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Acceleration Force Momentum Gravity Circular motion Friction Spin Pressure Buoyancy History
Waves Music Overtones Resonance
Electric force Magnetic force Batteries Resistance
Atoms Chemistry Nuclei Fusion Fission
Aerodynamic drag Pendulum Thermodynamics Spin II
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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
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Electric charge is also a fundamental unit and it is measured in Coulombs, but it won't appear in the laws of mechanics.
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
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)
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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 XSim: Position, velocity, and 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
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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 gravityIn the language of calculus,
Time = T Position = X = = V dT Velocity = V = ∂X/∂T = A dT Acceleration = A = ∂V/∂TExamples of position, velocity, and acceleration.
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Sim: Position, velocity, and acceleration #2
Mass = M Acceleration = A Force = F = M A (Newton's law)
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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 = gIf 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 forceFor all known forms of matter,
Mgrav = Minertial
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
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
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Mass = M Velocity = V Momentum = Q = M V
Suppose an object undergoes a constant force for time T.
Impulse = F T = M A T = M V = Momentum
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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 = 0The 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.
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 V2Newton'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 XThe 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
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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
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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 / InitialEnergyIf you know the value of K for a collision then you can solve it by writing down down equations for momentum and energy conservation.
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 = -V2iIf energy is not conserved then
V1f = -V1i K V2f = -V2i Kwhere K is the collision coefficient.
Power = Energy / Time = Force * Distance / Time = Force * VelocitySuppose 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
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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.
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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.
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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
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) = AIf Xi = Vi = 0 then the equations reduce to
X(T) = .5 A T2 V(T) = A T A(T) = AFor 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
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) M2If 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 TThe center of mass moves with constant velocity.
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Displacement of the spring = X Spring constant = K Force on the spring = F = K X (Hooke's law)
For a spring, the force is proportional to displacement.
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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
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 sThe "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
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 T2We can eliminate T from the trajectories and express Y in terms of X.
Y = X - X2 / 80The 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.
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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
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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.
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Radius of the circle = R Velocity = V Centripetal acceleraton = A = V2 / R
If artificial gravity is generated by spinning a spaceship, then according to en.wikipedia.org/wiki/Artificial_gravity, the spin period has to be at least 30 seconds for the inhabitants to not get dizzy.
Spin period = T = 2 R / V = 30 s Centripetal acceleration = A = V2 / R = 10 m/s2 Spin radius of the spaceship = R = T2 A / (2)2 = 228 meters Tangential velocity of the spaceship = V = (A R)1/2 = 48 m/s
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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 ∂RThe velocity of a circular orbit is obtained by setting gravitational force equal to centripetal force.
G M m / R = m V2 / RThe 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) Escape velocity =
For an object on a circular orbit,
Gravitational energy = -2 * Kinetic energyThe 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.
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]½Acceleration 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
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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 FcontactThe 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.
Villeneuve vs. Arnoux
At 0:49 Arnoux breaks before he hits the turn.
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.
Suppose an object with mass m rests on a ramp inclined by an angle
theta. The gravitational force on the object is
Pi is defined as the ratio of the circumference to the diameter.
If an object is at the edge of a record then the position is the
arc length around the circumference.
Suppose a billiard ball rolls across a table with a speed of 2 m/s.
For a mass moving around a circle, we can describe the motion in terms of either a position or an
angle.
Angular momentum is conserved. In the following figure, the angular momentum
is constant and V*R is constant.
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.
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,
The moment of inertia of an object depends on its mass and how far the mass
is from the axis of rotation.
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
Conservation arises from multiplying F=MA by various quantities.
For a rolling ball,
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.
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.
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.
The cat's first move is to maximize its moment of inertia to slow down its rotation.
Jupiter's spin makes it oblate.
For a spinning 3D object,
A typical set of parameters for a racing bike is
We assume a high gear, with 53 teeth on the front gear and 11 teeth on the rear gear.
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.
Suppose a bike is going uphill at large power and low velocity.
