Main site of science textbooks
Crowdfunding site for the free
online science textbooks project

Physics
Dr. Jay Maron


Units    Index of equations   

Velocity and acceleration    Newton's laws and momentum    Gravity    Energy    Circular motion    Friction    Spin    Pressure    Buoyancy    History   

Waves    Music    Overtones    Resonance   

Electromagnetism    Electric force    Magnetic force    Electric field    Magnetic field    Batteries    Resistance   

Atoms    Chemistry    Particles    Nuclei    Fusion    Fission   

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             = θ =  X / R
Velocity    = V              Angular velocity  = ω =  V / R
Acceleration= A              Angular accel.    = α =  A / R
Mass        = M              Moment of inertia = I =  M * R2
Force       = F              Torque            = Γ =  F * R
Momentum    = Q              Angular momentum  = L =  Q * R
Energy      = E
Time        = T
Radius      = R

                 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:           Γ = I α     L = I ω = Q R     E = .5 I ω2 = Γ θ    A = ω2 R       θ = ω T    α = ω T


Energy = .5 M V2 + .5 I ω2 + 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  =         =   V dT
Velocity     =  V  =  ∂X/∂T  =   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  =   F dX  =   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  =  2π R / V    =   30 s
Centripetal acceleration              =  A  =  V2 / R      =   10 m/s2
Spin radius of the spaceship          =  R  =  T2 A / (2π)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  =   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          = √2 (G M / R)1/2


Escape velocity  =  √2 * Circular orbit velocity


      Escape velocity   Circular orbit velocity
          (km/s)            (km/s)

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
Radius                  =  R
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

        Density  Radius   Gravity
        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(θ)
The force of gravity parallel to the ramp surface is
Framp = Fgrav sin(θ)
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(θ) = C Fgrav cos(θ)

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

Radians


θ  =  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  =  θ R

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

Full circle  =  360 degrees  = 2 π radians

1 radian  =  57.3 degrees
1 degree  = .0175 radians

Polar coordinates

Radius  =  R
Angle   =  θ
X coordinate  =  X  =  R cos(θ)
Y coordinate  =  Y  =  R sin(θ)

Spin and angular velocity

T  =  Time in seconds
θ  =  Angle  (dimensionless)
ω  =  Angular velocity in radians/second = 1.75 in the animated figure

Wikipedia: circular motion


Spin frequency
ω  =  Angular frequency
F  =  Spin frequency in Hertz or 1/second

ω  =  2 Pi F

1 Hertz  =  1 revolution/second  =  2π 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               =  ω   = 3.49  radians/second     (= 33.33 revolutions per minute)
Velocity of the outer edge  V  =  ω 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             =  ω  =  V/R   =   67 radians/second
Ball spin frequency in Hertz  =  F  =  ω/(2π)= 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
θ  =  Angle pointing to X

X  = θ R
Every linear quantity has a corresponding spin quantity, and every linear equation has a crresponding spin equation. By changing variables from X to θ we can translate a linear equation into a spin equation.

Linear variables             Spin variables

Position    = X              Angle             = θ =  X / R
Velocity    = V              Angular velocity  = ω =  V / R
Acceleration= A              Angular accel.    = α =  A / R
Mass        = M              Moment of inertia = I =  M * R2
Force       = F              Torque            = Γ =  F * R
Momentum    = Q              Angular momentum  = L =  Q * R
Energy      = E
Time        = T
Radius      = R

                 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:           Γ = I α     L = I ω = F T R    E = .5 I ω2 = Γ θ    A = ω2 R       θ = ω T    α = ω 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
Γ   = M R2 A/R   L   = M R2 V/R     E = .5 I ω2            E = Γ ω        θ   = ω T          ω  =  α T
Γ   = I α        L   = I ω

Angular velocity and angular acceleration
Angle             = θ  =  X / R
Angular velocity  = ω  =  V / R
Angular accel.    = α  =  A / R


Constant velocity:        X = V T
Constant spin:            θ = ω T

Constant acceleration:    V = A T
Constant angular accel.:  ω = α 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:  Γ = I α = F R

Momentum:                          Q = M V
Angular momentum:                  L = I ω = M V R

Energy:                            E = .5 M V2
Angular energy:                    L =  .5 I ω2
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 ω2

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
Radius             =  R
Mass               =  M
Angular frequency  =  ω  =  V/R
Moment of inertia  =  I  =  .4 M R2
Kinetic energy     =  Ek  =  .5 M V2
Spin energy        =  Es  =  .5 I ω2  =  .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             =  Γ  =  R C M g
Moment of inertia          =  I  = .4 M R2
Angular acceleration       =  α  =  Γ / I  =  2.5 C g / R

Time sliding               =  T
Rolling angular velocity   =  ω  =  α T  =  2.5 C g T / R
Ball velocity when rolling =  V  =  Vo + A T
                                 =  Vo - C g T
                                 =  Vo - .4 C g ω R / C / g
                                 =  Vo - .4 ω 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 * 2π)
                =  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
γ     =  Adiabatic constant             =  7/5 for air
Vtherm=  Thermal speed                   =  544 meters/second
Vsound=  Sound speed                     =  343 meters/second at 20 Celsius
     =  (γ P / D)1/2
     =  (γ/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
   =  (Miceice - Mwaterwater) / (Miceice)
   =  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.

According to legend Bach used a supersized viola, the "Viola Pomposa"



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.

Online tone generator


Standing waves

Two waves traveling in opposite directions create a standing wave.

Waves on a string simulation at phet.colorado.edu


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

Wikipedia: Overtones

Overtone simulation at phet.colorado.edu


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


The Chladni experiment

Chladni's original experiment


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


Audio file: Creating a triangle wave by adding 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.

Wikipedia: Damping


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).

Tacoma Narrows bridge collapse

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  =  23 e                    Form used in physics
   =  R Mol T / Vol            Form used in chemistry
   =  k N   T / Vol
   =  13 N M Vth2/ Vol
   =  13 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  =  32 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  =  13  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.


