        -       Fission chain reaction
```
The elements that can be used for fission energy are the ones with a critical mass. These are Uranium-233, Uranium-235, and Plutonium-239. Uranium-233 and Plutonium-239 can be created in a breeder reactor.
```Thorium-232  +  Neutron  ->  Uranium-233
Uranium-238  +  Neutron  ->  Plutonium-239
```
The "Fission" simulation at phet.colorado.edu illustrates the concept of a chain reaction.

Natural uranium is composed of .7% Uranium-235 and the rest is Uranium-238. Uranium-235 can be separated from U-238 using centrifuges, calutrons, or gas diffusion chambers. Uranium-235 is easy to detonate. A cannon and gunpowder gets it done.

Plutonium-239 is difficult to detonate, requiring a perfect spherical implosion. This technology is beyond the reach of most rogue states.

Uranium-233 cannot be used for a bomb and is hence not a proliferation risk.

Plutonium-238 emits alpha particles, which can power a radioisotope thermoelectric generator (RTG). RTGs based on Plutonium-238 generate 540 Watts/kg and are used to power spacecraft.

Teaching simulation for nuclear isotopes

Generating fission fuel in a breeder reactor

Creating Plutonium-239 and Uranium-233:

```Uranium-238 + Neutron  ->  Plutonium-239
Thorium-232 + Neutron  ->  Uranium-233

Detail:

Uranium-238 + Neutron  ->  Uranium-239
Uranium-239            ->  Neptunium-239 + Electron + Antineutrino    Halflife = 23 mins
Neptunium-239          ->  Plutonium-239 + Electron + Antineutrino    Halflife = 2.4 days

Thorium-232 + Neutron  ->  Thorium-233
Thorium-233            ->  Protactinium-233 + Electron + Antineutrino   Halflife = 22 mins
Protactinium-233       ->  Uranium-233      + Electron + Antineutrino   Halflife =
```

Nuclear fusion bombs

A nuclear fusion bomb contains deuterium and lithium-6 and the reaction is catalyzed by a neutron.

```N + Li6  ->  He4 + T +  4.87 MeV
T + D    ->  He4 + N + 17.56 MeV

Total energy released  =  22.43 MeV
Nucleons               = 8
Energy / Nucleon       = 22.434 / 8  =  2.80
```

Energy
```1 ton of TNT                  4*10^9  Joules
1 ton of gasoline             4*10^10 Joules
North Korea fission device    0.5 kilotons TNT
10 kg uranium fission bomb    10  kilotons TNT
10 kg hydrogen fusion bomb    10  megatons TNT
Tunguska asteroid strike      15  megatons TNT        50 meter asteroid
Chixulub dinosaur extinction  100 trillion tons TNT   10 km asteroid
```

History of nuclear physics
```1885       Rontgen discovers X-rays
1899       Rutherford discovers alpha and beta rays
1903       Rutherford discovers gamma rays
1905       E=mc^2. Matter is equivalent to energy
1909       Nucleus discovered by the Rutherford scattering experiment
1932       Neutron discovered
1933       Nuclear fission chain reaction envisioned by Szilard
1934       Fermi bombards uranium with neutrons and creates Plutonium. First
successful example of alchemy
1938       Fission discovered by Hahn and Meitner
1938       Bohr delivers news of fission to Princeton and Columbia
1939       Fermi constructs the first nuclear reactor in the basement of Columbia
1939       Szilard and Einstein write a letter to President Roosevelt advising
him to consider nuclear fission
1942       Manhattan project started
1942-1945  German nuclear bomb project goes nowhere
1945       Two nuclear devices deployed by the United States
```

History of nuclear devices
```           Fission Fusion

U.S.A.       1945  1954
Germany                  Attempted fission in 1944 & failed
Russia       1949  1953
Britain      1952  1957
France       1960  1968
China        1964  1967
India        1974        Uranium
Israel       1979   ?    Undeclared. Has both fission and fusion weapons
South Africa 1980        Dismantled in 1991
Iran         1981        Osirak reactor to create Plutonium. Reactor destroyed by Israel
Pakistan     1990        Centrifuge enrichment of Uranium. Tested in 1998
Built centrifuges from stolen designs
Iraq         1993        Magnetic enrichment of Uranium. Dismantled after Gulf War 1
Iraq         2003        Alleged by the United States. Proved to be untrue.
North Korea  2006        Obtained plutonium from a nuclear reactor. Detonation test fizzled
Also acquired centrifuges from Pakistan
Also attempting to purify Uranium with centrifuges
Syria        2007        Nuclear reactor destroyed by Israel
Iran         2009?       Attempting centrifuge enrichment of Uranium.
Libya         --         Attempted centrifuge enrichment of Uranium.  Dismantled before completion.
Cooperated in the investigation that identified
Pakistan as the proliferator of Centrifuge designs.
Libya        2010        Squabbling over nuclear material
Libya        2011        Civil war
```

Fusion power

A tokamak fusion reactor uses magnetic fields to confine a hot plasma so that fusion can occur in the plasma.

Deuterium + Tritium fusion

The fusion reaction that occurs at the lowest temperature and has the highest reaction rate is

```Deuterium  +  Tritium  ->  Helium-4  +  Neutron  +  17.590 MeV
```
but the neutrons it produces are a nuisance to the reactor.

A potential fix is to have "liquid walls" absorb the neutrons (imagine a waterfall of neutron-absorbing liquid lithium cascading down the walls of the reactor).

Fuel
Black: Carbon    White: Hydrogen    Red: Oxygen

Methane (Natural gas)
Ethane
Propane
Butane (Lighter fluid)
Octane (gasoline)
Dodecane (Kerosene)

Palmitic acid (fat)
Ethanol (alcohol)

Glucose (sugar)
Fructose (sugar)
Galactose (sugar)
Lactose = Glucose + Galactose
Starch (sugar chain)
Leucine (amino acid)

Phosphocreatine
Nitrocellulose (smokeless powder)
TNT
HMX (plastic explosive)

Lignin (wood)
Coal

Medival-style black powder
Modern smokeless powder
Capacitor
Lithium-ion battery
Nuclear battery (radioactive plutonium-238)
Nuclear fission
Nuclear fusion
Antimatter

Vehicle power

The energy sources that can be used by vehicles are:

```              Energy/Mass   Power/mass   Energy/\$   Rechargeable   Charge   Maximum charging
MJoule/kg     Watt/kg     MJoule/\$                  time          cycles

Gasoline            45                   60
Battery, aluminum    4.6       130                      No
Battery, lithium      .8      1600         .010         Yes        1 hour      1000
Supercapacitor        .016    8000         .00005       Yes        Instant     Infinite
Aluminum capacitor    .010   10000         .0001        Yes        Instant     Infinite
```

Energy and power sources

```                   Energy/Mass   Power/mass
MJoule/kg     Watt/kg

Antimatter          90000000000
Fusion bomb           250000000                 Max for d+t fusion
Fission bomb           83000000                 Max for a uranium bomb
Nuclear battery, Pu238  2265000          10     88 year half life
Nuclear battery, Sr90    589000          10     29 year half life
Hydrogen ( 0 carbons)       141.8
Methane  ( 1 carbon )        55.5               Natural gas
Ethane   ( 2 carbons)        51.9
Butane   ( 4 carbons)        49.5
Octane   ( 8 carbons)        47.8
Kerosene (12 carbons)        46
Diesel   (16 carbons)        46
Oil      (36 carbons)        46
Fat      (20 carbons)        37                 9 Calories/gram
Pure carbon                  32.8
Coal                         32                 Similar to pure carbon
Ethanol                      29                 7 Calories/gram
Wood                         22
Sugar                        17                 4 Calories/gram
Protein                      17                 4 Calories/gram
Plastic explosive             8.0               HMX
Smokeless powder              5.2               Modern gunpowder
TNT                           4.7
Black powder                  2.6               Medieval gunpowder
Phosphocreatine                .137             Recharges ATP
ATP                            .057             Adenosine triphosphate
Battery, aluminum-air         4.68      130
Battery, Li-S                 1.44      670
Battery, Li-ion                .8      1600
Battery, Li-polymer            .6      4000
Battery, Alkaline              .4
Battery, Lead acid             .15      150
Lithium supercapacitor         .054   15000
Supercapacitor                 .016    8000
Aluminum capacitor             .010   10000
Spring                         .0003
```

Aerodynamic drag

Newton length

The characteristic distance a ball travels before air slows it down is the "Newton length". This distance can be estimated by setting the mass of the ball is equal to the mass of the air the ball passes through.

```Mass of a soccer ball              =  M  =  .437  kg
Ball radius                        =  R  =  .110  meters
Ball cross-sectional area          =  A  =  .038  meters2
Ball density                       =  D  =  78.4  kg/meters3
Air density                        =  d  =   1.22 kg/meter3   (Air at sea level)
Ball initial velocity              =  V
Newton length                      =  L
Mass of air the ball passes through=  m  =  A L d

m  =  M

L  =  M / (A d)  =  (4/3) R D / d  =  9.6 meters
```
The depth of the penalty box is 16.45 meters (18 yards). Any shot taken outside the penalty box slows down substantially before reaching the goal.

Newton was also the first to observe the "Magnus effect", where spin causes a ball to curve.

Balls

The orange boxes depict the size of the court and the Newton length is the distance from the bottom of the court to the ball. Ball sizes are magnified by a factor of 20 relative to the court sizes.

```          Diameter  Mass  Drag  Shot   Drag/  Density   Ball   Max    Spin
(mm)    (g)   (m)   (m)    Shot   (g/cm3)   speed  speed  (1/s)
(m/s)  (m/s)
Ping pong    40      2.7   1.8    2.74    .64   .081     20    31.2    80
Squash       40     24    15.6    9.75   1.60   .716
Golf         43     46    25.9  200       .13  1.10      80    94.3   296
Badminton    54      5.1   1.8   13.4     .14   .062
Racquetball  57     40    12.8   12.22   1.0    .413
Billiards    59    163    48.7    2.7   18     1.52
Tennis       67     58    13.4   23.77    .56   .368     50    73.2   119
Baseball     74.5  146    27.3   19.4    1.4    .675     40    46.9    86
Whiffle      76     45     8.1                  .196
Football    178    420    13.8   20       .67   .142     20    26.8    18
Rugby       191    435    12.4   20       .62   .119
Bowling     217   7260   160     18.29   8.8   1.36
Soccer      220    432     9.3   16.5     .56   .078     40    59      29
Basketball  239    624    11.4    7.24   1.57   .087
Cannonball  220  14000   945   1000       .94  7.9
```
"Drag" is the Newton drag length and "Shot" is the typical distance of a shot, unless otherwise specified. "Density" is the density of the ball.

For a billiard ball, rolling friction is greater than air drag.

A bowling pin is 38 cm tall, 12 cm wide, and has a mass of 1.58 kg. A bowling ball has to be sufficiently massive to have a chance of knocking over 10 pins.

```Mass of 10 bowling pins  /  Mass of bowling ball  =  2.18
```

Bullet distance

To estimate the distance a bullet travels before being slowed by drag,

```Air density              =  Dair    =   .012 g/cm3
Water density            =  Dwater  =  1.0   g/cm3
Bullet density           =  Dbullet = 11.3   g/cm3
Bullet length            =  Lbullet =  2.0   cm
Bullet distance in water =  Lwater  ≈  Lbullet Dbullet / Dwater ≈ 23  cm
Bullet distance in air   =  Lair    ≈  Lbullet Dbullet / Dair  ≈ 185 meters
```

Density

```         g/cm3                                    g/cm3

Air        .00122  (Sea level)           Silver     10.5
Wood       .7 ± .5                       Lead       11.3
Water     1.00                           Uranium    19.1
Magnesium 1.74                           Tungsten   19.2
Aluminum  2.70                           Gold       19.3
Rock      2.6 ± .3                       Osmium     22.6   (Densest element)
Titanium  4.51
Steel     7.9
Copper    9.0
```

Kinetic energy penetrator

Massive Ordnance Penetrator
Bunker buster

```                         Cartridge  Projectile  Length  Diameter  Warhead  Velocity
(kg)      (kg)       (m)     (m)       (kg)     (m/s)

Massive Ordnance Penetrator   -       13608     6.2     .8        2404
PGU-14, armor piercing       .694     .395       .173   .030               1013
PGU-13, explosive            .681     .378       .173   .030               1020
```
The GAU Avenger armor-piercing shell contains .30 kg of depleted uranium.

The massive ordnamce penetrator typically penetrates 61 meters of Earth.

The PGU-13 and PGU-14 are used by the A-10 Warthog cannon.

The composition of natural uranium is .72% uranium-235 and the rest is uranium-238. Depleted uranium has less than .3% of uranium-235.

Drag force

The drag force on an object moving through a fluid is

```Velocity             =  V
Fluid density        =  D  =  1.22 kg/m2   (Air at sea level)
Cross-sectional area =  A
Drag coefficient     =  C  =  1            (typical value)
Drag force           =  F  =  ½ C D A V2
Drag power           =  P  =  ½ C D A V3  =  F V
Terminal velocity    =  Vt
```
"Terminal velocity" occurs when the drag force equals the gravitational force.
```M g  =  ½ C D A Vt2
```
Suppose we want to estimate the parachute size required for a soft landing. Let a "soft landing" be the speed reached if you jump from a height of 2 meters, which is Vt = 6 m/s. If a skydiver has a mass of 100 kg then the area of the parachute required for this velocity is 46 meters2, which corresponds to a parachute radius of 3.8 meters.
Drag coefficient

```               Drag coefficient

Bicycle car         .076        Velomobile
Tesla Model 3       .21         2017
Toyota Prius        .24         2016
Bullet              .30
Typical car         .33         Cars range from 1/4 to 1/2
Sphere              .47
Typical truck       .6
Formula-1 car       .9          The drag coeffient is high to give it downforce
Bicycle + rider    1.0
Skier              1.0
Wire               1.2
```

Fastest manned aircraft
```                  Mach

X-15              6.7      Rocket
Blackbird SR-71   3.5
X-2 Starbuster    3.2
MiG-25 Foxbat     2.83
XB-70 Valkyrie    3.0
MiG-31 Foxhound   2.83
F-15 Eagle        2.5
Aardvark F-111    2.5      Bomber
Sukhoi SU-27      2.35
F-22 Raptor       2.25     Fastest stealth aircraft
```

Drag power

Cycling power

```Fluid density    =  D
Cross section    =  A
Drag coef        =  C
Drag force       =  F  =  ½ C A D V2
Drag power       =  P  =  ½ C A D V3  =  K D V3  =  F V
Drag parameter   =  K  =  ½ C A

Speed   Density   Drag force   Drag power    Drag
(m/s)   (kg/m3)      (kN)       (kWatt)    parameter

Bike                 10       1.22      .035        .305   .50
Bike                 18       1.22      .103       1.78    .50
Bike, speed record   22.9     1.22      .160       3.66    .50
Bike, streamlined    38.7     1.22      .095       3.66    .104
Porche 911           94.4     1.22     7.00      661      1.29
LaFerrari            96.9     1.22     7.31      708      1.28
Lamborghini SV       97.2     1.22     5.75      559      1.00
Skydive, min speed   40       1.22      .75       30       .77        75 kg
Skydive, max speed  124       1.22      .75      101       .087       75 kg
Airbus A380, max    320        .28  1360      435200     94.9
F-22 Raptor         740        .084  312      231000      6.8
SR-71 Blackbird    1100        .038  302      332000      6.6
Sub, human power      4.1  1000         .434       1.78    .052
Blue Whale           13.9  1000      270        3750      2.8         150 tons, 25 Watts/kg
Virginia nuclear sub 17.4  1000     1724       30000     11.4
```
The drag coefficient is an assumption and the area is inferred from the drag coefficient.

For the skydiver, the minimum speed is for a maximum cross section (spread eagled) and the maximum speed is for a minimum cross section (dive).

Wiki: Energy efficiency in transportation

Altitude

Airplanes fly at high altitude where the air is thin.