Properties of atmospheres:
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.
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,
The following table estimates the average mass per air molecule.
We have neglected the argon molecules.
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:
For a ship floating in water,
For ice floating in water,
We estimate the number of helium balloons required to lift a person.
Helium is more expensive and more dense than hydrogen, but it is not flamable.
For a hot air balloon, the volume is fixed and the pressures on the inside and outside are equal
For example,
Jacques Charles made the first hot air balloon flight in 1783.
The properties of a wave are
A wave on a string moves at constant speed and reflects at the boundaries.
For a violin A-string,
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.
If we double both frequencies then it also sounds like an octave. The shape of
the blue wave is preserved.
If two notes are out of tune they produce dissonant beat frequencies.
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.
Violins, violas, and cellos are tuned in fifths. String basses, guitars, and bass
guitars are tuned in fourths. Pianos are tuned with equal tuning.
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"
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.
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.
Two waves traveling in opposite directions create a standing wave.
Waves on a string simulation at phet.colorado.edu
For example, the overtones of an A-string with a frequency of 440 Hertz are
Overtone simulation at phet.colorado.edu
An oboe and a clarinet are half-open pipes.
A cantilever has the same overtones as a half-open pipe.
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.
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.
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
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
A "formant" is a vocal resonance. Vowels can be identified by their
characteristic mode frequencies.
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.
Overtones are ubiquitous in vibrating systems. They are usually referred to as
"normal modes".
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.
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.
A bow produces a sequence of plucks at the fundamental frequency of the string.
In a brass instrument your lips take the function of a reed.
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.
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.
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.
Suppose a microphone samples a wave at fixed time intervals. The white curve
is the wave and the orange dots are the microphone samplings.
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.
Wikipedia:
Harmonic oscillator
Q factor
Resonance
Resonance
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)
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.
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.
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.
Wind can make a string vibrate (The von Karman vortex).
The Tacoma Narrows bridge collapse was caused by wind exciting resonances in the
bridge.
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
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.
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.
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.
Oscillators that are mechanically connected can transfer energy back and forth
between them.
Suppose you measure the frequency of a wave by counting the number of crests
and dividing by the time.
The equation for the uncertainty in a frequency measurement is
For a system in thermodynamic equilibrium each degree of freedom has
a mean energy of ½ k T. This is the definition of temperature.
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.
For air, γ = 7/5 and
We can change the sound speed by using a gas with a different value of M.
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.
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).
Carbon dioxide doesn't have a liquid state at standard temperature and pressure.
It sublimes directly from a solid to a vapor.
For an object to have an atmosphere, the thermal energy must be much less than the
escape energy.
The threshold for capturing an atmosphere appears to be around Z = 1/25, or
When an object collapses by gravity, its temperature increases such that
For an ideal gas,
The Earth's core is composed chiefly of iron. What is the temperature of an iron
atom moving at the Earth's escape speed?
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.
Time between collisions = 2 L / V
The average force on a wall is
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
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.
For a gas,
For a diatomic gas there are also two rotational degrees of freedom. In this
case d=5.
In general,
The fact that Gamma=7/5 for air was a clue for the existence of both atoms,
molecules, and quantum mechanics.
For dark energy,
As a substance expands it does work on its surroundings according to its pressure.
Dark matter consists of pointlike particles but they rarely interact with other
particles and so they exert no pressure.
For gases, the density at boiling point is used. Size data
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,
The "de Broglie wavelength" of a particle is
Charges of the same sign repel and charges of opposite sign attract.
A charge generates an electric field. The electric field points away from positive charges
and toward negative charges.
A moving charge is an "electric current". In an electric circuit, a battery moves electrons through a wire.
Moving charges and currents exert forces on each other. Parallel currents attract and antiparallel
currents repel.
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.
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).
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.
The direction of the force is the cross product "×" of V and B.
The direction is given by the "right hand rule".
In a superconductor, electrons move without interference.