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

Circuits

Battery

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

Voltage                           =  V  =  I R
Charge                            =  Q
Energy of a charge at voltage "V" =  E  =  Q V
Current                           =  I
Power                             =  P  =  Q V / T  =  I V
Resistance                        =  R

Ohm's law:  V =  I R

Power:      P  =  I V  =  I2 R  =  V2 / R

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

Batteries

Typical values for battery energies are:

                      Energy    Energy     Time     Power
                     (kJoule)  (WattHour)  (hour)   (Watt)

Smartphone               28.7     8         10        .80
Tablet                   57.6    16         10       1.60
Macbook air             129      36          5       7.2
Small external battery   42      11          -        -
Large external battery  160      44          -        -
All lithium batteries have a voltage of 3.7 Volts. If a lithium battery delivers 1 Amp of current for 1 hour,
Voltage of a lithium battery  =  V                    =  3.7 Volts
Energy of a lithium battery   =  E                    =  13320 Joules
Current                       =  I                    =  1 Amp
Time the battery lasts        =  T                    =  3600 seconds
Power                         =  P  =  E / T  =  V I  =  3.7 Watts
Battery energy is often quoted in WattHours.
1 WattHour  =  1 Watt * 3600 seconds  =  3600 Joules

Elements

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
Mass            = M              = 9.785e-26 kg
Number density  = N   =  D / M   = 9.096e28 atoms/m3
Cube volume     = Υcube= 1 / N    = 1.099e-29 m3          Volume/atom if the atoms are cubes
Cube length     = L   =  Υ1/3cube = 2.22e-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 = 8.138e-30 m3 = 43πR3  Volume/atom if the atoms are spheres
Sphere radius   = R              = 1.25e-10 m

Strength and melting point
Dot size   = (Shear Modulus)1/3      A measure of a material's strength
Dot color  =  Melting point

White  =  Highest melting points
Red    =  Lowest melting points
Blue   =  Elements that are a gas or a liquid at room temperature and pressure.
          Liquids and gases have a shear modulus of 0.
Rocket nozzles are made from materials with a high melting point, a high shear strength, and a high atomic mass. Tungsten is the element of choice, especially because it's vastly cheaper than Rhenium, Osmium, and Iridium.

For carbon, the values are given for diamond form.

Shear data Melt data


Strength and density
Color     =  Shear Modulus
Dot size  =  Density
Shear data Density data
Strength to weight ratio

Color      =  Shear Modulus / Density          A measure of a material's "strength to weight" ratio
Dot size   =  Density
The metals with the highest strength to weight ratio are Chromium, Ruthenium, and Beryllium.

Chromium is common in the Earth's crust and Ruthenium is rare.

Shear data Density data


High temperature materials
          Density Melt  Boil  Young Young   $/kg  ppm in metallic
          g/cm^3   K     K     GPa  /rho          asteroid
Tungsten   19.25  3693  5828   411   21.4     50     ~ 1
Rhenium    21.0   3459  5869   463   22.0   4600     ~ 1
Osmium     22.59  3306  5285   550   24.3  12000       2
Tantalum   16.7   3290  5731   186   11.1    400     ~  .5
Molybdenum 10.28  2896  4912   329   31.0     21     ~10
Niobium     8.75  2750  5017   105   12.0     40     ~ 3
Iridium    22.4   2739  4701   528   23.6  14000       2
Ruthenium  12.45  2607  4423   447   35.9   5500       5
Hafnium    13.31  2506  4876    78    5.9    500     ~ 1
Rhodium    12.41  2237  3968   380   30.6  90000       2
Thorium    11.7   2115  5061    79    6.8            ~  .5
Platinum   21.45  2041  4098   168    7.8  55000       9
Vanadium    6.0   2183  3680   128   21.3              0
Chromium    7.19  2180  2944   279   38.8              0
Zirconium   6.52  2128  4682    88   14.5              0
Titanium    4.51  1941  3560   116   25.7              0
Palladium  12.02  1828  3236   121   10.1            ~ ?
Iron        7.87  1811  3134   211   26.8         100000
Nickel      8.91  1728  3186   200   22.4          10000
Cobalt      8.90  1768  3200   209   23.5         ~ 1000
Uranium    19.1   1405  4404   208   10.9              1
Beryllium   1.85  1560  2742   287  155.1              0
Manganese   7.21  1519  2334   198   27.5              0
Aluminum    2.70   933  2792    70   25.9              0
Magnesium   1.74   923  1363    45   25.9              0

Conductivity

Red:   Lowest conductivities
White: Highest conductivities
If an element's conductivity is more than 2 orders of magnitude less than that of silver then it is left blank.

Resistivity data


History

Discovery
        Earliest   Shear    Melt  Density
        known use  Modulus  (K)   (g/cm^3)
        (year)     (GPa)
Wood    < -10000     15        -    .9
Rock    < -10000
Carbon  < -10000      -
Diamond < -10000    534     3800   3.5
Gold    < -10000     27     1337  19.3
Silver  < -10000     30     1235  10.5
Sulfur  < -10000
Copper     -9000     48     1358   9.0
Lead       -6400      6      601  11.3
Brass      -5000    ~40                    Copper + Zinc
Bronze     -3000    ~40                    Copper + Tin
Tin        -3000     18      505   7.3
Antimony   -3000     20      904   6.7
Mercury    -2000      0      234  13.5
Iron       -1200     82     1811   7.9
Arsenic     1649      8     1090   5.7
Cobalt      1735     75     1768   8.9     First metal discovered since iron
Platinum    1735     61     2041  21.4
Zinc        1746     43      693   7.2
Tungsten    1783    161     3695  19.2
Chromium    1798    115     2180   7.2


Stone age    antiquity
Copper age    -9000
Bronze age    -3500
Iron age      -1200
Bronze holds an edge better than copper and it is more corrosion resistant.

Gold was the densest known element until the discovery of platinum in 1735. This made it impossible to counterfeit as a currency.


Metals known since antiquity
Horizontal axis:  Density
Vertical axis:    Shear modulus / Density       (Strength-to-weight ratio)

Metals

Horizontal axis:  Density
Vertical axis:    Shear modulus / Density       (Strength-to-weight ratio)
Metals with a strength-to-weight ratio less than lead are not included, except for mercury.
Abundance

Solar abundance

Dot size =  log(SolarAbundance)
Elements with a dot size of zero have no stable isotope.