```                Altitude   Air density
(km)     (kg/m3)

Sea level          0       1.22
Denver (1 mile)    1.6      .85
Mount Everest      9.0      .45
Airbus A380       13.1      .25    Commercial airplane cruising altitude
F-22 Raptor       19.8      .084
SR-71 Blackbird   25.9      .038
```

Speed records

```                       m/s     Mach

Swim                    2.39
Boat, human power       5.14
Aircraft, human power  12.3
Run                    12.4
Boat, wind power       18.2
Bike                   22.9
Car, solar power       24.7
Bike, streamlined      38.7
Land animal            33               Cheetah
Bird, level flight     45               White-throated needletail
Aircraft, electric     69
Helicopter            111       .33
Train, wheels         160       .54
Train, maglev         168       .57
Aircraft, propeller   242       .82
Rocket sled, manned   282       .96
Aircraft, manned      981      3.33
Rocket plane, manned 2016      6.83
Rocket sled          2868      9.7
Scramjet             5901     20
```
Mach 1 = 295 m/s at high altitude.

Drag coefficient and Mach number

Commercial airplanes fly at Mach .9 because the drag coefficient increases sharply at Mach 1.

Turbulence and Reynolds number

The drag coefficient depends on speed.

```Object length    =  L
Velocity         =  V
Fluid viscosity  =  Q                  (Pascal seconds)
=  1.8⋅10-5 for air
=  1.0⋅10-3 for water
Reynolds number  =  R   =  V L / Q      (A measure of the turbulent intensity)
```
The drag coefficient of a sphere as a function of Reynolds number is:

Golf balls have dimples to generate turbulence in the airflow, which increases the Reynolds number and decrease the drag coefficient.

Drag coefficient and Reynolds number
```Reynolds  Soccer  Golf   Baseball   Tennis
number
40000   .49    .48      .49       .6
45000   .50    .35      .50
50000   .50    .30      .50
60000   .50    .24      .50
90000   .50    .25      .50
110000   .50    .25      .32
240000   .49    .26
300000   .46
330000   .39
350000   .20
375000   .09
400000   .07
500000   .07
800000   .10
1000000   .12             .35
2000000   .15
4000000   .18    .30
```
Data
Drafting

If the cyclists are in single file then the lead rider has to use more power than the following riders. Cyclists take turns occupying the lead.

A "slingshot pass" is enabled by drafting. The trailing car drops back by a few lengths and then accelerates. The fact that he is in the leading car's slipstream means he has a higher top speed. As the trailing car approaches the lead car it moves the side and passes.

Drag differential equation

For an object experiencing drag,

```Drag coefficient  =  C
Velocity          =  V
Fluid density     =  D
Cross section     =  A
Mass              =  M
Drag number       =  Z  =  ½ C D A / M
Drag acceleration =  A  =  -Z V2
Initial position  =  X0 =  0
Initial velocity  =  V0
Time              =  T
```
The drag differential equation and its solution are
```A  =  -Z V2
V  =  V0 / (V0 Z T + 1)
X  =  ln(V0 Z T + 1) / Z
```

Spin force (Magnus force)

Topspin

```1672  Newton is the first to note the Magnus effect while observing tennis players
at Cambridge College.
1742  Robins, a British mathematician and ballistics researcher, explains deviations
in musket ball trajectories in terms of the Magnus effect.
1852  The German physicist Magnus describes the Magnus effect.
```
For a spinning tennis ball,
```Velocity    =  V                          =    55 m/s             Swift groundstroke
Radius      =  R                          =  .067 m
Area        =  Area                       = .0141 m2
Mass        =  M                          =  .058 kg
Spin number =  S   =  W R / V             =   .25                 Heavy topspin
Spin rate   =  W   =    V / R             =   205 Hz
Air density =  Dair                       =  1.22 kg/m3
Ball density=  Dball
Drag coef   =  Cdrag                      =    .5                 For a sphere
Spin coef   =  Cspin                      =     1                 For a sphere and for S < .25
Drag force  =  Fdrag = ½ Cdrag Dair Area V2   =  13.0 Newtons
Spin force  =  Fspin = ½ Cspin Dair Area V2 S =   6.5 Newtons
Drag accel  =  Adrag                      =   224 m/s2
Spin accel  =  Aspin                      =   112 m/s2
Gravity     =  Fgrav = M g
```
For a rolling ball the spin number is S=1.

If the spin force equals the gravity force (Fspin = Fgrav),

```V2 S C R-1 Dair/Dball = .0383
```

Drag force

The drag force on an object moving through a fluid is

```Velocity             =  V
Fluid density        =  D  =  1.22 kg/m2   (Air at sea level)
Cross-sectional area =  A
Drag coefficient     =  C
Drag force           =  F  =  ½ C A D V2
Drag power           =  P  =  ½ C A D V3  =  F V
Drag parameter       =  K  =  C A
```
"Terminal velocity" occurs when the drag force equals the gravitational force.
```M g  =  ½ C D A V2
```
Suppose we want to estimate the parachute size required for a soft landing. Let a "soft landing" be the speed reached if you jump from a height of 2 meters, which is Vt = 6 m/s. If a skydiver has a mass of 100 kg then the area of the parachute required for this velocity is 46 meters2, which corresponds to a parachute radius of 3.8 meters.
Drag coefficient

```               Drag coefficient

Bicycle car         .076        Velomobile
Tesla Model 3       .21         2017
Toyota Prius        .24         2016
Bullet              .30
Typical car         .33         Cars range from 1/4 to 1/2
Sphere              .47
Typical truck       .6
Formula-1 car       .9          The drag coeffient is high to give it downforce
Bicycle + rider    1.0
Skier              1.0
Wire               1.2
```

Spin force (Magnus force)

Topspin

```1672  Newton is the first to note the Magnus effect while observing tennis players
at Cambridge College.
1742  Robins, a British mathematician and ballistics researcher, explains deviations
in musket ball trajectories in terms of the Magnus effect.
1852  The German physicist Magnus describes the Magnus effect.
```
For a spinning tennis ball,
```Velocity    =  V                          =    55 m/s             Swift groundstroke
Radius      =  R                          =  .067 m
Area        =  Area                       = .0141 m2
Mass        =  M                          =  .058 kg
Spin number =  S   =  W R / V             =   .25                 Heavy topspin
Spin rate   =  W   =    V / R             =   205 Hz
Air density =  Dair                       =  1.22 kg/m3
Ball density=  Dball
Drag coef   =  Cdrag                      =    .5                 For a sphere
Spin coef   =  Cspin                      =     1                 For a sphere and for S < .25
Drag force  =  Fdrag = ½ Cdrag Dair Area V2   =  13.0 Newtons
Spin force  =  Fspin = ½ Cspin Dair Area V2 S =   6.5 Newtons
Drag accel  =  Adrag                      =   224 m/s2
Spin accel  =  Aspin                      =   112 m/s2
Gravity     =  Fgrav = M g
```
For a rolling ball the spin number is S=1.

If the spin force equals the gravity force (Fspin = Fgrav),

```V2 S C R-1 Dair/Dball = .0383
```

Rolling drag

```Force of the wheel normal to ground  =  Fnormal
Rolling friction coefficient         =  Croll
Rolling friction force               =  Froll  =  Croll Fnormal

```
Typical car tires have a rolling drag coefficient of .01 and specialized tires can achieve lower values.
```                             Croll

Railroad                      .00035     Steel wheels on steel rails
Steel ball bearings on steel  .00125
Racing bicycle tires          .0025      8 bars of pressure
Typical bicycle tires         .004
18-wheeler truck tires        .005
Best car tires                .0075
Typical car tires             .01
Car tires on sand             .3
```

Rolling friction coefficient
```Wheel diameter          =  D
Wheel sinkage depth     =  Z
Rolling coefficient     =  Croll  ≈  (Z/D)½
```

Drag speed

For a typical car,

```Car mass                   =  M           = 1200 kg
Gravity constant           =  g           =  9.8 m/s2
Tire rolling drag coeff    =  Cr          =.0075
Rolling drag force         =  Fr = Cr M g =   88 Newtons

Air drag coefficient       =  Ca          =  .25
Air density                =  D           = 1.22 kg/meter3
Air drag cross-section     =  A           =  2.0 m2
Car velocity               =  V           =   17 m/s      (City speed. 38 mph)
Air drag force             =  Fa = ½CaADV2 =  88 Newtons

Total drag force           =  F  = Fr + Fa = 176 Newtons
Drag speed                 =  Vd           =  17 m/s     Speed for which air drag equals rolling drag
Car electrical efficiency  =  Q            = .80
Battery energy             =  E            =  60 MJoules
Work done from drag        =  EQ = F X     =  Cr M g [1 + (V/Vd)2] X
Range                      =  X  = EQ/(CrMg)/[1+(V/Vd)2] =  272 km
```
The range is determined by equating the work from drag with the energy delivered by the battery.   E Q = F X.

The drag speed Vd is determined by setting Fr = Fa.

```Drag speed  =  Vd  =  [Cr M g / (½ Ca D A)]½  =  4.01 [Cr M /(Ca A)]½  =  17.0 meters/second
```

Speed records

```                       m/s     Mach

Swim                    2.39
Boat, human power       5.14
Aircraft, human power  12.3
Run                    12.4
Boat, wind power       18.2
Bike                   22.9
Car, solar power       24.7
Bike, streamlined      38.7
Land animal            33               Cheetah
Bird, level flight     45               White-throated needletail
Aircraft, electric     69
Helicopter            111       .33
Train, wheels         160       .54
Train, maglev         168       .57
Aircraft, propeller   242       .82
Rocket sled, manned   282       .96
Aircraft, manned      981      3.33
Rocket plane, manned 2016      6.83
Rocket sled          2868      9.7
Scramjet             5901     20
```
Mach 1 = 295 m/s at high altitude.
Fastest manned aircraft
```                  Mach

X-15              6.7      Rocket
Blackbird SR-71   3.5
X-2 Starbuster    3.2
MiG-25 Foxbat     2.83
XB-70 Valkyrie    3.0
MiG-31 Foxhound   2.83
F-15 Eagle        2.5
Aardvark F-111    2.5      Bomber
Sukhoi SU-27      2.35
F-22 Raptor       2.25     Fastest stealth aircraft
```

Transport cost

Leitras velomobile
Loremo
Edison 2
BMW i8

The Saab 900, last of the boxy cars
Lamborghini Diablo
Ford Escape Hybrid
Hummer H2

```              Speed    Power    Force   Force   Force   Mass   Drag    Drag   Area  Drag  Roll  Year  100kph
(total) (fluid)  (roll)        (data)  (specs)       coef  coef         time
m/s     kWatt     kN      kN      kN     ton     m2      m2     m2                        s

eSkate            5.3       .11    .021    .013   .008    .08    .76                1.0   .01
eScooter Zoomair  7.2       .25    .035    .027   .008    .08    .85                1.0   .01
Bike             10         .30    .035    .030   .005    .10    .49                1.0   .005
Bike             18        1.78    .103    .098   .005    .10    .50                1.0   .005
eBike 250 Watt    8.9       .25    .028    .023   .005    .10    .48                1.0   .005
eBike 750 Watt   10.6       .75    .071    .066   .005    .10    .96                1.0   .005
eBike 1 kWatt    12.5      1.0     .080    .075   .005    .10    .79                1.0   .005
eBike 1.5 kWatt  15.3      1.5     .098    .093   .005    .10    .65                1.0   .005
eBike 3 kWatt    16.7      3.0     .180    .175   .005    .10   1.03                1.0   .005
eBike Stealth H  22.2      5.2     .234    .228   .006    .12    .76                1.0   .005
eBike Wolverine  29.2      7.0     .240    .234   .006    .12    .45                1.0   .005
Bike, record     22.9      3.66    .160    .155   .005    .10    .48                1.0   .005
Bike, steamline  38.7      3.66    .095    .090   .005    .10    .099                .11  .005

Loremo           27.8     45      1.62    1.58    .035    .47   3.39   .25    1.25   .20  .0075   2009
Mitsubishi MiEV  36.1     47      1.30    1.22    .081   1.08   1.53                 .35  .0075   2011
Aptera 2         38.1     82      2.15    2.09    .062    .82   2.36   .19    1.27   .15  .0075   2011
Nissan Leaf SL   41.7     80      1.92    1.81    .114   1.52   1.71   .72    2.50   .29  .0075   2012  10.1
Volkswagen XL1   43.9*    55      1.25    1.19    .060    .80   1.01   .28    1.47   .19  .0075   2013  11.9
Chevrolet Volt   45.3    210      4.64    4.52    .121   1.61   3.61   .62    2.21   .28  .0075   2014   7.3
Saab 900         58.3    137      2.35    2.25    .100   1.34   1.09   .66    1.94   .34  .0075   1995   7.7
Tesla S P85 249+ 69.2*   568      8.21    8.06    .150   2.00   2.76   .58    2.40   .24  .0075   2012   3.0
BMW i8           69.4*   260      3.75    3.63    .116   1.54   1.24   .55    2.11   .26  .0075   2015   4.4
Nissan GTR       87.2    357      4.09    3.96    .130   1.74    .85   .56    2.09   .27  .0075   2008   3.4
Lamborghini Dia  90.3    362      4.01    3.89    .118   1.58    .78   .57    1.85   .31  .0075   1995
Porsche 918      94.4    661      7.00    6.88    .124   1.66   1.27                 .29  .0075
LaFerrari        96.9    708      7.31    7.19    .119   1.58   1.26                      .0075
Lamborghini SV   97.2    559      5.75    5.62    .130   1.73    .98                      .0075
Bugatti Veyron  119.7    883      7.38    7.24    .142   1.89    .83   .74                .0075   2005
Hummer H2                242                      .218   2.90         2.46    4.32   .57  .0075   2003
Formula 1                                         .053    .702                       .9   .0075   2017