The maximum electric field is equal to the max field for air times
a dimensionless number characterizing the dielectric
A solenoid is a wire wound into a coil.
The critical electric field for electric breakdown for the following materials is:
Relative permittivity is the factor by which the electric field between
charges is decreased relative to vacuum. Relative permittivity is dimensionless.
Large permittivity is desirable for capacitors.
A ferromagnetic material amplifies a magnetic field by a factor called the "relative
permeability".
Resistivity in 10^-9 Ohm Meters
All of these particles are stable except for the neutron, which has a half life of 611 seconds.
An element has a fixed number of protons and a variable number of neutrons.
Each neutron number corresponds to a different isotope. Naturally-occuring elements
tend to be a mix of isotopes.
For a radioactive material,
Suppose an element has a half life of 2 years.
The weak force can convert a neutron into a proton, ejecting a high-energy electron.
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.
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.
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.
A neutron triggers the fission of Uranium-235 and plutonium-239,
releasing energy and more neutrons.
Fission releases neutrons that trigger more fission.
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 is more difficult to detonate than uranium. Plutonium detonation requires a spherical
implosion.
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 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.
Creating Plutonium-239 and Uranium-233:
A nuclear fusion bomb contains deuterium and lithium-6 and the reaction is
catalyzed by a neutron.
A tokamak fusion reactor uses magnetic fields to confine a hot plasma so that
fusion can occur in the plasma.
The fusion reaction that occurs at the lowest temperature and has the highest
reaction rate is
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).
For gasoline, we assume conversion to electricity by a generator with an efficiency of
1/4.
Cycling measurements are from Menaspa's (2013) analysis of Tour de France
sprints. For a 1020 Watt sprint the speed is 18.4 meters/second.
"Mass fraction" is the fraction of mass converted to energy, by E=MC2.
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.
Newton was also the first to observe the "Magnus effect", where spin causes
a ball to curve.
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.
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.
To estimate the distance a bullet travels before being slowed by drag,
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.
The drag force on an object moving through a fluid is
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
Airplanes fly at high altitude where the air is thin.
Commercial airplanes fly at Mach .9 because the drag coefficient increases sharply
at Mach 1.
The drag coefficient depends on speed.
Golf balls have dimples to generate turbulence in the airflow, which
increases the Reynolds number and decrease the drag coefficient.
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.
For an object experiencing drag,
If the spin force equals the gravity force (Fspin = Fgrav),
The drag force on an object moving through a fluid is
If the spin force equals the gravity force (Fspin = Fgrav),
For a typical car,
The drag speed Vd is determined by setting Fr = Fa.
Wiki: Energy efficiency in transportation
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.
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
Electric bikes are easy to make. All you have to do is replace a conventional
wheel with an electric wheel and attach a battery pack. Electric wheels come
in kits and you can make the battery pack yourself. Example configurations for
various motor powers:
Electric wheel prices are from Amazon.com.
An wing generates lift at the cost of drag. Lift exceeds drag.
The lift-to-drag coefficient Qw is proportional to wing length divided by wing width.
Wing width varies along the length of the wing. We define an effective width as
"Area" is the total for both wings, and "Length" is for one wing.
Aspect ratio is Length/Width.
Changing wing angle changes lift and drag. There is an optimum angle that maximizes
the lift-to-drag coefficient.
If the angle is larger than the optimal angle, you gain lift at the expense of drag.
If you make the angle of attack too large, lift ceases and the plane stalls.
The air drag force is
A glider is unpowered. The more efficient the glider, the
smaller the glide angle. The minimum glide angle is determined by the
wing lift/drag coefficient.
For the mass scalings, we assume that wing area scales as M2/3.
A wingtip creates a vortex as it moves. Wingtips are often equipped with a
vertical element to damp the vortex. The vertical element increases the effective
wing length and improves the lift-to-drag coefficient.
coefficient.
Birds fly in a "V" formation to use
the updraft from their neighbor's wingtip vortices.