Solar abundance data


Price

Color    =  Price per kilogram
Dot size = -log(SolarAbundance)      The smaller the dot, the more abundant the element.
Price 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


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

           Symbol  Protons  Neutrons  Half life

Electron      e       0        0      stable
Neutron       N       0        1      886 seconds
Proton        P       1        0      stable
Deuterium     D       1        1      stable
Tritium       T       1        2      12.3 years
Helium-3     He3      2        1      stable
Helium-4     He4      2        2      stable
Lithium-6    Li6      3        3      stable
Lithium-7    Li7      3        4      stable
Carbon-12    C12      6        6      stable
Oxygen-16    O16      8        8      stable
Uranium-235  U235    92      143       .70 billion years
Uranium-238  U238    92      146      4.47 billion years
An element has a fixed number of protons and a variable number of neutrons. Each neutron number corresponds to a different isotope.

Uranium is the heaviest naturally-occuring element.

Each number corresponds to the number of protons.

Teaching simulation for isotopes at phet.colorado.edu


Radiation

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).


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

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

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

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 =  D                          =  1.22 kg/m3
Drag coef   =  Cdrag                      =    .5                 For a sphere
Spin coef   =  Cspin                      =     1                 For a sphere and for S < .25
Drag force  =  Fdrag = ½ Cdrag D Area V2   =  13.0 N
Spin force  =  Fspin = ½ Cspin D Area V2 S =   6.5 N
Drag accel  =  Adrag                      =   224 m/s2
Spin accel  =  Aspin                      =   112 m/s2

Rolling drag

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

                             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
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

Cars

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

Vehicle drag

Leitras velomobile
Loremo
Edison 2
BMW i8

Nissan GTR
Lamborghini Diablo
Ford Escape Hybrid
Hummer H2

                  Drag   Area   Drag*  Engine  Mass  Top    Top   100 kph  Year
                  coef          Area                speed  speed   time
                          m2     m2     kWatt   kg   k/hr   m/s   s

Aptera 2            .15   1.27   .190    82    820   137   38.1        2011
Loremo              .20   1.25   .250    45          100   27.8        2009
Edison 2 VLC        .16   1.62   .259          450                     2010
Volkswagen XL1      .189  1.48   .279    55    795   158*        11.9  2011
BMW i8              .26   2.11   .548   260   1539   250*  69.4   4.4  2015
Nissan GTR          .27   2.09   .565   357   1740   314   87.2   3.4  2008
Lamborghini Diablo  .31   1.85   .573   362   1576   325   90.3        1995
Tesla Model S       .24   2.40   .576   568   2000   249   69.2   3.0  2012
Toyota Prius        .24   2.40   .576                                  2016
Chevrolet Volt      .281  2.21   .622                             9.5  2014
Mercedes Benz CLA   .30   2.17   .650                                  2013
Nissan Leaf         .29   2.50   .725                                  2012
Ford Escape hybrid  .40   2.62  1.05                                   2005
Hummer H2           .57   4.32  2.46    242   2900                     2003

*: The top speed is electronically limited
The Saab 900, last of the boxy cars

Energy and power

Power sources
                          MJ/kg   kWatt/kg   kWatts   kg

Diesel                    48        -
Gasoline                  45        -
Electric motor, maximum     -     10        200      19.9     EMRAX268 Brushless AC
Turbofan jet engine               10.0       83.2             GE90-115B Brayton
Supercapacitor, BaTiO3     1.47    8.0
ElectriFly brushless DC            7.78       1.04     .133   ElectriFly GPMG5220
Gasoline engine (BMW)              7.5      690               BMV V10 3L P84/5 2005
Battery, LiFePO4            .39    3.3
Model aircraft engine       -      2.8
Battery, lithium-ion        .95    1.5                        Typical commercial battery
Fuel cell, Honda            -      1.0
Battery, Li-S              1.21     .67
Typical diesel V8 turbo     -       .65
Battery, aluminum-air      4.68     .13
Solar cell, space station   -       .077
Solar cell, multilayer      -       .065
Solar cell, polycrystal     -       .040
Nuclear battery, Pu-238  2265000    .0051      .285  56       Teledyne Pu-238 GPHS-RTG

Power/Mass ratio
                     Engine   Engine  Engine  Vehicle  Vehicle   Year
                     (kWatt)   (hp)   (kW/kg)  (ton)   (kW/kg)

Human, sustained          .42     .6   .0067   .062    .0068            Alberto Contador
Human, maximum           2.0     3     .025    .080    .025
Sled dog, sustained       .16     .2   .004    .04     .004    -30000
Horse, sustained          .7     1     .004    .5      .004     -4000
Horse, maximum          11      15     .022    .5      .022     -4000
Car, first prototype      .56     .8   .0021   .265    .0021     1886   Benz Motorwagon. .95L
Car, Ford Model T       15      20             .54     .028      1908   2.9 Litres
Car, Mazda RX-8        184     247    1.5     1.34     .137      2003
Car, Toyota Prius       73      98            1.38     .053      2010   1.8 L
Car, Honda Accord      202     271            1.63     .124      2011
Car, Chevy Corv. C6    321     430            1.44     .223      2005
Car, Porche 911 GT2    390     523            1.44     .271      2007
Car, Lamborghini       493     661            1.55     .318      2009   Murcielago
Car, BMW 7             327     439            2.25     .145      2006   760Li 6 L V12
Car, Ferrari FXX       597     801            1.155    .517      2005
Car, Formula-1         690     925    7.5      .600   1.15       2005   Williams. BMW engine
Car, Hummer H1         224     300            3.56     .063      2006   6.6 L V8
Car, electric           30      40            1.038    .029      2008   Th!nk City
Car, electric           26      35             .635    .041      2008   Tata Nano. .62 L
Car, electric           47      63            1.08     .044      2009   Mitsubishi i MiEV
Car, electric, Tesla   215     288            1.235    .174      2011   Roadster
Car, drag racer       5960    8000            1.043   5.72       2008   John Force Racing
Tank, M1 Abrams       1120    1500           55.7      .020      1980
Motorcycle, Kawasaki    26      35             .182    .143      1987   KLR650 .65 L
Motorcycle, Suzuki      50      67             .194    .258      2004   V-Strom 650 .65 L
Motorcycle, Honda      177     237             .148   1.19       2005   Honda RC211V
Plane, Wright Bros       9      12             .274    .033      1903
Plane, Junkers Jumo    647     867    1.10    6.7      .097      1934
Plane, B-50 Bomber    3210    4300    1.83   38.4      .084      1944
Plane, B-29 Bomber    2540    3400    2.09   33.8      .075      1937
Plane, 747-400     4x44700  4x59900   5.67  178.8     1.00       1971
Plane, 777         2x83200 4x111526  10.0   134.8     1.23       1992
Space shuttle      3x53700   72000  153      68.6     2.35       1981   H2 + O2 rocket
Train, steam, 1829     015             .0035  4.32     .0035     1829   Stephenson's Rocket
Cargo ship           80100  108920     .030  55400     .0014     2006   Emma Maersk
Snowmobile, Polaris    115     154             .221    .523      2009
Model aircraft         930       1.2  2.8                               O.S. Engines 49-PL Type 2
Typical V8 turbo       250     340     .65     .38
Electric motor, max                  10
Nuclear, Pu-238                        .00051                           Galileo spacecraft
1 horsepower = 745.7 Watts
Engine efficiency
Gasoline engine      .15
Diesel engine        .20
Human muscles        .22
Electric car engine  .80
Biomass plant        .25
Natural gas plant    .35
Solar cell           .20     Crystalline type
Solar cell           .40     Multilayer type
Turboprop, Mach .4   .80     Turboprops work up to Mach .5
Turbojet, Mach .4    .40
Turbofan, Mach .4    .68
Turbojet, Mach .9    .77
Turbofan, Mach .9    .90
For an electric vehicle the overall efficiency is similar to that of a diesel engine.
Overall efficiency  =  Power plant efficiency  *  Vehicle efficiency  =  .35 * .80 =  .28