Bus (2 decks)            138                            12.6                              .005    2012
Subway (R160)    24.7    448                            38.6                              .0004   2006

Airbus A380     320   435200   1360    1360                    21.8
F-22 Raptor     740   231000    312     312                      .93
Blackbird SR71 1100   332000    302     302                      .41

Skydive, min     40       30       .75     .75    0       .075   .77                1.0   0
Skydive, max    124      101       .75     .75    0       .075   .080               1.0   0

Sub, human power  4.1      1.78    .434                          .051
Blue Whale       13.9   3750    270                             2.74
Sub, nuke        17.4  30000   1724                            11.2                            Virginia Class

*:               The top speed is electronically limited
Drag (data)      Drag parameter obtained from the power and top speed
Data (specs)     Drag parameter from Wikipedia
Force (total)    Total drag force  =  Fluid drag force +  Roll drag force
Force (fluid)    Fluid drag force
Force (roll)     Roll drag force
Area             Cross section from Wikipedia
Drag coef        Drag coefficient from Wikipedia
Roll coef        Roll coefficient. Assume .0075 for cars and .005 for bikes.
100kph time      Time to accelerate to 100 kph
```
For the skydiver, the minimum speed is for a maximum cross section (spread eagled) and the maximum speed is for a minimum cross section (dive).

Cycling power

Wiki: Energy efficiency in transportation

Drafting

If the cyclists are in single file then the lead rider has to use more power than the following riders. Cyclists take turns occupying the lead.

A "slingshot pass" is enabled by drafting. The trailing car drops back by a few lengths and then accelerates. The fact that he is in the leading car's slipstream means he has a higher top speed. As the trailing car approaches the lead car it moves the side and passes.

Fast cars
```           0-100kph  400m  400m    Top   Power  Mass   Top
(s)    (s)   speed  speed  (kw)   (kg)  speed
(kph)  (kph)               (m/s)

Porche 918      2.2    9.8  233    340    661   1704   94.4
LaFerrari       2.4    9.7  240    349    708   1255   96.9
Bugatti Veyron  2.5    9.7  224    431    883   1888  119.7
Tesla S         2.6   10.9  198    249    568   2000   69.2
Lamborghini SV  2.6   10.4  218    350    559   1769   97.2
Porche 997 S    2.7   10.9  205    315    390   1570   87.5
```

Formula 1

If everything seems under control, you're just not going fast enough. -- Mario Andretti

I will always be puzzled by the human predilection for piloting vehicles at unsafe velocities -- Data

The car

```Car minimum mass           =  702 kg        Includes the driver and not the fuel
Engine volume              =  1.6 litres    Turbocharged. 2 energy recovery systems allowed
Energy recovery max power  =  120 kWatts
Energy recovery max energy =  2 Megajoules/lap
Engine typical power       =  670 kWatts  =  900 horsepower
Engine cylinders           =  6
Engine max frequency       =  15000 RPM
Engine intake              =  450 litres/second
Fuel consumption           =  .75 litres/km
Fuel maximum               =  150 litres
Forward gears              =  8
Reverse gears              =  1
Gear shift time            =  .05 seconds
Lateral accelertion        =  6 g's
Formula1 1g downforce speed=  128 km/h       Speed for which the downforce is 1 g
Formula1 2g downforce speed=  190 km/h       Speed for which the downforce is 2 g
Indycar 1g downforce speed =  190 km/h
Rear tire max width        =  380 mm
Front tire max width       =  245 mm
Tire life                  =  300 km
Brake max temperature      = 1000 Celsius
Deceleration from 100 to 0 kph = 15 meters
Deceleration from 200 to 0 kph = 65 meters    (2.9 seconds)
Time to 100 kph            = 2.4 seconds
Time to 200 kph            = 4.4 seconds
Time to 300 kph            = 8.4 seconds
Max forward acceleration   = 1.45 g
Max breaking acceleration  = 6 g
Max lateral acceleration   = 6 g
Drag at 250 kph            = 1 g
```

Budget

Timeline
```1950  Formula-1 begins. Safety precautions were nonexistent and death was considered
an acceptable risk for winning races.
1958  Constructor's championship established
1958  First race won by a rear-engine car. Within 2 years all cars had rear engines.
1966  Aerodynamic features are required to be immobile (no air brakes).
1977  First turbocharged car.
1978  The Lotus 79 is introduced, which used ground effect to accelerate air
under the body of the car, generating downforce. It was also the first
instance of computer-aided design. It was unbeatable until the introduction
of the Brabham Fancar.
1978  The Brabham "Fancar" is introduced, which used a fan to extract air from
underneath the car and enhance downforce. It won the race decisively.
The rules committee judged it legal for the rest of the season but the
team diplomatically
Wiki
1982  Active suspension introduced.
1983  Ground effect banned. The car underside must be flat.
1983  Cars with more than 4 wheels banned.
1989  Turbochargers banned.
1993  Continuously variable transmission banned before it ever appears.
1994  Electronic performance-enhancing technology banned, such as active suspension,
traction control, launch control, anti-lock breaking, and 4-wheel steering.
(4-wheel steering was never implemented)
1999  Flexible wings banned.
2001  Traction control allowed because it was unpoliceable.
2001  Beryllium alloys in chassis or engines banned.
2002  Team orders banned after Rubens Barrichello hands victory to Michael
Schumacher at final corner of the Austrian Grand Prix.
2004  Automatic transmission banned.
2007  Tuned mass damper system banned.
2008  Traction control banned. All teams must use a standard electrontrol unit.
2009  Kinetic energy recovery systems allowed.
```

Circuits

Catalunya
Suzuka
Magny Cours

Points
```Place   Points          Place   Points

1       25              6       8
2       18              7       6
3       15              8       4
4       12              9       2
5       10             10       1
```

Electric bikes

Electric bikes are easy to make. All you have to do is replace a conventional wheel with an electric wheel and attach a battery pack. Electric wheels come in kits and you can make the battery pack yourself. Example configurations for various motor powers:

```Power    Max    Range   Motor   Battery   Battery
speed           cost     cost     energy
kWatt    mph    miles    \$         \$      MJoule

.75   30      10     160       40       .5
1.5    35      20     240       60      1.2
3      45      40     570      100      1.8
6      55      80    1150      200      3.6
```
The bikes have one electric wheel and one conventional wheel except for 6 kWatt bike, which has 2 electric wheels with 3 kWatt each.

Electric wheel prices are from Amazon.com.

Electric bike speed limits
```             Speed   Power   License
mph    kWatt   required?

Connecticut    30    1.5     Yes
California     28     .75    No
Massachusetts  25     .75    Yes
Oregon         20    1.0     No
Washington     20    1.0     No
Pennsylvania   20     .75    No
Delaware       20     .75    No
Maryland       20     .5     No
DC             20     ?      No
```

Flight

Lift

```Air density            =  D
Velocity               =  V
Wing area              =  Awing
Wing drag coefficient  =  Cwing
Drag force on the wing =  Fdrag = ½ CWing Awing D V2

Cwing

F-4 Phantom   .021     (subsonic)
Cessna 310    .027
Airbus A380   .027
Boeing 747    .031
F-4 Phantom   .044     (supersonic)
```

Lift-to-drag ratio
```Flift  =  Lift force (upward)
Fdrag  =  Drag force (rearward)
Qlift  =  Lift-to-drag coefficient  =  Flift / Fdrag

Qlift

U-2            23     High-altitude spy plane
Albatross      20     Largest bird
Gossamer       20     Gossamer albatross, human-powered aircraft
Hang glider    15
Tern           12
Herring Gull   10
Airbus A380     7.5
Concorde        7.1
Boeing 747      7
Cessna 150      7
Parachute       5
Sparrow         4
Wingsuit        2.5
Flying lemur    ?     Most capable gliding mammal.  2 kg max
Flying squirrel 2.0
```

Gliding

A glider is an airplane without an engine. The more efficient the glider, the smaller the glide angle. The minimum glide angle is determined by the wing lift/drag coefficient.

```Wing lift/drag coefficient =  Qlift  =  Flift / Fdrag
Glider horizontal velocity =  Vx
Glider vertical velocity   =  Vz
Drag force                 =  Fdrag
Gravitational force        =  Fgrav
Lift force                 =  Flift  =  Fgrav
Drag power                 =  Pdrag  =  Fdrag Vx
Power from gravit          =  Pgrav  =  Fgrav Vz
```
If the glider descends at constant velocity,
```Pdrag  =  Pgrav
```
The goal of a glider is to maximize the glide ratio Vx / Vz.
```Vx / Vz  =  (Pdrag / Fdrag)  /  (Pgrav / Fgrav)
=  Fgrav / Fdrag
=  Qlift
```
The glide ratio is equal to the lift coefficient Qlift.

Level flight

```D    =  Air density
Awing =  Wing area
Cwing =  Wing drag coefficient
Fdrag =  Drag force on the wing   =  ½ Cwing D Awing V^2
Qwing =  Wing lift coefficient    =  Flift / Fdrag
Flift =  Lift force from the wing =  Qwing Fdrag
M    =  Aircraft mass
Feng  =  Engine force
Fgrav =  Gravity force            =  M g
Pdrag =  Drag power               =  Fdrag V  =  ½ Cwing D Awing V3
V    =  Cruising speed
Agility= Power-to-weight ratio    =  Pdrag / M  =  V g / Q      (derived below)
```
For flight at constant velocity,
```Feng  =  Fdrag              Horizontal force balance

Flift =  Fgrav              Vertical force balance

Agility =  Pdrag   / M
=  Fdrag V / M
=  Flift V / M / Q
=  M g  V / M / Q
=  V g / Q
```
We can use this equation to solve for the minimum agility required to fly.
```Pdrag  =  M g V / Q  =  ½ Cwing D Awing V3

Agility  =  g3/2 M½ Q-3/2 (½ C D A)-½
```
If we assume that mass scales as size cubed and wing area scales as size squared, then
```Awing   ~  M2/3

Agility ~  g3/2 M1/6 Q-3/2 C-½ D-½
```

Aircraft data

Cessna 150
Boeing 747
Airbus 380

SR-71 Blackbird
U-2 spy plane
Concorde
Concorde temperature at Mach 2

```         Vcruise  Vmax  Mass  Takeoff  Ceiling  Density  Force  Wing   Len   Wing   Range
m/s   m/s   ton    ton      km      kg/m3     kN     m2     m     m      km

Cessna 150    42   56     .60     .73  4.3   .79      1.34   15     7.3  10.1    778
Boeing 747   254  274  178.1   377.8  11.0   .36   1128     525    70.6  64.4  14200
Boeing 787-9 251  262  128.9   254.0  13.1   .26    640     360.5  62.8  60.1  14140
Airbus A380  243  262  276.8   575    13.1   .26   1360     845    72.2  79.8  15200
Concorde     599  605   78.7   190.5  18.3   .115   560     358.2  61.7  25.6   7223
F-22 Raptor  544  740   19.7    38.0  19.8   .091   312      78.0  18.9  13.6   2960
U-2          192  224    6.49   18.1  21.3   .071    84.5    92.9  19.2  31.4  10308
SR-71        954  983   30.6    78.0  25.9   .034   302     170    32.7  16.9   5400
```
Mach 1 = 298 m/s.

Altitude

Commercial airplanes fly at high altitude where the air is thin. The thinner the air, the less the drag force and the less the energy required to travel a given distance.

```                Altitude   Density
(km)     (kg/m3)

Sea level          0       1.22
Cessna 150         3.0      .79
Boeing 747        11.0      .36
Airbus A380       13.1      .26
Concorde          18.3      .115
F-22 Raptor       19.8      .091
U-2               21.3      .071
SR-71 Blackbird   25.9      .034
```

Solar powered aircraft
```                Cruise  Max  Ceiling  Mass  Cruise  Motor  Solar  Cells  Battery
m/s    m/s    kW     tons    kw     kW    cells   m2     tons
kW

Aquila           35.8          27.4     .40   5.0                          .2
Solar Impulse 2  25.0   38.9   12      2.3           52     66    269.5    .633
```

The Loon balloon is 15 meters wide, 12 meters, tall, and .076 mm thick. The solar panels generate 100 Watts and the payload is 10 kg. It is too large to be self-propelled and relies and buoyancy modulation and air currents to maneuver.

History
```1961  Piggott accomplishes the first human-powered flight, covering a distance
of 650 meters.
1977  The "Gossomer Condor 2" flies 2172 meters in a figure-eight and wins
the Kremer Prize.  It was built by Paul MacCready and piloted by amateur
cyclist and hang-glider pilot Bryan Allen.
It cruised at 5.0 m/s with a power of 260 Watts.
1988  The MIT Daedalus 88 piloted by Kanellos Kanellopoulos flies from Crete
to Santorini (115.11 km), setting the distance record, which still stands.
```
Human-powered helicopters can only reach a height of 3 meters and can only hover for 20 seconds.

Agility
```               Mass    Power   Agility
(kg)    (kW)   (Watts/kg)

Human             75    2500     33
BMW i8          1485     170    114
Cessna 150       600      75    125
Airbus A380   276000   49000    178
Formula-1 car    642     619    964
SR-71          30600   33000   1078
F-22 Raptor    19700   33000   1675
```
If you put a wing on a BMW i8, it would be able to go fast enough to take off.
Wing shape
```Xwing =  Length of the wing, from the fuselage to the tip
Ywing =  Wing dimension in the direction of flight,
measured along the point of attachment with the fuselage
Awing =  Wing area
Rwing =  Wing aspect ratio   =  Xwing / Ywing
Qlift =  Wing lift-drag ratio

QLift  Rwing     Wing     Xwing
area
(m2)        (m)
U-2         23     10.6                        High-altitude spy plane
Albatros    20                       1.7       Largest bird
Gossamer    20             41.34    14.6       Gossamer albatross, human-powered aircraft
Airbus A380  7.5    7.5   845       36.3
Concorde     7.1          358.2     11.4
Boeing 747   7      7.9   525       29.3
Cessna 150   7             15        4.5
Wingsuit     2.5    1       2        1.0
```
QLift tends to be proportional to Rwing.

Wingtip vortex

A wingtip creates a vortex as it moves. Birds fly in a "V" formation to use the updraft from their neighbor's wingtip vortex.

Flight on other worlds

The minimum agility required to fly scales as

```Agility  ~  g3/2 M1/6 Q-3/2 Cwing-½ D-½
```
We can normalize the Earth to 1 and estimate the minimum agility for other planets. For example,
```MarsAgility / EarthAgility  =  (MarsGravity / EarthGravity)3/2 * (MarsDensity / EarthDensity)-½

Gravity  Atmosphere   Agility     Power/     Maximum
density     normalized   mass      mass for
(m/s^2)   (kg/m^3)    to Earth   (Watts/kg)  flight (kg)

Earth     9.78      1.22       1.0        400              20
Mars      3.8        .020      1.89       756                .44
Titan     1.35      5.3         .025       10        >1000000
Venus     8.87     67           .12        48        >1000000
Pandora   7.8       1.46        .65       261             265
```
For the "Power/mass" column we assume that the power required for human flight is 400 Watts and estimate the power required for flight on other planets.

On Titan you can fly with a wingsuit. A creature as massive as a whale can fly.

"Pandora" is the fictional moon from the film "Avatar".

The largest flying birds on the Earth have a mass of 20 kg. We can use the agility scaling to estimate the maximum mass for flight on other planets.