Airplanes fly at high altitude where the air is thin.
An aircraft moving at Mach 2 and turning with a radius of 1.2 km has a g force of
7 g's.
For propellers,
The parameters are not independent. They're related through the blade aspect ratio.
A drive propeller has to move substantially faster than the aircraft to be effective.
This distinguishes it from a hovering propeller, which is designed to minimze propeller
speed.
A commonly-appearing quantity is the power/mass ratio, which is inversely
proportional to the force/power ratio.
The properties of a flying car are determined by the properties of propellers
and lithium-ion batteries. Typical parameters for a 1-person car are:
For hovering, the more rotors the better. The hovering time scales as rotor number to
the 1/6 power. Adding rotors also increases stability and failsafe.
A electric propeller-driven aircraft can hover for more than an hour. The
hovering time is determined by the battery energy per mass and by the rotor radius.
Example values:
The power per mass required to hover is determined by the
physics of rotors. For a 1 kg vehicle with a 1/4 meter radius rotor,
Curtis LeMay: Flying fighters is fun. Flying bombers is important.
High explosives have a large shock velocity.
The best oxidizer is liquid oxygen, and the exhaust speed for various fuels when
burned with oxygen is:
We use kerosene as a standard fuel and show the rocket speed for various oxidizers.
Some of the oxidizers can be used by themselves as monopropellants.
Above 550 Celsius, potassium nitrate decomposes. 2 KNO3 ↔ 2 KNO2 + O2.
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
Maximum cornering acceleration = C g
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()
The force of gravity parallel to the ramp surface is
Framp = Fgrav sin()
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() = C Fgrav cos()
C = tan()
This is a handy way to measure the coefficient of friction. Tilt the ramp until
the object slides and measure the angle.
= 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 = R
Full circle = 360 degrees = 2 radians
1 radian = 57.3 degrees
1 degree = .0175 radians
Radius = R
Angle =
X coordinate = X = R cos()
Y coordinate = Y = R sin()
T = Time in seconds
= Angle (dimensionless)
= Angular velocity in radians/second = 1.75 in the animated figure
= Angular frequency
F = Spin frequency in Hertz or 1/second
= 2 Pi F
1 Hertz = 1 revolution/second = 2 radians/second
T = Period in seconds
F = Frequency in Hertz or 1/second
F T = 1
Time = T
Radius = R = .15 meters (for a vinyl record)
Angular velocity = = 3.49 radians/second (= 33.33 revolutions per minute)
Velocity of the outer edge V = R = .523 meters
Ball velocity = V = 2 meters/second
Ball radius = R = .03 meters
Angular frequency = = V/R = 67 radians/second
Ball spin frequency in Hertz = F = /(2)= 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.
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
R = Radius of the circle
X = Position of the mass on the circumference of the circle
= Angle pointing to X
X = R
Every linear quantity has a corresponding spin quantity, and every linear equation has
a crresponding spin equation.
By changing variables from X to we can translate a
linear equation into a spin equation.
Linear variables Spin variables
Position = X Angle = = X / R
Velocity = V Angular velocity = = V / R
Acceleration= A Angular accel. = = A / R
Mass = M Moment of inertia = I = M * R2
Force = F Torque = = 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: = I L = I = F T R E = .5 I = A = R = T = 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
= M R2 A/R L = M R2 V/R E = .5 I E = = T = T
= I L = I
Angle = = X / R
Angular velocity = = V / R
Angular accel. = = A / R
Constant velocity: X = V T
Constant spin: = T
Constant acceleration: V = A T
Constant angular accel.: = T
The equivalent of Newton's law for spin is
Newton's law for linear motion: F = M A
Newton's law for circular motion: = I = F R
Momentum: Q = M V
Angular momentum: L = I = M V R
Energy: E = .5 M V2
Angular energy: L = .5 I
L = R x Q
= |R| |Q| sin(Theta)
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
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
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
Energy of matter = M C2 C = Speed of light = 3.00e8 m/s
Force = Mass * Acceleration
Force * Time = Momentum
Force * Distance = Energy
Force * Radius = Torque
Force * Time * Radius = Angular Momentum
Velocity = V
Radius = R
Mass = M
Angular frequency = = V/R
Moment of inertia = I = .4 M R2
Kinetic energy = Ek = .5 M V2
Spin energy = Es = .5 I = .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
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 = = R C M g
Moment of inertia = I = .4 M R2
Angular acceleration = = / I = 2.5 C g / R
Time sliding = T
Rolling angular velocity = = T = 2.5 C g T / R
Ball velocity when rolling = V = Vo + A T
= Vo - C g T
= Vo - .4 C g R / C / g
= Vo - .4 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.