Energy loss
Energy fraction    High speed    Low speed    Bicycle
                     car           car       at 10 m/s

Air drag            .14            .08       .16
Rolling drag        .08            .10       .03
Engine loss         .72            .75       .78
Drivetrain loss     .06            .07       .01
For a cyclist, "engine loss" is the cyclist's muscle energy loss.

For a bicycle, the "drivetrain" is the chain, gears, and derailleurs. The loss from derailleurs is greater than the loss from gears, which is why sprinting bikes have only one gear.

Wiki: Bicycle performance


Fuel efficiency
                       Speed   l/km   l/km/   Passengers
                        m/s           person

Walk                    1.4     .0065  .0065   1         60 Watts
Run                             .009   .009    1
Bike                    4.4     .0032  .0032   1
Bike, aerodynamic      13.9     .0005  .0005   1
Car, solar power                .067   .067    1
Car, electric, Tesla            .015   .004    4
Car, electric, GEM NER 10.8     .012   .003    4
Car, electric, GE EV1           .026   .006    4
Car, electric, Volt             .026   .006    4
Car, VW Bluemotion              .038   .010    4
Car, Honda Insight              .049   .012    4
Car, Toyota Prius               .051   .013    4
Car, Cadillac Wagon             .17    .028    6         6.2L engine
Car, Bugatti Veyron             .24    .12     2
Train, Switzerland              .17    .0026  65
Train, Japan                    .65    .011   59
Plane, Dieselis        44.4     .019   .010    2
Plane, Pipistrel Sinus 62.5     .048   .024    2
Plane, Tecnam Sierra   65.8     .072   .036    2
Plane, DynAero MCR-4S  61.1     .088   .022    4         100 hp
Plane, Boeing 747-400                 3.1    660
Plane, Concorde                      16.6    128
Plane, Airbus A380                    3.0    835
Ship, Queen Elizabeth        300       .17  1777
Ship, Cargo            12.8 1070       -       -         Emma Maersk. 170000 tons
Helicopter, Sikorsky   72.2    1.43    .12    12         Model S-76
1 litre gasoline = 31.7 MJoules
U.S. transportation averages
                  MJ/km/    Passengers
                  person    per vehicle

Train, Switzerland  .085       65
Train, Japan        .35        59
Car, electric      1.2          1.5
Train, city        1.60        30.9
Train, intercity   1.65        24.5
Motorcycle         1.61         1.16
Air                1.85        99.3
Car, gasoline      2.32         1.55
Bus                2.78         9.2
Taxi              10.3          1.55

Freight
                  MJ/km/ton

Ship, U.S. local    .16
Ship, ocean cargo   .22     Emma Maersk. 170000 tons
Train               .21
Truck              2.43
Air                6.9

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 cars

Range

Tesla Model S

The range of a typical electric car is

Battery energy in MJoules  =  E
Battery cost in dollars    =  B
Battery energy/$           =  Z  =  .0070 MJoules/$
Drag speed                 =  V0  ≈  17 meters/second
Range in kilometers        =  9.1 E / [1 + (V/V0)2]
                           = .064 B / [1 + (V/V0)2]
The drag speed V0 is the speed for which air drag equals the rolling drag and is typically in the range of 17 m/s. To calculate V0 for a typical electric car,
Air drag coefficient      =  Cd         =  .25
Aerodynamic cross-section =  A          =  2.0 m2
Air density               =  D          = 1.22 kg/m3
Car velocity              =  V          =   15 m/s      (City speed. 34 mph)
Aerodynamic drag force    =  Fd = ½CdADV2=   69 Newtons
Rolling drag coefficient  =  Cr         =.0075          (Typical value for car tires)
Car mass                  =  M          = 1200 kg
Gravity constant          =  g          =  9.8 m/s2
Rolling drag force        =  Fr = Cr M g =  88 Newtons
Total drag force          =  F  = Fd + Fr= 157 Newtons
Setting the air drag equal to the rolling drag (Fd=Fr),
V0  =  [Cr M g / (½ Cd D A)]½  =  4.01 (Cr M / Cd A)2  =  17.0 meters/second
For a typical lithium-ion battery,
Battery mass              =  Mb           =  100 kg
Battery energy/mass       =  e            =  .60 MJoules/kg
Battery energy            =  E  = Mb e    =   60 MJoules
Battery energy/$          =  Z            =.0070 MJoules/$
Battery cost                              = 8570 $
Vehicle energy efficiency =  Q            =  .80
Vehicle energy expended   =  E  = F X / Q =   60 MJoules    (Battery energy = energy expended)
Vehicle range             =  X  = E Q / F =  306 km
Vehicle range/$                           = .036 km/$
The range in meters as a function of speed is
Range  =  X  =  E Q / (Fd + Fr)  =  E Q / [½ Cd A D V2 + Cr M g]  =  E Q / [Cr M g] / [1 + (V/V0)2]
       = .0091 E / [1 + (V/V0)2]

Electric car batteries

Mitsubishi i-MiEV
Nissan Leaf

The battery is the dominant cost in an electric car.