```Agility ~  g3/2 M1/6 Q-3/2 Cwing-½ D-½

M       ~  g-9  D3
```

Downforce

The wing on a Formula-1 car is an upside-down aircraft wing that generates downforce, to help with friction.

```M   =  Mass
V   =  Velocity
A   =  Acceleration (in any direction)
Cfri=  Friction coefficient
C↓  =  Wing coefficient for downforce
Fgrav=  Gravitational force on the car
=  M g
F↓  =  Downforce from the wing
=  M g C↓ V2
Ffri=  Maximum friction force
=  Cfri (Fgrav + F↓)
=  Cfri M g (1 + C↓ V2)
```
A formula-1 car generates 1 g of downforce at 50 m/s, hence C = 1/502. At the top speed of 100 m/s the downforce is 4 g. The maximum accelerations incurred by the driver are of order 5 g.

The maximum cornering speed for a circle of radius R is:

```Ffri  =  M V2/R  =  M g Cfri (1 + V2/Cfri2)

V2 = g R Cfri / (1 - R/Cfri2)
```

Angle of attack

The angle of attack is the angle of the plane's noze with respect to level fight. As the angle of attack increases the lift increases, with an accompanying increase in drag. If the angle of attack is too high then lift drops and the plane stalls.

Altitude

Airplanes fly at high altitude where the air is thin.

```                Altitude   Air density
(km)     (kg/m3)

Sea level          0       1.22
Denver (1 mile)    1.6      .85
Mount Everest      9.0      .45
Airbus A380       13.1      .25    Commercial airplane cruising altitude
F-22 Raptor       19.8      .084
SR-71 Blackbird   25.9      .038
```

Combat aircraft

F-22 Raptor
F-35 Lightning
F-15 Eagle

F-15 Eagle cockpit
F-16 Falcon
MiG-25 Foxbat

```               Speed  Mass  Takeoff  Ceiling  Thrust  Range  Cost  Number Year Stealth
Mach   ton     ton      km       kN     km     M\$

SR-71 Blackbird  3.3   30.6   78.0     25.9    302    5400          32   1966
MiG-25 Foxbat    2.83  20.0   36.7     20.7    200.2  1730        1186   1970
MiG-31 Foxhound  2.83  21.8   46.2     20.6    304    1450         519   1981
F-22A Raptor     2.51  19.7   38.0     19.8    312    2960   150   195   2005   *
F-15 Eagle       2.5   12.7   30.8     20.0    211.4  4000    28   192   1976
F-14 Tomcat      2.34  19.8   33.7     15.2    268    2960         712   1974
MiG-29 Fulcrum   2.25  11.0   20.0     18.0    162.8  1430    29  1600   1982
Su-35            2.25  18.4   34.5     18.0    284    3600    40    48   1988
F-4 Phantom II   2.23  13.8   28.0     18.3           1500        5195   1958
Chengdu J-10     2.2    9.8   19.3     18.0    130    1850    28   400   2005
F-16 Falcon      2.0    8.6   19.2     15.2    127    1200    15   957   1978
Chengdu J-7      2.0    5.3    9.1     17.5     64.7   850        2400   1966
Dassault Rafale  1.8   10.3   24.5     15.2    151.2  3700    79   152   2001
Euro Typhoon     1.75  11.0   23.5     19.8    180    2900    90   478   2003
F-35A Lightning  1.61  13.2   31.8     15.2    191    2220    85    77   2006   *
B-52              .99  83.2  220       15.0    608   14080    84   744   1952
B-2 Bomber        .95  71.7  170.6     15.2    308   11100   740    21   1997   *
A-10C Warthog     .83  11.3   23.0     13.7     80.6  1200    19   291   1972
Drone RQ-180          ~15              18.3          ~2200               2015   *
Drone X-47B       .95   6.4   20.2     12.2           3890           2   2011   *  Carrier
Drone Avenger     .70          8.3     15.2     17.8  2900    12     3   2009   *
Drone RQ-4        .60   6.8   14.6     18.3     34   22800   131    42   1998
Drone Reaper      .34   2.2    4.8     15.2      5.0  1852    17   163   2007
Drone RQ-170                           15                           20   2007   *

India HAL AMCA   2.5   14.0   36.0     18.0    250    2800     ?     0   2023   *
India HAL FGFA   2.3   18.0   35.0     20.0    352    3500     ?     0  >2020   *
Mitsubishi F-3   2.25   9.7     ?        ?      98.1  3200     ?     1   2024   *
Chengdu J-20     2.0   19.4   36.3       ?     359.8     ?   110     4   2018   *
Sukhoi PAK FA    2.0   18.0   35.0     20.0    334    3500    50     6   2018   *
Shenyang J-31    1.8   17.6   25.0       ?     200    4000     ?     0   2018   *

Mach 1 = 295 m/s
```
5th generation fighters: F-22, F-35, X-2, HAL AMCA, J-20, J-31, Sukhoi PAK FA

An aircraft moving at Mach 2 and turning with a radius of 1.2 km has a g force of 7 g's.

Drones

X-47B
X-47B

RQ-170 Sentinel
MQ-9 Reaper

Missiles

Air to air missiles

F-22 and the AIM-120
AIM-9
Astra
Predator and Hellfire
Helfire in a transparent case

```                Mach   Range  Missile  Warhead  Year  Engine
km      kg       kg

Russia  R-37      6      400    600      60    1989   Solid rocket
Japan   AAM-4     5      100    224       ?    1999   Ramjet
India   Astra     4.5+   110    154      15    2010   Solid rocket
EU      Meteor    4+     200    185       ?    2012   Ramjet
Russia  R-77-PD   4      200    175      22.5  1994   Ramjet
USA     AIM-120D  4      180    152      18    2008   Solid rocket
Israel  Derby-IR  4      100    118      23           Solid rocket
Israel  Rafael    4       50    118      23    1990   Solid rocket
France  MICA      4       50    112      12    1996   Solid rocket
Israel  Python 5  4       20    105      11           Solid rocket
Russia  K-100     3.3    400    748      50    2010   Solid rocket
UK      ASRAAM    3+      50     88      10    1998   Solid rocket
Germany IRIS-T    3       25     87.4          2005   Solid rocket
USA     AIM-9X    2.5+    35     86       9    2003   Solid rocket
USA     Hellfire  1.3      8     49       9    1984   Solid rocket  AGM-114
```

Ground to air missiles

David's Sling
Terminal High Altitude Area Defense (THAAD)

SM-3
SM-3
Chu-SAM
RIM-174

```                 Mach   Range  Missile  Warhead  Year  Engine     Stages   Anti
km      kg       kg                              missile

USA     SM-3      15.2   2500   1500       0    2009   Solid rocket  4       *
Israel  Arrow      9      150   1300     150    2000   Solid rocket  2
USA     THAAD      8.24   200    900       0    2008   Solid rocket          *
USA     David      7.5    300                   2016   Solid rocket          *
Russia  S-400      6.8    400   1835     180    2007   Solid rocket          *
India   Prithvi    5     2000   5600            2006   Solid, liquid 2       *
India   AAD Ashwin 4.5    200   1200       0    2007   Solid rocket  1
Taiwan  Sky Bow 2  4.5    150   1135      90    1998   Solid rocket
China   HQ-9       4.2    200   1300     180    1997   Solid rocket  2
USA     Patriot 3  4.1     35    700      90    2000   Solid rocket          *
China   KS-1       4.1     50    900     100    2006   Solid rocket          *
USA     RIM-174    3.5    460   1500      64    2013   Solid rocket  2
India   Barak 8    2      100    275      60    1015   Solid rocket  2
Japan   Chu-SAM                  570      73    2003   Solid rocket
Korea   KM-SAM             40    400            2015   Solid rocket
```

Ground to ground missiles

Tomahawk
Tomahawk

```                Mach   Range  Missile  Warhead  Year  Engine        Launch
km      kg       kg                         platform

USA     Tomahawk   .84  2500   1600     450    1983   Turbofan      Ground
USA     AGM-129    .75  3700   1300     130    1990   Turbofan      B-52 Bomber
USA     AGM-86     .73  2400   1430    1361    1980   Turbofan      B-52 Bomber
```

Hypersonic missiles

HTV-2
X-51
DARPA Falcon HTV-3

```                   Speed   Mass  Payload  Range  Year
mach    tons   tons     km

USA      SR-72         6                          Future. Successor to the SR-71 Blackbird
USA      HSSW          6                    900   Future. High Speed Strike Weaspon
USA      HTV-2        20           5500   17000   2 Test flights
USA      X-41          8           450            Future
USA      X-51          5.1  1.8             740   2013    Tested. 21 km altitude. Will become the HSSW
Russia   Object 4202  10                          Tested
India    HSTDV        12                          Future
China    Wu-14        10                          2014   7 tests.  also called the DZ-ZF
```
The SR-72 has two engines: a ramjet for below Mach 3 and a ramjet/scramjet for above Mach 3. The engines share an intake and thrust nozzle.
Intercontinental ballistic missiles

First ICBM: SM-65 Atlas, completed in 1958
Titan 2
Peacekeeper
Minuteman 3
Minuteman 3

Trident 2
Peacekeeper
Minuteman 3

```                     Payload  Paylod   Range  Mass    Launch   Year
(tons)   (Mtons)  (km)   (tons)

USA     Titan 2               9        15000   154     Silo    1962   Inactive
USA     Minuteman 3            .9      13000    35.3   Silo    1970
USA     Trident 2              .95     11300    58.5   Sub     1987
USA     Titan                 3.75     10200   151.1   Silo    1959   Inactive
USA     Peacekeeper           3         9600    96.8   Silo    1983   Inactive
Russia  RS-24                 1.2      12000     49    Road    2007
Russia  Voevoda         8.7   8        11000    211.4  Silo    1986
Russia  Layner                         11000     40    Sub     2011
Russia  RS-28 Sarmat   10              10000   >100    Silo    2020   Liquid rocket
Russia  Bulava                 .9      10000     36.8  Sub     2005
France  M51.1                 1        10000    52     Sub     2006
China   DF-5B                 8        15000    183    Silo    2015
China   DF-5A                 4        15000    183    Silo    1983
China   JL-2                  6        12000     42    Sub     2001
China   DF-5                  5        12000    183    Silo    1971
China   DF-31A                3        12000     42    Road
China   DF-31                 1         8000     42    Road    1999
China   DF-4                  3.3       7000     82    Silo    1974
India   Surya          15              16000     70    Road    2022
India   Agni-VI        10              12000     70    Road    2017
India   Agni-V          6               8000     50    Road    2012
India   K-4             2.5             3500     17    Sub     2016   Solid. Arihant nuclear sub
India   K-15           ~6.5              750      1.0  Sub     2010   Solid. 2 stages. Arihant nuclear sub
Israel  Jericho 3        .75           11500     30    Road    2008
N. Kor. Taepodong-2                     6000     79.2  Pad     2006
Pakis.  Shaheen 3                       2750           Road    2015   Solid. 2 stages.
Pakis.  Shaheen 2                       2000     25    Road    2014   Solid. 2 stages.
Pakis.  Ghauri 2        1.2             1800     17.8  Road
Pakis.  Ghauri 1         .7             1500     15.8  Road    2003   Liquid. 1 stage.
Iran    Shabab 3        1.0             1930                   2003
```
Payload in "tons" represents the mass of the payload.
Payload in "Mtons" is the nuclear detonation payload in terms of tons of TNT.
Propeller lift and power

The properties of a propeller are:

```Rotor radius     =  R
Air density      =  D  =  1.22 kg/meter3
Rotor tip speed  =  V
Rotor lift force =  Fl =  D W R2 V2
Rotor drag force =  Fd
Rotor lift param =  W  =  Fl D-1 R-2 V-2
Rotor lift/drag  =  Q  =  Fl / Fd
Rotor power      =  P  =  Fd V  =  F V / Q
Rotor quality    =  q  =  Q W½ D½
=  Fl3/2 P-1 R-1
Rotor force/power=  Z  =  Fl/ P
=  Q / V
=  D½ W½ Q R F-½
=  q R Fl-½
```
The physical parameters of a propeller are {R,Q,W,q}, with typical values of
```Q  = 5.5
W  =  .045
q  = 1.29
```
Most propellers have 2 blades and some have 3. If there are 4 or more blades then q declines.

A measurement of Fl and V determines W.
A measurement of P, Fl, and V determines Q.
A measurement of Fl, P, and r determines q.

Q and W are not independent. They are related to the blade aspect ratio.

```Q  ≈  Aspect ratio
W  ≈  Q-½
q  ≈  Q½
```

Power/Mass ratio

A commonly-appearing quantity is the power/mass ratio, which is inversely proportional to the force/power ratio.

```Mass              =  M
Gravity           =  g
Hover force       =  F  =  M g
Hover power       =  P
Force/Power ratio =  Z  =  F/P
Power/Mass ratio  =  p  =  P/M  =  g/Z
```

Flying electric cars

Design

The larger the propeller radius, the less power that is required to fly. One hence makes the propeller as large as possible, subject to the constraint that if the propeller is too large, the mass is too large.

The most efficient copter has one large rotor for lift plus a small rotor for stability (a "monocopter"). If we increase the number of lift rotors while preserving the total rotor mass then the radius of each rotor is smaller and it takes more power to fly.

Increasing the rotor number increases stability and redundancy. Most drones use 4, 6, or 8 rotors. We construct a design for a flying car using 2 large rotors and 2 small rotors. The large rotors carry most of the weight and the small rotors add stability and failsafe.

The large propellers are mounted forward and aft and the foward propeller tilts forward for horizontal flight. The small propellers are to the right and left.

We assume the total vehicle mass is 400 kg and we use the properties of propellers to calculate the power required to hover. We use a peak power that is comfortably larger than the hover power. The battery mass required to supply this power is 1/4 the total vehicle mass.