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.
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
Number of links in the front gear = Nf = 53
Number of links in the rear gear = Nr = 11
Length of one link of a bicycle chain = L = .0127 m = .5 inches
Radius of the front gear = Rf = Nf L / (2 π) = .107 m
Radius of the rear gear = Rr = Nr L / (2 π) = .0222 m
Torque balance:
Ground force * Wheel radius = Chain force * Rear gear radius
Pedal force * Pedal radius = Chain force * Front gear radius
Chain force = Ground force * Wheel radius / Rear gear radius
= 128 * .311 / .0222
= 1793 Newtons
Pedal force = Ground force * Wheel radius / Pedal radius * Front gear radius / Rear gear radius
= Ground force * Wheel radius / Pedal radius * Front gear teeth / Rear gear teeth
= 128 * .311 / .17 * 53 / 11
= 1128 Newtons
Radius Force Torque Gear
(m) (N) (Nm) teeth
Pedal crank .170 1128 191.9 -
Front gear .107 1793 191.9 53
Rear gear .0222 1793 39.8 11
Rear wheel .311 128 39.8 -
Wheel frequency = Velocity / (Radius * 2Pi)
= 20 / (.311 * 2)
= 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.
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.
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.
Surface area = A
Force = F
Pressure = P = F / A (Pascals or Newtons/meter2 or Joules/meter3)
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
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.
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
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.
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
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
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
P = Pressure = 101325 Pascals
D = Density = 1.22 kg/m3
= Adiabatic constant = 7/5 for air
Vtherm= Thermal speed = 544 meters/second
Vsound= Sound speed = 343 meters/second at 20 Celsius
= ( P / D)1/2
= (/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.
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
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
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.
Gravitational acceleration = g = 9.8 m/s2
Density of ice = Dice = 920 kg/m3
Density of water = Dwater = 1000 kg/m3
Volume of the iceberg = Υice
Volume of water displaced by iceberg = Υwater
Mass of the iceberg = Mice = Dice Υice
Mass of water displaced by iceberg = Mwater = Dwater Υwater
Gravity force on the iceberg = Fgrav = Mice g
Buoyant force on the iceberg = Fbuoy = Mwater g
Fraction of iceberg above surface = f = (Υice - Υwater) / Υice
Fgrav = Fbuoy (Principle of bouyancy)
Mice = Mwater (Principle of Archimedes)
f = (Dice - Dwater) / Dice
= (Mice/Υice - Mwater/Υwater) / (Mice/Υice)
= 1 - Υice / Υwater
= .08
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.
kg/m3
Hydrogen .0899
Helium .179
Air (hot) 1.12 320 Kelvin
Air (room temp) 1.22 293 Kelvin
Ice 920
Water 1000
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
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
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
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 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
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)
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.
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
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.
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.
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.
Whan a wave on a string encounters an endpoint it reflects with the waveform
preserved and the amplitude reversed.
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
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
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".
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}
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)
A string has the same overtones as a closed pipe.
An instrument of length L has overtones with frequency
Frequency = Z * Wavespeed / (2 * Length)
Z corresponds to the white numbers in the figure above.
Violin 440, 2*440, 3*440, 4*440, ...
Oboe 440, 3*440, 5*440, 7*440, ...