                  Engine  Battery  Battery  Battery  Battery  Battery  Battery  Battery  Car    Car   Car
                  power    power   energy    mass                       cost             mass  range  cost
                  kWatt    kWatt   MJoule     kg      MJ/kg    kW/kg      $      MJ/$    kg     km     $

Tesla S P85D         568    397     306       540     .57      .74     44000    .0070    2239   426  115000
Ford Focus Electric  107     92      82.8     295     .281     .31     12000    .0069    1674   122   22495
Nissan Leaf           80     80      76.7     218     .35      .37      5500    .0139    1493   172   22360
Mitsubishi i-MiEV     47     47      58       201     .288     .23                       1080   100   16345

Energy source

Tesla Roadster

The performance of a car depends on its power source. Electric cars outperform gasoline cars in every category except range.

                         Energy/Mass   Power/mass  Energy/$   Recharge    Max
                          MJoule/kg     kWatt/kg   MJoule/$   time       charges

Diesel fuel                   48          -        41          -
Lithium battery                 .60       1.5        .014     hour       104
Lithium-ion supercapacitor      .054     15                   seconds    105
Supercapacitor                  .036     10          .001     seconds    106
Double layer capacitor          .014     10                   seconds    106
Aluminum electrolyte capacitor  .0011   100                   seconds     ∞
Electric motors are simpler, more flexible, and have a larger power/mass ratio than gasoline motors.
Electric motors can reach a power/mass of 10 kWatts/kg.
Supercapacitors are a rapidly-improving technology and lithium batteries are a mature technology.
Motors

Electric motors and gasoline motors have a similar power/mass.

                        MJ/kg  kWatt/kg  kWatts  kg

Supercapacitor, Li-ion    .054  15
Electric motor, maximum   -     10       200     19.9    EMRAX268 Brushless AC
Turbofan jet engine       -     10.0      83.2    8.32   GE90-115B Brayton
Electric motor, DC        -      7.8       1.04    .133  ElectriFly GPMG5220 brushless DC
Gasoline engine (BMW)            7.5     690             BMV V10 3L P84/5 2005
Model aircraft engine     -      2.8
Battery, lithium-ion      .75    1.5
Fuel cell, Honda          -      1.0
Typical diesel V8 turbo   -       .65
Solar cell, space station -       .077

Rolling drag

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

Mass of car              =  M
Gravity coefficient      =  g  =  9.8 meters/second2
Car downward force       =  Fg =  M g
Rolling drag coefficient =  Cr
Rolling drag             =  Fr =  C Fg
The tires with the lowest rolling drag coefficient are:
Tire                  Rolling drag coefficient

Bridgestone B381          .00615
Michelin Symmetry         .00650
Michelin Tiger Paw        .00683
Bridgestone Dueller       .00700
BFGoodrich Rugged Trail   .00709
Michelin LTX              .00754
Goodyear Integrity        .00758

Railroad                  .00035       Steel wheels on steel rails
Racing bicycle tires      .0025        8 bars of pressure
Typical bicycle tires     .004
18-wheeler truck tires    .005
Typical car tire          .01
Data
Batteries

Energy        =  E               Joules
Mass          =  M               kg
Volume        =  Vol             meters3
Time          =  T               seconds      Time required for the battery to drain
Power         =  P   =  E / T    Watts        Power delivered by the battery
Energy/Volume =  Evol =  E / Vol
Energy/Mass   =  Emass=  E / M
Voltage       =  V   = 3.7       Volts        Voltage for a lithium-ion battery
Current       =  I               Amps
Power         =  P   = I V       Watts
1 WattHour    =  1 Watt            * 3600 Seconds  =   3600 Joules
1 AmpHour     =  1 Amp * 3.7 Volts * 3600 Seconds  =  13320 Joules


                 Energy   Mass   Vol   Price  Energy  Energy  Energy  Len  Wid  Hei  Density  Energy  Energy
                                              /mass    /vol     /$
                   MJ      kg    cm3     $    MJ/kg    MJ/m3   kJ/$   mm   mm   mm    g/cm3     Wh      Ah

Anker Astro E3     .137   .232   151.5   22    .59     900     6.2   137   67  16.5    1.53    10      37
Poweradd Pilot Pro .426   .862   630    130    .49     680     3.3   185  122  27.9    1.37    32     118.4
Ravpower 23000     .306   .590   466    100    .52     650     3.1   185  124  20.3    1.27    23      85.1
Samsung Galaxy S5  .0388  .038    17.6    7   1.02    2200     5.5    84   42   5.0    2.16    10.78    2.8
Samsung Galaxy S6  .042           17.2   13                    3.2    99   44   3.94           11.55    3.0
Samsung Galaxy S7  .050                                                                        13.86    3.6
Iphone 7           .040                                                                        11.1     2.9

1 kJ  =  103 Joules
1 MJ  =  106 Joules
Data for batteries from Amazon.com. The Anker, Poweradd, and Ravpower batteries are external batteries for charging phones and tablets. The IPhone battery is a 3.82 Volt lithium-polymer battery. The Galaxy S6 Edge battery is a 3.85 Volt lithium-polymer battery. Samsung Galaxy S7 batteries are renowned for exploding.
Future batteries
              MJoules   kWatts  Wh/$  Voltage  Voltage  Anode   Cathode   Recharge  Year
                /kg      /kg           (eff)    (max)