```Total aircraft mass =  M        = 400    kg           (Includes passenger)
# of large rotors   =  N        =   2
Rotor radius        =  R        =   1.5  meters
Gravity constant    =  g        =   9.8  meters/second2
Rotor force         =  F = Mg/N =1960    Newtons
Rotor quality       =  q        =   1.02
Air density         =  D        =   1.22 kg/meter3
Rotor power         =  Pr=(qDR)-1F3/2= 46.2  kWatts
Hover power         =  Ph= N Pr =  92.4  kWatts
Peak power          =  P        = 150    kWatts
Battery power/mass  =  p        =1600    Watts/kg
Battery energy/mass =  e        =    .8  MJoules/kg
Battery mass        =  m = P/p  =  94    kg
Battery energy      =  E = e m  =  75    MJoules
Hover time          =  T = E/Ph = 812    seconds  =  14 minutes
```
The properties of propellers are discussed in the
propeller section. The rotor tip speed is
```Rotor lift/drag  =  Q  =  5.5
Rotor tip speed  =  V  =  P Q / F  =  130 m/s
```
The maximum horizontal speed is around 1/3 of the rotor tip speed. If we assume a horizontal speed of 40 meters/second then the range is 32 km.
Mass

For the large propellers,

```Propeller radius          =  R  =  1.5 meters
Propeller mass parameter  =  C  =  5   kg/meter3
Propeller mass            =  M  =  C R3  =  17 kg
```
For the motors on the large propellers,
```Motor power      = 60 kWatts
Motor power/mass =  8 kWatts/kg
Motor mass       =7.5 kg
```
The masses of the components in kg is
```2 large 1.5 meter rotors       32
2 small 1.0 meter rotors       10
2 motors fo rthe large rotors  16
2 motors for the small rotors  10
Battery                       100
Cabin                          50
Fuselage                       50
Pilot                          80
Cargo                          20

Total                         380
```

Flight time

The flight time of a drone is determined by:
*) The battery energy/mass.
*) The power/mass required to hover.
*) The ratio of the battery mass to the drone mass.

Typical parameters for a drone are:

```Drone mass         =  M          =  1.0 kg
Battery mass       =  Mbat        =  .5 kg           (The battery is the most vital component)
Battery energy     =  E          =  .38 MJoules
Battery energy/mass=  ebat= E/Mbat=  .75 MJoules/kg   (Upper range for lithium batteries)
Drone energy/mass  =  e  =  E/M  =  .38 MJoules/kg
Drone power/mass   =  p  =  P/M  =   60 Watts/kg    (Practical minimum to hover. Independent of mass)
Drone power        =  P  =  p M  =   60 Watts       (Power required to hover)
Flight time        =  T  =  E/P  = 6250 seconds  =  104 minutes
```
The flight time in terms of component parameters is
```T  =  (ebat/p) * (Mbat/M)
```

Drone power system

One has to choose a wise balance for the masses of the motor, battery, fuselage, and payload. The properties of the electrical components are:

```                    Energy/Mass  Power/mass  Energy/\$  Power/\$  \$/Mass
MJoule/kg    kWatt/kg   MJoule/\$  kWatt/\$   \$/kg

Electric motor          -         10.0        -        .062     160
Lithium-ion battery     .75        1.5        .009     .0142    106
Lithium supercapacitor  .008       8          .0010    .09       90
Aluminum capacitor      .0011    100
```
If the battery and motor have equal power then the battery has a larger mass than the motor.
```Mass of motor            =  Mmot
Mass of battery          =  Mbat
Power                    =  P             (Same for both the motor and the battery)
Power/mass of motor      =  pmot  =  P/Mmot  =   8.0 kWatt/kg
Power/mass of battery    =  pbat  =  P/Mbat  =   1.5 kWatt/kg
Battery mass / Motor mass=  R    =Mbat/Mmot  =  pmot/pbat  =  5.3
```
The "sports prowess" of a drone is the drone power divided by the minimum hover power. To fly, this number must be larger than 1.
```Drone mass               =  Mdro
Motor mass               =  Mmot
Motor power/mass         =  pmot =  8000 Watts/kg
Hover minimum power/mass =  phov =    60 Watts/kg
Drone power              =  Pdro =  pmot Mmot
Hover minimum power      =  Phov =  phov Mdro
Sports prowess           =  S   =  Pdro/Phov  =  (pmot/phov) * (Mmot/Mdro)  =  80 Mmot/Mdro
```
If S=1 then Mmot/Mdro = 1/80 and the motor constitutes a negligible fraction of the drone mass. One can afford to increase the motor mass to make a sports drone with S >> 1.

If the motor and battery generate equal power then the sports prowess is

```S  =  (pbat/phov) * (Mbat/Mdro)  =  25 Mbat/Mdro
```
If Mbat/Mdro = ½ then S=12.5, well above the minimum required to hover.

Suppose a drone has a mass of 1 kg. A squash racquet can have a mass of as little as .12 kg. The fuselage mass can be much less than this because a drone doesn't need to be as tough as a squash racquet, hence the fuselage mass is negligible compared to the drone mass. An example configuration is:

```              kg

Battery       .5
Motors        .1   To match the battery and motor power, set motor mass / battery mass = 1/5
Rotors       <.05
Fuselage      .1
Camera        .3
Drone total  1.0
```
Supercapacitors can generate a larger power/mass than batteries and are useful for extreme bursts of power, however their energy density is low compared to batteries and so the burst is short. If the supercapacitor and battery have equal power then
```Battery power/mass         =  pbat  =  1.5 kWatts/kg
Supercapacitor power/mass  =  psup  =  8.0 kWatts/kg
Battery power              =  P
Battery mass               =  Mbat  =  P / pbat
Supercapacitor mass        =  Msup  =  P / psup
Supercapacitor/Battery mass=  R     =Msup/ Mbat  =  pbat/psup  =  .19
```
The supercapacitor is substantially ligher than the battery. By adding a lightweight supercapacitor you can double the power. Since drones already have abundant power, the added mass of the supercapacitor usually makes this not worth it.

If a battery and an aluminum capacitor have equal powers,

```Aluminum capacitor mass  /  Battery mass  =  .015
```
If a battery or supercapacitor is operating at full power then the time required to expend all the energy is
```Mass          =  M
Energy        =  E
Power         =  P
Energy/Mass   =  e  =  E/M
Power/Mass    =  p  =  P/M
Discharge time=  T  =  E/P  =  e/p

Energy/Mass  Power/Mass   Discharge time   Mass
MJoule/kg    kWatt/kg       seconds        kg

Lithium battery         .75          1.5          500           1.0
Supercapacitor          .008         8.0            1.0          .19
Aluminum capacitor      .0011      100               .011        .015
```
"Mass" is the mass required to provide equal power as a lithium battery of equal mass.
World War 2 bombers

Avro Lancaster
B-29 Superfortress
Heinkel He 177

Handley Page Halifax
B-17 Flying Fortress
B-17 Flying Fortress

focke-Wulf Condor
Mitsubishi Ki-67
Mitsubishi G4M

Yokosuka Ginga
Tupolev Tu-2

```                            Max    Mass   Max   Bombs  Max   Engine   Range    #    Year
speed          mass         alt                   Built
kph    ton    ton    ton   km    kWatt     km

UK       Avro Lancaster        454  16.6   32.7  10.0   6.5   4x 954   4073   7377  1942
USA      B-29 Superfortress    574  33.8   60.6   9.0   9.7   4x1640   5230   3970  1944
Germany  Heinkel He 177        565  16.8   32.0   7.2   8.0   2x2133   1540   1169  1942
UK       Short Stirling        454  21.3   31.8   6.4   5.0   4x1025   3750   2371  1939
UK       Handley Page Halifax  454  17.7   24.7   5.9   7.3   4x1205   3000   6176  1940
Germany  Fokke-Wulf Condor     360  17.0   24.5   5.4   6.0   4x 895   3560    276  1937
Soviet   Tupolev Tu-2          528   7.6   11.8   3.8   9.0   2x1380   2020   2257  1942
USA      B-17 Flying Fortress  462  16.4   29.7   3.6  10.5   4x 895   3219  12731  1938
Japan    Mitsubishi Ki-67      537   8.6   13.8   1.6   9.5   2x1417   3800    767  1942
Soviet   Petlyakov Pe-2        580   5.9    8.9   1.6   8.8   2x 903   1160  11427  1941
Japan    Yokosuka P1Y Ginga    547   7.3   13.5   1.0   9.4   2x1361   5370   1102  1944
Japan    Mitsubishi G4M        428   6.7   12.9   1.0   8.5   2x1141   2852   2435  1941

Curtis LeMay: Flying fighters is fun. Flying bombers is important.
```

World War 2 heavy fighters

A-20 Havoc
F7F Tigercat
P-38 Lightning

P-61
P-38
Airspeed chart

Fairey Firefly
Beaufighter
Mosquito
Fairey Fulmar
Defiant

Messerschmitt 410
Heinkel He-219
Junkers Ju-88

Do-217
Me-110

Kawasaki Ki-45
J1N

Gloster Meteor
Me-262 Swallow
Heinkel He-162

```                       Max   Climb  Mass   Max   Bombs  Max   Engine   Range   #   Year
speed                mass         alt                  Built
kph    m/s   ton    ton    ton   km    kWatt     km

USA    P51 Black Widow  589  12.9  10.6   16.2   2.9   10.6  2x1680   982    706  1944
USA    A-20 Havoc       546  10.2   6.8   12.3    .9    7.2  2x1200  1690   7478  1941
USA    F7F Tigercat     740  23     7.4   11.7    .9   12.3  2x1566  1900    364  1944
USA    P-38 Lightning   667  24.1   5.8    9.8   2.3   13.0  2x1193        10037  1941
UK     Fairey Firefly   509   8.8   4.4    6.4    .9    8.5  1x1290  2090   1702  1943
UK     Mosquito         668  14.5   6.5   11.0   1.8   11.0  2x1103  2400   7781  1941
UK     Beaufighter      515   8.2   7.1   11.5    .3    5.8  2x1200  2816   5928  1940
UK     Fairie Fulmar    438         3.2    4.6    .1    8.3  1x 970  1255    600  1940
UK     Defiant          489   9.0   2.8    3.9   0      9.2  1x 768   749   1064  1939
Japan  Dragon Slayer    540  11.7   4.0    5.5   0     10.0  2x 783         1701  1941  Ki-45
Japan  Flying Dragon    537   6.9   8.6   13.8   1.6    9.5  2x1417  3800    767  1942  Ki-67
Japan  J1N Moonlight    507   8.7   4.5    8.2   0           2x 840  2545    479  1942
Ger.   Hornet           624   9.3   6.2   10.8   1.0   10.0  2x1287  2300   1189  1943
Ger.   Flying Pencil    557   3.5   9.1   16.7   4.0    7.4  2x1287  2145   1925  1941  Do-217
Ger.   Heinkel He-219   616               13.6   0      9.3  2x1324  1540    300  1943
Ger.   Junkers Ju-88    360        11.1   12.7   0      5.5  2x1044  1580  15183  1939
Ger.   Me-110           595  12.5          7.8   0     11.0  2x1085   900   6170  1937
SU     Petlyakov Pe-3   530  12.5   5.9    8.0    .7    9.1  2x 820  1500    360  1941
UK     Gloster Meteor   965  35.6   4.8    7.1    .9   13.1   Jet     965   3947  1944
Ger.   Me-262 Swallow   900 ~25     3.8    7.1   1.0   11.5   Jet    1050   1430  1944
Ger.   Heinkel He-162   840  23.4   1.7    2.8   0     12.0   Jet     975    320  1945

Me-262 Swallow jet  =  2x 8.8 kNewtons
Heinkel He-162 jet  =  1x 7.8 kNewtons
Gloster Meteor jet  =  2x16.0 kNewtons
```

World War 2 light fighters

P-39 Airacobra
P-40 Warhawk
P-43 Lancer

P-47 Thunderbolt
P-51 Mustang
P-63 Kingcobra

F2A Buffalo
F4F
F4U

F6F Hellcat
F8F Bearcat

Ki-27
Ki-43
Ki-44

Ki-61
Ki-84
Ki-100

A5M
Mitsubishi A6M Zero
A6M2

J2M
N1K

Hawker Tempest
Hawker Hurricane
Hawker Typhoon

Submarine Seafire
Submarine Spitfire

Fw-190
Bf-109

YaK-1
Yak-7
Yak-9
Polykarpov I-16

MiG-3
LaGG-3
La-5
La-7

```                       Max   Climb  Mass   Max   Bombs  Max   Engine   Range   #    Year
speed                mass         alt                  Built
kph    m/s   ton    ton    ton   km    kWatt     km

USA    P-39 Airacobra   626  19.3   3.0    3.8    .2   10.7  1x 894   840   9588  1941
USA    P-63 Kingcobra   660  12.7   3.1    4.9    .7   13.1  1x1340   725   3303  1943
USA    F2A Buffalo      517  12.4   2.1    3.2   0     10.1  1x 890  1553    509  1939
USA    P-40 Warhawk     580  11.0   2.8    4.0    .9    8.8  1x 858  1100  13738  1939
USA    P-51 Mustang     703  16.3   3.5    5.5    .5   12.8  1x1111  2755 >15000  1942
USA    F4F Wildcat      515  11.2   2.7    4.0   0     10.4  1x 900  1337   7885  1940
USA    F6F Hellcat      629  17.8   4.2    7.0   1.8   11.4  1x1491  1520  12275  1943
USA    F8F Bearcat      730  23.2   3.2    6.1    .5   12.4  1x1678  1778   1265  1945
USA    P-43 Lancer      573  13.0   2.7    3.8   0     11.0  1x 895  1046    272  1941
USA    P-47 Thunderbolt 713  16.2   4.5    7.9   1.1   13.1  1x1938  1290  15677  1942
USA    F4U Corsair      717  22.1   4.2    5.6   1.8   12.6  1x1775  1617  12571  1942
Japan  Zero             534  15.7   1.7    2.8    .3   10.0  1x 700  3104  10939  1940
Japan  N1K Strong Wind  658  20.3   2.7    4.9    .5   10.8  1x1380  1716   1532  1943
Japan  Ki-84 "Gale"     686  18.3   2.7    4.2    .7   11.8  1x1522  2168   3514  1943
Japan  Ki-61            580  15.2   2.6    3.5    .5   11.6  1x 864   580   3078  1942
Japan  Ki-100           580  13.9   2.5    3.5   0     11.0  1x1120  2200    396  1945
Japan  A5M              440         1.2    1.8   0      9.8  1x 585  1200   1094  1936
Japan  A6M2             436  12.4   1.9    2.9    .1   10.0  1x 709  1782    327  1942
Japan  J2M Thunderbolt  655  23.4   2.8    3.2    .1   11.4  1x1379   560    671  1942
Japan  Ki-27            470  15.3   1.1    1.8    .1   12.2  1x 485   627   3368  1937
Japan  Ki-43            530         1.9    2.9    .5   11.2  1x 858  1760   5919  1941
Japan  Ki-44            605  19.5   2.1    3.0   0     11.2  1x1133         1225  1942
UK     Hawker Hurricane 547  14.1   2.6    4.0    .5   11.0  1x 883   965  14583  1943
UK     Hawker Tempest   700  23.9   4.2    6.2    .9   11.1  1x1625  1190   1702  1944
UK     Hawker Typhoon   663  13.6   4.0    6.0    .9   10.7  1x1685   821   3317  1941
UK   Submarine Seafire  578  13.4   2.8    3.5          9.8  1x1182   825   2334  1942
UK   Submarine Spitfire 595  13.2   2.3    3.0   0     11.1  1x1096   756  20351  1938
Ger.   Fw-190           685  17.0   3.5    4.8    .5   12.0  1x1287   835 >20000  1941
Ger.   Bf-109           640  17.0   2.2    3.4    .3   12.0  1x1085   850  34826  1937
SU     MiG-3            640  13.0   2.7    3.4    .2   12.0  1x 993   820   3172  1941
SU     Yak-1            592  15.4   2.4    2.9   0     10.0  1x 880   700   8700  1940
SU     Yak-3            655  18.5   2.1    2.7   0     10.7  1x 970   650   4848  1944
SU     Yak-7            571  12.0   2.4    2.9   0      9.5  1x 780   643   6399  1942
SU     Yak-9            672  16.7   2.5    3.2   0     10.6  1x1120   675  16769  1942
SU     LaGG-3           575  14.9   2.2    3.2    .2    9.7  1x 924  1000   6528  1941
SU     La-5             648  16.7   2.6    3.4    .2   11.0  1x1385   765   9920  1942
SU     La-7             661  15.7   3.3           .2   10.4  1x1230   665   5753  1944
SU     Polykarpov I-16  525  14.7   1.5    2.1    .5   14.7  1x 820   700   8644  1934
```

World War 2 aircraft carriers

U.S. Essex Class
U.S. Independence Class

Shokaku Class
Hiyo Class
Chitose Class

Unryu Class
Zuiho Class