Aluminum air     4.68    .130         1.2                                    N
Zinc air         1.59                 1.1      1.6      Zn        O2         N
Lithium sulfur   1.44    .67                   2.0                           Y      Lab
LiFeS2           1.07                 1.5      1.8      Li        FeS2       N
Molten salt      1.04    .22    4.54           2.58                          Y      Lab
LiMnO2           1.01                 3.0               Li        MnO2       N
Lithium-ion       .954  1.5     2.5   3.6      3.8                           Y
LiNiCoAlO2        .79                 3.6      4.3      Graphite  LiNiCoAlO2 Y      1999
LiNiMnCoO2        .74                 3.6      4.2      Graphite  LiNiMnO2O2 Y      2008
LiCoO2            .70           2.84  3.7      4.2      Graphite  LiCoO2     Y      1991
Alkaline          .59                 1.15     1.5      Zn        MnO2       N
LiMn2O4           .54           2.84  3.9      4.2      Graphite  LiMn2O4    Y      1999
Sodium sulfur     .54                                                        Y      Lab
LiFePO4           .47    .20          3.2      3.65     Graphite  LiFePO4    Y      1996
Silver zinc       .47                 1.5      1.85     Zn        Ag2O       N
AgZn              .46                 1.5      1.86                          Y      Lab
Rechargeable alk. .4                  1.57     1.6      Zn        MnO2       Y      1992
NiMH              .36   1.0     3.41  1.2      1.3      MH        NiO(OH)    Y      1990
NiZn              .36                 1.65     1.85     Zn        NiO(OH)    Y      2009
NiH2              .23    .2           1.55              H         NiO(OH)    Y      1975
Lead acid         .144   .18   18.0   2.1      2.32     Pb        PbO2       Y      1881
NiCd              .14    .2           1.2      1.3      Cd        NiO(OH)    Y      1960
NiFe              .09    .1     5.68  6.0      6.2      Fe        NiO(OH)    Y      1901

Supercapacitors

Graphene
Supercapacitor capacitance as a function of frequency

Capacitors can deliver 10 times more power/mass than batteries and are useful for enhancing a car's acceleration. They can also be charged instantaneously and are hence useful for recovering braking energy.

Capacitance      =  C               (Farads)
Voltage          =  V               (Volts)
Energy           =  E  =  ½ C V2    (Joules)
Resistance       =  R
Max power        =  P  =  ¼ V2 / R  (Watts)
Mass             =  M               (kg)
Energy/mass      =  e  =  E/M       (Joules/kg)
Power/mass       =  p  =  P/M       (Watts/kg)
The following table lists emerging supercapacitor technologies. For reference, commercially-available supercapacitors have an energy density in the range of .036 MJoules/kg.
                               MJ/kg  kW/kg

Ruthenium dioxide + graphene   .501    128
Magnesium dioxide nanoflakes   .396
3D porous graphene electrode   .353
Aerogel                        .325     20
Curved graphene sheets         .311
Graphene                       .308
KOH + graphite oxide           .306
Ni + CoAsS + carbon aerogel    .191       .0022
Conjugated microporous polymer .191
NiOH nanoflakes + carbon tubes .182
Ruthenium dioxide              .096
Lithium-ion supercapacitor     .054     15
Carbon nanotubes               .050     37
Commercial supercapacitor      .036     10
Carbide-derived carbon         .036
Aluminum electrolyte           .00108

Recovering breaking energy

Energy recovered from breaking can be stored with supercapacitors. Example values:

Car mass                   =  M   =  800 kg
Car velocity               =  V   =  15 m/s
Car kinetic energy         =  E   =  90000 Joules
Supercapacitor energy/mass =  e   =  36000 Joules/kg
Supercapacitor mass        =  E/e =  2.5 kg
2.5 kg of supercapacitors are required.
Flying cars

Fantrainer
Fantrainer
Terra Fugia

Optica
Ducted fan

The principal challenge for flying cars is noise. There is no such thing as a quiet flying car.

Fixed wing flight is at least 6 times more efficient than helicopter flight.

The larger the propeller the less noise. The sound power of a propeller scales as the 5th power of tip speed. A flying car should have a propeller as large as possible. A single large propeller is better than multiple small propellers.

A ducted (shielded) propeller is substantially quieter than an unshielded propeller, and is more efficient in producing thrust.

Nominal configuration for a quiet flying car:

A single large ducted fan mounted on the rear
50 kWatt gasoline engine
10 kg lithium battery for takeoff and emergency power
Gyrofans and gyroscopes for stability
Wings that fold for driving
Telescoping wing section within the main wing (wings should be as long as possible)
A cockpit with a low cross-section, like a velobike. The passengers sit behind the pilot
Thin tires for a low cross section
A 10 kg vehicle parachute for emergency landing
2 kg parachutes for passengers
A total mass in the range of 500 kg


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)
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 D

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 D
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)


        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 D

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)

Orville and Wilbur Wright

Orville Wright
Wilbur Wright

The Wright brothers designed, manufactured, and sold their own line of bicycles and used the profits to fund their aircraft research.

They began by designing wings and gliders.

They were the first to use wind tunnels to test wings. Their wing design eclipsed the European designs.

They innovated the design of steering and stability systems

They advanced the design of propellers.

First flight
82nd flight: 2.75 miles and 304 seconds

Orville was injured in a crash and a passenger was killed
The original aircraft


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.


Engines


Turboprop

In a turboprop, incoming air is compressed by a fan and then mixed with fuel. The exploding air powers a crankshaft that turns the propeller. This is the most efficient engine up to Mach .5.


Turbofan
In a turbofan the exploding fuel+air mixture powers a fan. Some of the incoming air is reacted with fuel and most of it is bypassed, channelled instead to the fan and propelled aftward.

Turbofans are the most efficient kind of engine from Mach .5 to 1.0. All commercial aircraft that fly close to Mach 1 are turbofans.


Turbojet

A turbojet uses most of its intake air for combustion and most of the thrust comes from jet action rather than from a fan. Such engines can go beyond Mach 1.


Afterburner

F/A-18 Hornet
SR-71 Blackbird
SR-71 Blackbird engine test

A turbofan in afterburner mode injects fuel aft of the fan. The fuel explodes and adds thrust. This enables an aircraft to go beyond Mach 1.


Ramjet

If the aircraft is moving faster than Mach 1 then the incoming air doesn't need to be compressed with a fan. The ram pressure from slowing down the air in the combustion chamber is enough.

Ramjets work up to Mach 6, at which point the incoming compressed air becomes too hot.

Ramjets are simpler than turbofans because they don't have any moving parts. They are often used for missiles.


Scramjet

NASA X-23
Turbofan, ramjet, scramjet

In a scramjet the incoming air is compressed enough to make it explodable but not enough to slow it down below Mach 1. Such an engine sidesteps the heating concern of a ramjet and can go up to Mach 15.


SR-71 Blackbird engine


de Laval nozzle

In a de Laval nozzle, incomming subsonic air accelerates as it approaches the nozzle. The nozzle is shaped so that the air becomes supersonic at the narrowest point, after which it continues accelerating as it expands aftward.


Specific impulse

Exhaust velocity  =  V
Gravity constant  =  g
Specific impulse  =  I  =  V/g

Air compression

A jet engine compresses air before mixing it with fuel. For adiabatic compression,

Adiabatic index =  A  =  1.4      (for air)
Density         =  D
Pressure        =  P  ≈  DA  ≈  D T
Temperature     =  T  ≈  DA-1
The larger the air density the more efficient the engine. Increased density comes with increased temperature and the performance of an engine is determined by the quality of the high-temperature alloys.

The following table shows the properties of an adiabatically compressed gas in relative units and in Kelvin. In practice the gas compression is not adiabatic and the pressure and temperature values are larger.

Density  Pressure   Temperature   Temperature
                                   (Kelvin)
   1      1             1            250         Ambient air at 10 km altitude
   2      2.64          1.32         330
   3      4.66          1.55         388
   4      6.96          1.74         435
   6     12.3           2.05         512
   8     18.4           2.30         574
  12     32.4           2.70         675
  16     48.5           3.03         758
  24     85.6           3.57         891
  32    128.0           4.0         1000
Aircraft engines:
                                   Pressure ratio

Airbus A350   Rolls-Royce Trent XWB     52
Boeing 747    General Electric CF6      42
Boeing 777    General Electric GE90     42
Airbus A380   Rolls-Royce Trent 900     39
F-15          General Electric F110     30
Concorde      Rolls-Royce 593           15.5

Bird flight

Flapping the wings propels the bird upward and then the bird glides downward until the next flap. Forward propulsion comes more from gravitational descent than from flapping.


Flight lab


Wings

Construct a wing and a wind tunnel and measure the following:

Wing drag force       =  Fdrag  =  Rearward force on the wing
Wing lift force       =  Flift  =  Upward lift force on the wing
Wing lift coefficient =  Qlift  =  Flift / Fdrag
The larger the value of Qlift the better. You can measure the lift coefficient by measuring wing forces as above or you can measure the glide ratio, discussed below.

You will have to adjust the wing's "angle of attack" to get the optimal lift coefficient.


Gliders

Build a glider using any materials available and try to minimze the glide angle, which is defined as the change in height divided by the change in lateral distance. Try different kinds of designs and see if you can optimize the design.

To measure the glide ratio, launch the glider with zero initial velocity from a height close to the ceiling and measure how far it travels before hitting the floor.

Z  =  Glide ratio  =  Distance traveled / Initial height
The larger the glide ratio the better.

The glide ratio is equal to the lift coefficient.

Z  =  Qlift

Propellers

First electric helicopter, 2011

Construct a propeller powered by a rubber band and construct a helicopter to support the propeller. Place the helicopter on the ground and engage the propeller and measure the maximum height that the helicopter reaches.

Maximum height of the helicopter  =  H
The larger the value of H the better.
Propeller aircraft

The propeller provides forward force and the wing provides upward force.

Combine a glider with a rubber-band-powered propeller to form a propeller aircraft. Release it from at rest on the ground and measure the distance it flies.

Distance the plane flies  =  X
The larger the value of X the better.

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
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.
Helicopter drones

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  Battery  Drone  Battery  Battery  Drone  Drone  Flight  Price  Wireless
                mass   energy           mass             power          time           range
                 kg    MJoule   MJ/kg    kg      MJ/kg   Watt    W/kg  minutes   $      km

Mota Jetjat Nano .011    .00160  .145                     3.3    303      8      40    .02
Nihui NH-010     .0168   .00200  .119   .0047    .43      6.7    397      5      24    .03
ByRobot Fighter  .030    .0040   .133                     6.7    222     10     120    .1
Blade mQX        .0751   .0067   .089                    11.2    149     10     115
XDrone Zepto     .082    .0067   .081                     4.6     56     24      40    .05
SKRC D20         .092    .0080   .087                    11.1    121     12      20    .3
SKRC Q16         .111    .0067   .060                     7.4     67     15      40    .2
Walkera QR Y100  .146    .0213   .146   .0413    .52     17.8    122     20     100    .1
MJX Bugs 3       .485    .0480   .099                    44.4     92     18     100    .5
Bayangtoys X16   .50     .088    .176                    92      183     16     110    .8
DJI Mavic Pro    .725    .157    .217   .24      .65    109      150     24    1000   7.0
XK Detect X380C 1.18     .216    .183                   120      102     30     522   1.0
Xiaomi Mi       1.376    .319    .214                   197      143     27     380   1.0
DJI Phantom 4   1.38     .293    .212   .426     .69    174      126     28    1000   5.0
XK X500         1.8      .288    .160                   160       89     30     303   1.0
JYU Spider X    2.1      .360    .171   .812     .44    200       95     30     155   4.0
MD4-1000        2.65    1.039    .392                   197       74     88    2000    .5
DJI Inspire     2.935    .360    .123   .67      .54    333      114     18    2000   7.0
Altura Zenith   3.5     1.327    .379                   491      140     45    2000   1.0
Walkera QR X800 3.9      .799    .205  1.134     .70    222       57     60    2700   2.0
AEE F100        6.0     1.598    .266                   380       63     70   58000  10.0
Chaos HL48      6.8     1.758    .259                   651       96     45   20000  20.0
Ehang 184     200      51.8      .259                 37500      188     23  300000   3.5
The minimum power requirement for quadcopter flight is of order 60 Watts/kg.
1 MJ = 1 MJoule = 106 Joules

Electric power outperformes gasoline power in all categories except energy density. Electric motors are lighter, simpler, cheaper, more flexible, and more reliable than combustion motors.