```       Class        Speed   Power  Length  Displace  Planes     #     Year
kph    MWatt    m       kton             built

USA    Essex         60.6   110     263      47       100      24     1942
USA    Independence  58      75     190      11        33       9     1942
Japan  Shokaku       63.9   120     257.5    32.1      72       2     1941
Japan  Hiyo          47.2    42     219.3    24.2      53       3     1944
Japan  Unryu         63     113     227.4    17.8      65       3     1944
Japan  Chitose       53.5    42.4   192.5    15.5      30       2     1944
Japan  Zuiho         52      39     205.5    11.4      30       2     1940
```

Explosives

Medieval-style black powder
Modern smokeless powder

```               MJoules  Rocket  Shock  Density  Boil
/kg     km/s   km/s   g/cm3  Kelvin

Beryllium+ O2    23.2   5.3
Aluminum + O2    15.5
Magnesium+ O2    14.8
Hydrogen + O2    13.2   4.56             .07    20
Kerosene + O3    12.9
Octanitrocubane  11.2          10.6     1.95
Methane  + O2    11.1   3.80             .42   112
Octane   + O2    10.4                    .70   399
Kerosene + O2    10.3   3.52             .80   410
Dinitrodiazeno.   9.2          10.0     1.98
C6H6N12O12        9.1                   1.96        China Lake compound
Kerosene + H2O2   8.1   3.2
Kerosene + N2O4   8.0   2.62
HMX (Octogen)     8.0   3.05    9.1     1.86
RDX (Hexagen)     7.5   2.5     8.7     1.78
Al + NH4NO3       6.9
Nitroglycerine    7.2           8.1     1.59        Unstable
PLX               6.5                   1.14        95% CH3NO2 + 5% C2H4(NH2)2
Composition 4     6.3           8.04    1.59        91% RDX. "Plastic explosive"
Kerosene + N2O    6.18
Dynamite          5.9           7.2     1.48        75% Nitroglycerine + stabilizer
PETN              5.8           8.35    1.77
Smokeless powder  5.2           6.4     1.4         Used after 1884. Nitrocellulose
TNT               4.7           6.9     1.65        Trinitrotoluene
Al + Fe2O3        4.0                               Thermite
H2O2              2.7   3.1             1.45   423  Hydrogen peroxide
Black powder      2.6   3.08     .6     1.65        Used before 1884
Al + NH4ClO4            2.6
NH4ClO4                 2.5
N2O               1.86  1.76
N2H4              1.6   2.2             1.02   387  Hydrazine
NH4NO3            1.4   2.0     2.55    1.12        Ammonium nitrate
Bombardier beetle  .4                               Hydroquinone + H2O2 + protein catalyst
N2O4               .10                  1.45   294

Rocket: Rocket exhaust speed
Shock:  Shock speed
```
Nitrocellulose
TNT
RDX
HMX
PETN
Octanitrocubane

Nitrocellulose
TNT
RDX
HMX
PETN
Octanitrocubane

Dinitrodiazenofuroxan
Nitromethane

High explosives

High explosives have a large shock velocity.

```
MJoules   Shock  Density
/kg     km/s    g/cm3

Octanitrocubane    11.2   10.6     1.95
Dinitrodiazeno.     9.2   10.0     1.98
C6H6N12O12          9.1            1.96    China Lake compound
HMX (Octogen)       8.0    9.1     1.86
RDX (Hexagen)       7.5    8.7     1.78
PLX                 6.5            1.14    95% CH3NO2 + 5% C2H4(NH2)2
Composition 4       6.3    8.04    1.59    91% RDX. "Plastic explosive"
Dynamite            5.9    7.2     1.48    75% Nitroglycerine + stabilizer
PETN                5.8    8.35    1.77
```

Liquid oxygen

The best oxidizer is liquid oxygen, and the exhaust speed for various fuels when burned with oxygen is:

```                Exhaust  Energy   Density of fuel + oxidizer
speed   /mass
km/s    MJ/kg      g/cm3

Hydrogen   H2      4.46   13.2    .32
Methane    CH4     3.80   11.1    .83
Ethane     C2H6    3.58   10.5    .9
Kerosene   C12H26  3.52   10.3   1.03
Hydrazine  N2H4    3.46          1.07
```
Liquid hydrogen is usually not used for the ground stage of rockets because of its low density.
Oxidizer

We use kerosene as a standard fuel and show the rocket speed for various oxidizers. Some of the oxidizers can be used by themselves as monopropellants.

```    Energy/Mass       Energy/Mass        Rocket           Rocket         Boil    Density
with kerosene   as monopropellant  with kerosene  as monopropellant  Kelvin   g/cm3
MJoule/kg         MJoule/kg          km/s             km/s

O3        12.9           2.97                                              161
O2        10.3           0                  3.52             0             110     1.14
H2O2       8.1           2.7                3.2              1.6           423     1.45
N2O4       8.00           .10               2.62                           294     1.44
N2O        6.18          1.86                                1.76          185
N2H4       -             1.58                                2.2           387     1.02
```

Solid rocket fuel
```               MJoules  Rocket   Density
/kg     km/s    g/cm3

C6H6N12O12        9.1             1.96        China Lake compound
HMX (Octogen)     8.0   3.05      1.86
RDX (Hexagen)     7.5   2.5       1.78
Al + NH4ClO4            2.6
NH4ClO4                 2.5
NH3OHNO3                2.5       1.84        Hydrxyammonium nitrate
Al + NH4NO3       6.9
NH4NO3            1.4   2.0       1.12        Ammonium nitrate
```

History
```~808  Qing Xuzi publishes a formula resembling gunpower, consisting of
6 parts sulfur, 6 parts saltpeter, and 1 part birthwort herb (for carbon).
~850  Incendiary property of gunpower discovered
1132  "Fire lances" used in the siege of De'an, China
1220  al-Rammah of Syria publishes "Military Horsemanship and Ingenious War
Devices", describes the purification of potassium nitrate by
adding potassium carbonate with boiling water, to precipitate out
magnesium carbonate and calcium carbonate.
1241  Mongols use firearms at the Battle of Mohi, Hungary
1338  Battle of Arnemuiden.  First naval battle involving cannons.
1346  Cannons used in the Siege of Calais and the Battle of Crecy
1540  Biringuccio publishes "De la pirotechnia", giving recipes for gunpowder
1610  First flintlock rifle
1661  Boyle publishes "The Sceptical Chymist", a treatise on the
distinction between chemistry and alchemy.  It contains some of the
earliest modern ideas of atoms, molecules, and chemical reaction,
and marks the beginning of the history of modern chemistry.
1669  Phosphorus discovered
1774  Lavoisier appointed to develop the French gunpowder program.  By 1788
French gunpowder was the best in the world.
1832  Braconnot synthesizes the first nitrocellulose (guncotton)
1846  Nitrocellulose published
1847  Sobrero discovers nitroglycerine
1862  LeConte publishes simple recipes for producing potassium nitrate.
1865  Abel develops a safe synthesis of nitrocellulose
1867  Nobel develops dynamite, the first explosive more powerful than black powder
It uses diatomaceous earth to stabilize nitroglycerine
1884  Vieille invents smokeless gunpowder (nitrocellulose), which is 3 times
more powerful than black powder and less of a nuisance on the battlefield.
1902  TNT first used in the military.  TNT is much safer than dynamite
1930  RDX appears in military applications
1942  Napalm developed
1949  Discovery that HMX can be synthesized from RDX
1956  C-4 explosive developed (based on RDX)
1999  Eaton and Zhang synthesize octanitrocubane and heptanitrocubane
```

Above 550 Celsius, potassium nitrate decomposes. 2 KNO3 ↔ 2 KNO2 + O2.

```Black powder           =  .75 KNO3  +  .19 Carbon  +  .06 Sulfur
1 kg TNT equivalent    =   4.184   MJ
Fission bomb           =   9.20e13 J     =  22000 tons of TNT equivalent
Fission bomb           =   420     kg
Fission bomb           =   2.19e11 J/kg
Fusion bomb maximum    =   2.51e13 J/kg   (Maximum theoretical efficiency)
Fusion bomb practical  =   1.0 e13 J/kg   (Practical efficiency achieved in real bombs)
```

Cryogenics
```            Boil   Density
Kelvin   g/cm3

Hydrogen     20      .07
Oxygen       90     1.14
Methane     112      .42
NO          121
Ozone       161
Ethane      185      .54
N2O         185
Propane     231      .60
Butane      274      .60
N2O4        294     1.44
Pentane     309      .63
Hexane      342      .65
Heptane     372      .68
N2H4        387     1.02
Octane      399      .70
H2O2        423     1.45
Dodecain    489      .75
```

Black powder

Sulfur
Sulfur
Saltpeter
Saltpeter

Charcoal
Icing sugar and KNO3
Mortar and pestle
Mortar and pestle

```Potassium nitrate  KNO3     75%       (Saltpeter)
Charcoal           C7H4O    15%
Sulfur             S        10%

Oversimplified equation:  2 KNO3 + 3 C + S  →  K2S + N2 + 3 CO2

Realistic equation:       6 KNO3 + C7H4O + 2 S  →  KCO3 + K2SO4 + K2S + 4 CO2 + 2 CO + 2 H2O + 3 N2
```
Nitrite (NO3) is the oxidizer and sulfur lowers the ignition temperature.
Fuel air explosives
```                   MJoules
/kg

Hydrogen + Oxygen     13.16
Gasoline + Oxygen     10.4

Mass   Energy    Energy/Mass
kg      MJ         MJ/kg

MOAB    9800   46000        4.7               8500 kg of fuel
```

Phosphorus
White phosphorus
White, red, violet, and black phosphorus
Red phosphorus

Violet phosphorus
Black phosphorus
Black phosphorus

```Form      Ignition    Density
(Celsius)

White        30        1.83
Red         240        1.88
Violet      300        2.36
Black                  2.69
```
Red phosphorus is formed by heating white phosphorus to 250 Celsius or by exposing it to sunlight. Violet phosphorus is formed by heating red phosphorus to 550 Celsius. Black phosphorus is formed by heating white phosphorus at a pressure of 12000 atmospheres. Black phosphorus is least reactive form and it is stable below 550 Celsius.
Matches

Striking surface
P4S3

The safety match was invented in 1844 by Pasch. The match head cannot ignite by itself. Ignitition is achieved by striking it on a rough surface that contains red phosphorus. When the match is struck, potassium chlorate in the match head mixes with red phosphorus in the abrasive to produce a mixture that is easily ignited by friction. Antimony trisulfide is added to increase the burn rate.

```Match head                 Fraction             Striking surface   Fraction

Potassium chlorate    KClO3  .50                Red phosphorus      .5
Silicon filler        Si     .4                 Abrasive            .25
Sulfur                S      small              Binder              .16
Antimony3 trisulfide  Sb2S3  small              Neutralizer         .05
Neutralizer                  small              Carbon              .04
Glue                         small
```
A "strike anywhere" match has phosphorus in the match head in the form of phosphorus sesquisulfide (P4S3) and doesn't need red phosphorus in the striking surface. P4S3 has an ignition temperature of 100 Celsius.
Flint

Before the invention of iron, fires were started by striking flint (quartz) with pyrite to generate sparks. Flintlock rifles work by striking flint with iron. With the discovery of cerium, ferrocerium replaced iron and modern butane lighters use ferrocerium, which is still referred to as "flint".

```Cerium        .38      Ignition temperature of 165 Celsius
Lanthanum     .22
Iron          .19
Neodymium2    .04
Praseodymium  .04
Magnesium     .04
```

Nitrous oxide engine

Nitrous oxide is stored as a cryogenic liquid and injected along with gaoline into the combustion chamber. Upon heating to 300 Celsius the nitrous oxide decomposes into nitrogen and oxygen gas and releases energy. The oxygen fraction in this gas is higher than that in air (1/3 vs. .21) and the higher faction allows for more fuel to be consumed per cylinder firing.

```Air density                  =  .00122 g/cm3
Nitrous oxide gas density    =  .00198 g/cm3
Diesel density               =  .832   g/cm3
Gasoline density             =  .745   g/cm3
Diesel energy/mass           =  43.1   MJoules/kg
Gasoline energy/mass         =  43.2   MJoules/kg
Nitrous oxide boiling point  = -88.5   Celsius
Air oxygen fraction          =  .21
Nitrous oxide oxygen fraction=  .33
Nitrous oxide decompose temp =  300    Celsius
Nitrous oxide liquid pressure=   52.4  Bars     Pressure required to liquefy N2O at room temperature
```

Bombardier beetle

Hydroquinone
P-quinone

Hydroquinone and peroxide are stored in 2 separate compartments are pumped into the reaction chamber where they explode with the help of protein catalysts. The explosion vaporizes 1/5 of the liquid and expels the rest as a boiling drop of water, and the p-quinone in the liquid damages the foe's eyes. The energy of expulsion pumps new material into the reaction chamber and the process repeats at a rate of 500 pulses per second and a total of 70 pulses. The beetle has enough ammunition for 20 barrages.