Fixed-wing drones
           Drone  Battery  Drone  Drone  Drone  Flight  Cost  Wireless  Wing  Wing  Cruise   Max   Flight
           mass   energy    E/M   power   E/M    time          range    area  span  speed   speed  range
            kg    MJoule   MJ/kg  Watts  W/kg   minutes  $       km      m2    m     m/s     m/s    km

Sky Hawk      .355  .0346  .097    19.2  54.1     30     400    1.0           1.0
Chaipirinha   .62   .088   .142                          154                   .85
AgDrone      2.25   .639   .284   161    71.7     66            4.5            .124  46.7     82   50
Trimble UX5  2.5    .320   .128   107    42.7     50   10000    2.5     .34    .10   80            60

Power
For a typical drone,
Drone mass             =  M
Battery mass           =  Mbat
Payload mass           =  Mpay
Climbing speed         =  Vcli
Max horizontal speed   =  Vmax
Hover constant         =  H            =  60 Watts/kg       Power/mass required to hover
Hover power            =  Phov=  H M
Hover power for payload=  Ppay=  H Mpay
Gravity constant       =  g            =  9.8 m/s
Power to climb         =  Pcli=  M g V
Drag coefficient       =  C
Air density            =  D            = 1.22 kg/meter3
Drone cross-section    =  A
Drag power             =  Pdrag= ½ C D A V3
If the climbing power is equal to the hover power,
V = H / g  =  6 meters/second
If the climbing power is equal to the drag power,
M g V = ½ C D A V3

V  =  [2Mg/(CDA)]½  =  4.0 [M/(CA)]½ meters/second
                Mass    Hover  Battery  Battery  Payload  Climb   Max speed  Power/mass
                 kg     Watts    MJ       kg       kg      m/s       m/s      Watts/kg

DJI Mavic Pro      .725   109    .157    .24                5       17.9
Phantom 4         1.38    174    .293    .426               6       20
JYU Spider X      2.1     200    .360    .812       2.3     5        8         516
MD4-1000          2.65    197   1.039               1.2             12
AEE F100          6.0     380   1.598               2.5             27.7
Ehang wingspan  200     37500  51.8                99.8             28
The JYU Spider X has the largest value for (payload mass) / (drone mass). The battery power/mass is
Battery power/mass  =  Phov ⋅ (M + Mpay) / M / Mbat  =  516 Watts/kg

Electric motors
                  Mass    Power   Power/Mass   Price  Power/Price
                   kg     kWatt     kW/kg        $      Watt/$

FXC1806-14 2300kv   .018     .15    8.3         25       6.0
RCX H1806 2400kv    .02      .144   7.22        10      14.4
Tesla Model S     31.8    269.9     8.49

Supercapacitors
                Mass     Energy      E/M    Power   P/M   Price  Energy/$    C     Voltage
                 kg      kJoule     kJ/kg   kWatt  kW/kg    $    kJoule/$  Farads   Volts

PM-5R0V105-R      .000454    .0062  13.8                    5.7   .0011       1      5.0
Adafruit          .135      .984     7.3                   20     .049      630      2.5
BMOD0006E160B02  5.2      37.1       7.1    2.08   .40   1170     .032        5.8  160
XLM-62R1137-R   15       125.3       8.4  124.2   8.3    1396     .090      130     62.1


Capacitance      =  C            (Farads)
Voltage          =  V            (Volts)
Total energy     =  E0 =  ½ C V2  (Joules)
Effective energy =  E  =  ¼ C V2  (Joules)
Max current      =  I            (Amperes)
Max power        =  P  =  I V    (Watts)
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. In the data table we use the effective energy.
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  Speed   Density  C  H  N  O
                     /kg    (km/s)  (g/cm3)

Bombardier beetle      .4                                 Hydroquinone + H2O2 + protein catalyst
Ammonium nitrate      2.0    2.55    1.12    0  4  2  3
Black powder          2.6     .6     1.65                 Used before 1884
Smokeless powder      5.2    6.4     1.4     6  9  1  7   Used after 1884. Nitrocellulose
TNT                   4.7    6.9     1.65    7  5  3  6   Trinitrotoluene
PETN                  5.8    8.35    1.77    5  8  4 12
Dynamite              5.9    7.2     1.48    3  5  3  9   75% Nitroglycerine + stabilizer
Composition 4         6.3    8.04    1.59    3  6  6  6   91% RDX. "Plastic explosive"
PLX                   6.5            1.14    1  3  1  2   95% CH3NO2 + 5% C2H4(NH2)2
Nitroglycerine        7.2    8.1     1.59    3  5  3  9   Unstable
RDX (Hexagen)         7.5    8.7     1.78    3  6  6  6
HMX (Octogen)         8.0    9.1     1.86    4  8  8  8
Dinitrodiazeno.       9.2   10.0     1.98    4  0  8  8
Octanitrocubane      11.2   10.6     1.95    8  0  8 16
Gasoline + Oxygen    10.4                    8 18  0 13
Hydrogen + Oxygen    13.16                   0  2  0  1
Uranium bomb     219000
Hydrogen bomb        10 mil
Antimatter        90000 mil

Speed: Detonation speed
C:     Carbon atoms
H:     Hydrogen atoms
N:     Nitrogen atoms
O:     Oxygen atoms
Nitrocellulose
TNT
RDX
HMX
PETN
Octanitrocubane

Nitrocellulose
TNT
RDX
HMX
PETN
Octanitrocubane

Dinitrodiazenofuroxan
Nitromethane

~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)

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.
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
ω  =  Angular frequency

X  =  X position     =  R    cos(ω T)
Vx =  X velocity     = -R ω  sin(ω T)
Ax =  X acceleration = -R ω2 cos(ω T)

Y  =  Y position     =  R    sin(ω T)
Vy =  Y velocity     =  R ω  cos(ω T)
Ay =  Y acceleration = -R ω2 sin(ω T)

V2  =  Vx2 + Vx2  =  R2 Kt2          ->  V = R ω
A2  =  Ax2 + Ay2  =  R2 Kt2          ->  A = R ω2

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)


Main page

Support the free online science textbooks project






© Jason Maron, all rights reserved.