```2 H2O2  →  2 H2O +  O2           (with protein catalyst)
C6H4(OH)2  →  C6H4O2 + H2        (with protein catalyst)
O2 + 2 H2  →  2 H2O

Firing rate                     = 500 pulses/second
Number of pulses in one barrage =  70
Firing time                     = .14 seconds
Number of barrages              =  20
```

Flame speed

A turbojet engine compresses air before burning it to increase the flame speed and make it burn explosively. A ramjet engine moving supersonically doesn't need a turbine to achieve compression.

Turbojet
Ramjet

```Airbus A350 compression ratio  =  52
Air density at sea level       = 1    bar
Air density at 15 km altitude  =  .25 bar
Air density in A350 engine     =  13  bar
```
From the thermal flame theory of Mallard and Le Chatelier,
```Temperature of burnt material    =  Tb
Temperature of unburnt material  =  Tu
Temperature of ignition          =  Ti
Fuel density                     =  Dfuel
Oxygen density                   =  Doxygen
Reaction coefficient             =  C
Reaction rate                    =  R  =  C Dfuel Doxygen
Thermal diffusivity              =  Q  = 1.9⋅10-5 m2/s
Flame speed                      =  V

V2  =  Q C Dfuel Doxygen (Tb - Ti) / (Ti - Tu)
```

Shocks

Spherical implosion
Mach < 1,    Mach = 1,     Mach > 1

If the pressure front moves supersonically then the front forms a discontinuous shock, where the pressure makes a sudden jump as the shock passes.

Energy boost

Metal powder is often included with explosives.

```        Energy/mass    Energy/mass
not including  including
oxygen         oxygen
(MJoule/kg)    (MJoule/kg)

Hydrogen    113.4      12.7
Gasoline     46.0      10.2
Beryllium    64.3      23.2
Aluminum     29.3      15.5
Magnesium    24.5      14.8
Carbon       12.0       3.3
Lithium       6.9       3.2
Iron          6.6       4.6
Copper        2.0       1.6
```

Fireworks

Li
B
Na
Mg
K
Ca
Fe

Cu
Zn
As
Sr
Sb
Rb
Pb

BaCl (green), CuCl (blue), SrCl (red)
Zero gravity
Bunsen burner, O2 increases rightward
Methane

Oxygen candle

Sodium chlorate

An oxygen candle is a mixture of sodium chlorate and iron powder, which when ignited smolders at 600 Celsius and produces oxygen at a rate of 6.5 man-hours of oxygen per kilogram of mixture. Thermal decomposition releases the oxygen and the burning iron provides the heat. The products of the reaction are NaCl and iron oxide.

Pendulum

History

Foucault pendulum

```
-2000 System of hours, minutes, and seconds developed in Sumer
-300  Water clock developed in Ancient Greece
100  Zhang Heng constructs a seismometer using pendulums that was capable of
detecting the direction of the Earthquake.
1300  First mechanical clock deveoped.
1400  Spring-based clocks developed.
1500  Pendulums are used for power, for machines such as saws, bellows, and pumps.
1582  Galileo finds that the period of a pendulum is independent of mass
and oscillation angle, if the angle is small.
1636  Mersenne and Descartes find that the pendulum was not quite isochronous.
Its period increased somewhat with its amplitude.
1656  Huygens builds the first pendulum clock, delivering a precision of
15 seconds per day.  Previous devices had a precision of 15 minutes per day.
Fron this point on pendulum clocks were the most accurate timekeeping devices
until the development of the quartz oscillator was developed in 1921.
1657  Balance spring developed by Hooke and Huygens, making possible portable
pocketwatches.
1658  Huygens publishes the result that pendulum rods expand when heated.
This was the principal error in pendulum clocks.
1670  Previous to 1670 the verge escapement was used, which requires a large angle.
The anchor escapement mechanism is developed in 1670, which allows for a smaller
angle.  This increased the precision because the oscillation period is
independent of angle for small angles.
1673  Huygens publishes a treatise on pendulums.
1714  The British Parliament establishes the "Longitude Prize" for anyne
who could find an accurate method for determing longitude at sea.
At the time there was no clock that could measure time on a moving ship
accurately enough to determine longitude.
1721  Methods are developed for compensating for thermal expansion error of a pendulum.
1726  Gridiron pendulum developed, improving precision to 1 second per day.
1772  Harrison builds a clock which James Cook used in his exploration of the Pacific.
Cook's log is full of praise for the watch and the charts of the Pacific
Ocean were remarkably accurate.
1772  Harrison gives one of his clocks to King George III, who personally tested it and
found it to be accurate to 1/3 of one second per day.  King George III advised
Harrison to petition Parliament for the full Longitude Prize after threatening
to appear in person to dress them down.
1851  Foucault shows that a pendulum can be used to measure the rotation period of
the Earth.  The penulum swings in a fixed frame and the Earth rotates with
respect to this frame.  In the Earth frame the pendulum appears to precess.
1921  Quartz electronic oscillator developed
1927  First quartz clocks developed, which were more precise than pendulum clocks.

L  =  Length of the pendulum
g  =  Gravity constant
=  9.8 meters/second2
T  =  Period of the pendulum
Z  =  Angle of maximum amplitude, in radians.
```
If the angle Z is small (Z << 1) then the period of oscillation is
```T  = 2 Pi SquareRoot(L/g)
```
As the angle increases the period of oscillation increases.

Angle = 30 degrees
Angle = 60 degrees
Angle = 120 degrees
Angle = 170 degrees

For a leg with a center of mass that is .5 meters below the hip joint the pendulum period is 1.4 seconds, similar to your walking candence. The pendulum frequency of your arms is slightly shorter.

Gravity escapement
Anchor escapement
Grasshopper escapement

Derivaton of circular acceleration

Suppose an object starts at (X,Y) = (0,-R) and moves with a speed V around the circle

```X2 + Y2 = R2
```
Approximating the motion near (X,Y) = (0,-R)
```Y  = -(R2 - X2)1/2
= -R (1 - X2/R2)1/2
~ -R (1 - .5 X2/R2
~ -R + .5 X2/R

X  =  V T

Y  =  -R + .5 V2 T2 / R
```
This has the form
```Y  =  -R + .5 A T2
```
where
```A  =  V2/R
```
This is the acceleration of an object moving with constant velocity around a circle.
Circular motion

Suppose a particle moves around a circle.

```R  =  Radius of the circle
$\omega$  =  Angular frequency

X  =  X position     =  R    cos($\omega$ T)
Vx =  X velocity     = -R $\omega$  sin($\omega$ T)
Ax =  X acceleration = -R $\omega$2 cos($\omega$ T)

Y  =  Y position     =  R    sin($\omega$ T)
Vy =  Y velocity     =  R $\omega$  cos($\omega$ T)
Ay =  Y acceleration = -R $\omega$2 sin($\omega$ T)

V2  =  Vx2 + Vx2  =  R2 Kt2          ->  V = R $\omega$
A2  =  Ax2 + Ay2  =  R2 Kt2          ->  A = R $\omega 2$
```

Thermodynamics

Kinetic energy of a gas

The pressure in a gas arises from kinetic energy of gas molecules.

```Number of gas molecules              =  N
Mass of a gas molecule               =  M
Volume of the gas                    =  Vol
Number of gas molecules per volume   =  n  =  N / Vol
Thermal speed of gas molecules       =  Vth
Mean kinetic energy per gas molecule =  E  =  .5 M Vth2     (Definition)
Kinetic energy per volume            =  e  =  E / Vol
Boltzmann constant                   =  k  =  1.38e-23 Joules/Kelvin
Density                              =  D  =  N M / Vol
```
The characteristic thermal speed of a gas molecule is defined in terms of the mean energy per molecule.
```E  =  .5 M Vth2
```
The ideal gas law can be written in the following forms:
```P  =  2/3 e
=  8.3 Mol T / Vol
=  k T N     / Vol
=  1/3 N M Vth2/ Vol
=  1/3 D Vth2
=  k T D / M
```

Boltzmann constant

For a system in thermodynamic equilibrium each degree of freedom has a mean energy of .5 k T. This is the definition of temperature.

A gas molecule moving in 3 dimensions has 3 degrees of freedom and so the mean kinetic energy is

```E  =  1.5 k T  =  .5 M V2
```

Molecules
```       Melt   Boil   Solid    Liquid   Gas        Mass
(K)    (K)   density  density  density    (AMU)
g/cm^3   g/cm^3   g/cm^3

O2      54      90             1.14    .00143     32.0
N2      63      77              .81    .00125     28.0
H2O    273     373     .917    1.00    .00080     18.0
CO2    n/s     195    1.56      n/a    .00198     44.0
H2      14      20              .070   .000090     2.0
CH4     91     112              .42    .00070     16.0
CH5OH  159     352              .79    .00152     34.0      Alcohol
```
Gas density is for 20 Celsius and 1 Bar.

Carbon dioxide doesn't have a liquid state at standard temperature and pressure. It sublimes directly from a solid to a vapor.

Atmospheric height

```M  =  Mass of a gas molecule
Vth=  Thermal speed
H  =  Characteristic height of an atmosphere
g  =  Gravitational acceleration
```
Suppose a molecule at the surface of the Earth is moving upward with speed V and suppose it doesn't collide with other air molecules. It will reach a height of
```M H g  =  .5 M Vth2
```
This height H is the characteristic height of an atmosphere.

The density of the atmosphere scales as

```Density  ~  Density At Sea Level * exp(-E/E0)
```
where E is the gravitational potential energy of a gas molecule and E is the characteristic thermal energy given by
```E = M H g = 1/2 M Vth2
```
Expressed in terms of altitude h,
```Density  ~  Density At Sea Level exp(-h/H)
```
For oxygen,
```E  =  1.5 k T
```
E is the same for all molecules regardless of mass, and H depends on the molecule's mass. H scales as
```H  ~  M-1
```

Derivation of the ideal gas law

We first derive the law for a 1D gas and then extend it to 3D.

Suppose a gas molecule bounces back and forth between two walls separated by a distance L.

```M  = Mass of molecule
V  = Speed of the molecule
L  = Space between the walls
```
With each collision, the momentum change = 2 M V
Time between collisions = 2 L / V

The average force on a wall is

```Force  =  Change in momentum  /  Time between collisions  =  M  V^2  /  L
```
Suppose a gas molecule is in a cube of volume L^3 and a molecule bounces back and forth between two opposite walls (never touching the other four walls). The pressure on these walls is
```Pressure  =  Force  /  Area
=  M  V^2  /  L^3
=  M  V^2  /  Volume

Pressure  Volume  =  M  V^2
```
This is the ideal gas law in one dimension. For a molecule moving in 3D,
```Velocity^2  = (Velocity in X direction)^2
+ (Velocity in Y direction)^2
+ (Velocity in Z direction)^2
```
Characteristic thermal speed in 3D = 3 * Characteristic thermal speed in 1D.
```To produce the 3D ideal gas law, replace  V^2  with  1/3 V^2  in the 1D equation.

Pressure  Volume  =  1/3  M  V^2               Where V is the characteristic thermal speed of the gas
```
This is the pressure for a gas with one molecule. If there are n molecules,
```Pressure  Volume  =  n  1/3  M  V^2            Ideal gas law in 3D
```
If a gas consists of molecules with a mix of speeds, the thermal speed is defined as
```Kinetic dnergy density of gas molecules  =  E  =  (n / Volume) 1/2 M V^2
```
Using this, the ideal gas law can be written as
```Pressure  =  2/3  E
=  1/3  Density  V^2
=  8.3  Moles  Temperature  /  Volume
```
The last form comes from the law of thermodynamics: M V^2 = 3 B T
Atmospheric escape

The "Balloons and Buoyancy" simulation at phet.colorado shows a gas with a mix of light and heavy molecules.

```S = Escape speed
T = Temperature
B = Boltzmann constant
= 1.38e-23 Joules/Kelvin
g = Planet gravity at the surface

M = Mass of heavy molecule                    m = Mass of light molecule
V = Thermal speed of heavy molecule           v = Thermal speed of light molecule
E = Mean energy of heavy molecule             e = Mean energy of light molecule
H = Characteristic height of heavy molecule   h = Characteristic height of light molecule
= E / (M g)                                   = e / (m g)
Z = Energy of heavy molecule / escape energy  z = Energy of light molecule / escape energy
= .5 M V^2 / .5 M S^2                         = .5 m v^2 / .5 m S^2
= V^2 / S^2                                   = v^2 / S^2

For an ideal gas, all molecules have the same mean kinetic energy.

E     =     e      =  1.5 B T

.5 M V^2  =  .5 m v^2  =  1.5 B T
```
The light molecules tend to move faster than the heavy ones. This is why your voice increases in pitch when you breathe helium. Breathing a heavy gas such as Xenon makes you sound like Darth Vader.

For an object to have an atmosphere, the thermal energy must be much less than the escape energy.

```V^2 << S^2        <->        Z << 1

Escape  Atmos    Temp    H2     N2      Z        Z
speed   density  (K)    km/s   km/s    (H2)     (N2)
km/s    (kg/m^3)
Jupiter   59.5             112   1.18    .45   .00039   .000056
Saturn    35.5              84   1.02    .39   .00083   .00012
Neptune   23.5              55    .83    .31   .0012    .00018
Uranus    21.3              53    .81    .31   .0014    .00021
Earth     11.2     1.2     287   1.89    .71   .028     .0041
Venus     10.4    67       735   3.02   1.14   .084     .012
Mars       5.03     .020   210   1.61    .61   .103     .015
Titan      2.64    5.3      94   1.08    .41   .167     .024
Europa     2.02    0       102   1.12    .42   .31      .044
Moon       2.38    0       390   2.20    .83   .85      .12
Ceres       .51    0       168   1.44    .55  8.0      1.14
```
Even if an object has enough gravity to capture an atmosphere it can still lose it to the solar wind. Also, the upper atmosphere tends to be hotter than at the surface, increasing the loss rate.

Titan is the smallest object with a dense atmosphere, suggesting that the threshold for capturing an atmosphere is on the order of Z = 1/25, or

Thermal Speed < 1/5 Escape speed

Heating by gravitational collapse

When an object collapses by gravity, its temperature increases such that

```Thermal speed of molecules  ~  Escape speed
```
In the gas simulation at phet.colorado.edu, you can move the wall and watch the gas change temperature.

For an ideal gas,

```3 * Boltzmann_Constant * Temperature  ~  MassOfMolecules * Escape_Speed^2
```
For the sun, what is the temperature of a proton moving at the escape speed? This sets the scale of the temperature of the core of the sun. The minimum temperature for hydrogen fusion is 4 million Kelvin.

The Earth's core is composed chiefly of iron. What is the temperature of an iron atom moving at the Earth's escape speed?

```      Escape speed (km/s)   Core composition
Sun        618.             Protons, electrons, helium
Earth       11.2            Iron
Mars         5.03           Iron
Moon         2.38           Iron
Ceres         .51           Iron
```

Virial theorem

A typical globular cluster consists of millions of stars. If you measure the total gravitational and kinetic energy of the stars, you will find that

```Total gravitational energy  =  -2 * Total kinetic energy
```
just like for a single satellite on a circular orbit.

Suppose a system consists of a set of objects interacting by a potential. If the system has reached a long-term equilibrium then the above statement about energies is true, no matter how chaotic the orbits of the objects. This is the "Virial theorem". It also applies if additional forces are involved. For example, the protons in the sun interact by both gravity and collisions and the virial theorem holds.

```Gravitational energy of the sun  =  -2 * Kinetic energy of protons in the sun
```

Spin

Frequency-force relationship

Position, velocity, and acceleration

Suppose an object undergoes periodic motion.

```M    =  Mass of the object
S    =  Amplitude of the oscillation
t    =  Time
T    =  Period of the oscillation
F    =  Frequency of the oscillation
S    =  Position of the object as a function of time
=  S sin(2 Pi t/T)
V    =  Peak velocity of the object
=  2 Pi S / T
A    =  Peak acceleration of the object
=  4 Pi2 S / T2
Force=  Peak force during the oscillation
=  M A
=  4 Pi2 M S / T2
```
The peak force and the period are related by
```Force = 4 Pi2 M S / T2
```
The brain is good at measuring frequencies. Whenever possible, convert force measurements into frequency measurements.

If you shake a sword like a pendulum then we can translate force to torque and mass to moment of inertia.

```R  =  Distance of the rotating object from the axis of rotation.
I  =  Moment of inertia
=  M R2

Torque  =  4 Pi2 I S / (R T2)
```

Gyroscope

Gyroscope
A gyroscope maintains its internal spin axis regardless of the motion of the exterior object.
Torque on a gyroscope
Torque on a spinning top

For a spinning top, the forces of gravity and the ground on the top generate a torque, which causes the spin to precess.

Precession of the spin axis from an external torque.

Foucault's gyroscope
Hubble telescope

The gyroscope was invented by Serson in 1743 and it was used by Foucault in 1852 to measure the Earth's spin.

The Hubble telescope uses gyroscopes to orient itself. The gyroscopes periodically fail, requiring a servicing mission.

In the 1860s, the advent of electric motors made it possible for a gyroscope to spin indefinitely; this led to the first prototype heading indicators and the gyrocompass. The first functional gyrocompass was patented in 1904 by German inventor Hermann Anschutz-Kaempfe. The American Elmer Sperry followed with his own design later that year, and other nations soon realized the military importance of the invention -- in an age in which naval prowess was the most significant measure of military power -- and created their own gyroscope industries. The Sperry Gyroscope Company quickly expanded to provide aircraft and naval stabilizers as well.

Coriolis acceleration

```Omega =  Rotation speed in radians/second
V     =  Velocity of the object
A     =  Coriolis acceleration
=  -2 Omega V
```

Centripetal potential

For a central force, we can isolate the radial component of the motion. Using conservation of angular momentum we can write the radial force in terms of angular momentum and then convert it into an effective potential for centripetal acceleration.

For a satellite orbiting a central potential,

```Gravity constant      =  G
Mass of central object=  M
Mass of satellite     =  m
Radius                =  R                  Distance of satellite from the center
Tangential velocity   =  V                  Velocity transverse to the radius vector
Angular momentum      =  L  =  m R V
Centripetal accel     =  AC =  V2 R-1  =    L2 m-2 R-3  =  -∂R ΦC
Centripetal potential =  ΦC =  .5 L2 m-2 R-2
Gravity acceleration  =  AG =  -G M R-2  =  -∂R ΦG
Gravity potential     =  ΦG =  -G M R-1
Total potential       =  Φ  =  ΦG + ΦC  =   - G M R-1 + L2 m-2 R-2
```

Non-commutivity of rotations

Suppose you rotate an airplane 90 degrees upward in the pitch direction and then roll it 90 degrees in the rightward direction, and then take note of its final position. Now start over and do the rotations in reverse order. The final orientation depends on order.

A rotation can be expressed as a matrix and matrix multiplication is non-commutative. In the following, "!=" stands for "not necessarily equal to".

```A   =  Rotation matrix (3x3 matrix)
Qo  =  Original orientation of an object (3D vector)
Qf  =  Final orientation of an object after rotating using matrix "A". (3D vector)
I   =  Identity matrix (diagonal elements equal to 1, off-diagonal elements equal to 0)

Qf  =  A * Qo            (Using "A" to rotate from "Qo" to "Qf")
Qo  =  I * Qo            (The identity matrix does not rotate an object)

Ai  =  Inverse matrix of "A"
B   =  3x3 rotation matrix that is not necessarily the same as "A".
Bi  =  Inverse matrix of "B"

A * Ai  =  Ai * A  =  I
B * Bi  =  Bi * B  =  I

B * A                       (This stands for rotating by "A" and then rotating by "B")
A * B                       (This stands for rotating by "B" and then rotating by "A")

B * A  !=  A * B            (If there are 2 rotations then order matters)

Ai * Bi * B * A  =  I       (Doing 2 rotations & then unwinding them in order restores
the original orientation)
B * Ai * B * A  !=  I       (Doing 2 rotations & then unwinding them out of order does
not necessarily restore the original orientation)

```

Appendix

Drag coefficient and Mach number

Commercial airplanes fly at Mach .9 because the drag coefficient increases sharply at Mach 1.

Turbulence and Reynolds number

The drag coefficient depends on speed.

```Object length    =  L
Velocity         =  V
Fluid viscosity  =  Q                  (Pascal seconds)
=  1.8⋅10-5 for air
=  1.0⋅10-3 for water
Reynolds number  =  R   =  V L / Q      (A measure of the turbulent intensity)
```
The drag coefficient of a sphere as a function of Reynolds number is:

Golf balls have dimples to generate turbulence in the airflow, which increases the Reynolds number and decrease the drag coefficient.

Drag coefficient and Reynolds number
```Reynolds  Soccer  Golf   Baseball   Tennis
number
40000   .49    .48      .49       .6
45000   .50    .35      .50
50000   .50    .30      .50
60000   .50    .24      .50
90000   .50    .25      .50
110000   .50    .25      .32
240000   .49    .26
300000   .46
330000   .39
350000   .20
375000   .09
400000   .07
500000   .07
800000   .10
1000000   .12             .35
2000000   .15
4000000   .18    .30
```
Data

Drag differential equation

For an object experiencing drag,

```Drag coefficient  =  C
Velocity          =  V
Fluid density     =  D
Cross section     =  A
Mass              =  M
Drag number       =  Z  =  ½ C D A / M
Drag acceleration =  A  =  -Z V2
Initial position  =  X0 =  0
Initial velocity  =  V0
Time              =  T
```
The drag differential equation and its solution are
```A  =  -Z V2
V  =  V0 / (V0 Z T + 1)
X  =  ln(V0 Z T + 1) / Z
```

Current density

Current density
Resistor

```                  Electric quantities             |                Thermal quantities
|
Q  =  Charge                 Coulomb              |   Etherm=  Thermal energy          Joule
I  =  Current                Amperes              |   Itherm=  Thermal current         Watts
E  =  Electric field         Volts/meter          |   Etherm=  Thermal field           Kelvins/meter
C  =  Electric conductivity  Amperes/Volt/meter   |   Ctherm=  Thermal conductivity    Watts/meter/Kelvin
A  =  Area                   meter^2              |   A     =  Area                    meter^2
Z  =  Distance               meter                |   Z     =  Distance                meter^2
J  =  Current flux           Amperes/meter^2      |   Jtherm=  Thermal flux            Watts/meter^2
=  I / A                                       |         =  Ittherm / A
=  C * E                                       |         =  Ctherm * Etherm
V  =  Voltage                Volts                |   Temp  =  Temperature difference  Kelvin
=  E Z                                         |         =  Etherm Z
=  I R                                         |         =  Itherm Rtherm
R  =  Resistance             Volts/Ampere = Ohms  |   Rtherm=  Thermal resistance      Kelvins/Watt
=  Z / (A C)                                   |         =  Z / (A Ct)
H  =  Current heating        Watts/meter^3        |
=  E J                                         |
P  =  Current heating power  Watts                |
=  E J Z A                                     |
=  V I                                         |
```

Electrical and thermal conductivity of a wire
```L  =  Length of wire            meters
A  =  Cross section of wire     meters^2
_______________________________________________________________________________________________________
|
Electric quantities             |                Thermal quantities
|
Q  =  Charge                 Coulomb              |   Etherm=  Thermal energy          Joule
I  =  Current                Amperes              |   Itherm=  Thermal current         Watts
E  =  Electric field         Volts/meter          |   Etherm=  Thermal field           Kelvins/meter
C  =  Electric conductivity  Amperes/Volt/meter   |   Ctherm=  Thermal conductivity    Watts/meter/Kelvin
A  =  Area                   meter^2              |   A     =  Area                    meter^2
Z  =  Distance               meter                |   Z     =  Distance                meter^2
J  =  Current flux           Amperes/meter^2      |   Jtherm=  Thermal flux            Watts/meter^2
=  I / A                                       |         =  Ittherm / A
=  C * E                                       |         =  Ctherm * Etherm
V  =  Voltage                Volts                |   Temp  =  Temperature difference  Kelvin
=  E Z                                         |         =  Etherm Z
=  I R                                         |         =  Itherm Rtherm
R  =  Resistance             Volts/Ampere = Ohms  |   Rtherm=  Thermal resistance      Kelvins/Watt
=  Z / (A C)                                   |         =  Z / (A Ct)
H  =  Current heating        Watts/meter^3        |
=  E J                                         |
P  =  Current heating power  Watts                |
=  E J Z A                                     |
=  V I                                         |
```

Continuum
```Continuum quantity       Macroscopic quantity

E             <->      V
C             <->      R = L / (A C)
J = C E       <->      I = V / R
H = E J       <->      P = V I
```

Superconductors

```                 Critical    Critical  Type
temperature  field
(Kelvin)    (Teslas)

Magnesium-Boron2     39        55       2   MRI machines
Niobium3-Germanium   23.2      37       2   Field for thin films.  Not widely used
Magnesium-Boron2-C   34        36           Doped with 5% carbon
Niobium3-Tin         18.3      30       2   High-performance magnets.  Brittle
Vanadium3-Gallium    14.2      19       2
Niobium-Titanium     10        15       2   Cheaper than Niobium3-Tin.  Ductile
Niobium3-Aluminum

Technetium           11.2               2
Niobium               9.26       .82    2
Vanadium              5.03      1       2
Tantalum              4.48       .09    1
Lead                  7.19       .08    1
Lanthanum             6.3               1
Mercury               4.15       .04    1
Tungsten              4                 1    Not BCS
Tin                   3.72       .03    1
Indium                3.4        .028
Rhenium               2.4        .03    1
Thallium              2.4        .018
Thallium              2.39       .02    1
Aluminum              1.2        .01    1
Gallium               1.1
Protactinium          1.4
Thorium               1.4
Thallium              2.4
Molybdenum             .92
Zinc                   .85       .0054
Osmium                 .7
Zirconium              .55
Ruthenium              .5
Titanium               .4        .0056
Iridium                .1
Lutetium               .1
Hafnium                .1
Uranium                .2
Beryllium              .026
Tungsten               .015

HgBa2Ca2Cu3O8       134                 2
HgBa2Ca Cu2O6       128                 2
YBa2Cu3O7            92                 2
C60Cs2Rb             33                 2
C60Rb                28         2       2
C60K3                19.8        .013   2
C6Ca                 11.5        .95    2    Not BCS
Diamond:B            11.4       4       2    Diamond doped with boron
In2O3                 3.3       3       2
```
The critical fields for Niobium-Titanium, Niobium3-Tin, and Vanadium3-Gallium are for 4.2 Kelvin.

All superconductors are described by the BCS theory unless stated otherwise.

```         Boiling point (Kelvin)

Water      273
Ammonia    248
Freon R12  243
Freon R22  231
Propane    230
Acetylene  189
Ethane     185
Xenon      165.1
Krypton    119.7
Oxygen      90.2
Argon       87.3
Nitrogen    77.4     Threshold for cheap superconductivity
Neon        27.1
Hydrogen    20.3     Cheap MRI machines
Helium-4     4.23    High-performance magnets
Helium-3     3.19
```
The record for Niobium3-Tin is 2643 Amps/mm^2 at 12 T and 4.2 K.

Titan has a temperature of 94 Kelvin, allowing for superconducting equipment. The temperature of Mars is too high at 210 Kelvin.

Superconducting critical current

The maximum current density decreases with temperature and magentic field.

Maximum current density in kAmps/mm2 for 4.2 Kelvin (liquid helium):

```
Teslas               16    12     8      4    2

Niobium3-Tin         1.05  3
Niobium3-Aluminum           .6   1.7
Niobium-Titanium            -    1.0    2.4   3
Magnesium-Boron2-C          .06   .6    2.5   4
Magnesium-Boron2            .007  .1    1.5   3

```
Maximum current density in Amps/mm2 for 20 Kelvin (liquid hydrogen):
```
Teslas               4     2

Magnesium-Boron2-C   .4   1.5
Magnesium-Boron2     .12  1.5
```

History of superconductivity
```1898  Dewar liquefies hydrogen (20 Kelvin) using regenerative cooling and
his invention, the vacuum flask, which is now known as a "Dewar".
1908  Helium liquified by Onnes. His device reached a temperature of 1.5 K
1911  Superconductivity discovered by Onnes.  Mercury was the first superconductor
found
1935  Type 2 superconductivity discovered by Shubnikov
1953  Vanadium3-Silicon found to be superconducting, the first example of a
superconducting alloy with a 3:1 chemical ratio.  More were soon found
1954  Niobium3-Tin superconductivity discovered
1955  Yntema builds the first superconducting magnet using niobium wire, reaching
a field of .7 T at 4.2 K
1961  Niobium3-Tin found to be able to support a high current density and
magnetic field (Berlincourt & Hake). This was the first material capable of
producing a high-field superconducting magnet and paved the way for MRIs.
1962  Niobium-Titanium found to be able to support a high current density and
magnetic field.  (Berlincourt & Hake)
1965  Superconducting material found that could support a large
current density (1000 Amps/mm^2 at 8.8 Tesla)
(Kunzler, Buehler, Hsu, and Wernick)
1986  Superconductor with a high critical temperature discovered in a ceramic
(35 K) (Lanthanum Barium Copper Oxide) (Bednorz & Muller).
More ceramics are soon found to be superconducting at even higher temperatures.
1987  Nobel prize awarded to Bednorz & Muller, one year after the discovery of
high-temperature superconductivity.  Nobel prizes are rarely this fast.
```

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