Flute 440, 3*440, 5*440, 7*440, ...
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.
The interior of a football is a spherical resonator.
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)
As a sound waves travels back and forth along the clarinet it forces the reed to
vibrate with the same frequency.
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.
Lift incrases with wing angle, unless the angle is large enough for the airflowto stall.
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.
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.
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.
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
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.
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.
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.
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.
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
Amplitude of the resonance = Constant * q
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.
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.
dF T >= 1
Pressure = P (Pascals or Newtons/meter2 or Joules/meter3)
Temperature = T (Kelvin)
Volume = Vol (meters3)
Total gas kinetic energy = E (Joules)
Kinetic energy per volume = e = E/Vol (Joules/meter3)
Number of gas molecules = N
Mass of a gas molecule = M
Gas molecules per volume = n = N / Vol
Gas density = D = N M / Vol
Avogadro number = Avo= 6.022⋅1023 moles-1
Moles of gas molecules = Mol= N / Avo
Boltzmann constant = k = 1.38⋅10-23 Joules/Kelvin
Gas constant = R = k Avo = 8.31 Joules/Kelvin/mole
Gas molecule thermal speed = Vth
Mean kinetic energy / gas molecule= ε = E / n = ½ M Vth2 (Definition of the mean thermal speed)
Gas pressure arises from the kinetic energy of gas molecules and has units of
energy/volume.
The ideal gas law can be written in the following forms:
P = 2⁄3 e Form used in physics
= R Mol T / Vol Form used in chemistry
= k N T / Vol
= 1⁄3 N M Vth2/ Vol
= 1⁄3 D Vth2
= k T D / M
Gas simulation at phet.colorado.edu
Derivation of the ideal gas law
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
Molecule mass = M
Thermal speed = Vth
Boltzmann constant = k = 1.38⋅10-23 Joules/Kelvin
Molecule mean kinetic energy = ε
A gas molecule moving in N dimensions has N degrees
of freedom. In 3D the mean energy of a gas molecule is
ε = 3⁄2 k T = ½ M V2th
Adiabatic constant = γ
= 5/3 for monatomic molecules such as helium, neon, krypton, argon, and xenon
= 7/5 for diatomic molecules such as H2, O2, and N2
= 7/5 for air, which is 21% O2, 78% N2, and 1% Ar
≈ 1.31 for a triatomic gas such as CO2
Pressure = P
Density = D
Sound speed = Vsound
Mean thermal speed = Vth
K.E. per molecule = ε = ½ M Vth2
V2sound = γ P / D = 1⁄3 γ V2th
The sound speed depends on temperature and not on density or pressure.
Vsound = .68 Vth
These laws are derived in the appendix.
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.
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.
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.
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
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.
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.
Thermal Speed < 1/5 Escape speed
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.
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.
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
M = Mass of molecule
V = Speed of the molecule
L = Space between the walls
With each collision, the momentum change = 2 M V
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
Total gravitational energy = -2 * Total kinetic energy
just like for a single satellite on a circular orbit.
Gravitational energy of the sun = -2 * Kinetic energy of protons in the sun
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.
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).
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, yielding a sound speed of
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.
E = Energy
dE = Change in energy
e = Energy density
Vol= Volume
P = Pressure
The volume expands as the universe expands.
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.
Density = D = 8900 kg/m3
Atomic mass unit= M0 = 1.661⋅10-27 kg
Atomic mass = MA = 63.55 Atomic mass units
Mass = M = MA⋅M0 = 9.785⋅10-26 kg
Number density = N = D / M = 9.096⋅1028 atoms/m3
Cube volume = Υcube= 1 / N = 1.099⋅10-29 m3 Volume/atom if the atoms are cubes
Cube length = L = Υ1/3cube = 2.22⋅10-10 m Side length of the cube
Sphere fraction = f = π/(3√2) = .7405 Fraction of volume occupied by spheres in a stack o spheres
Sphere volume = Υsph= Υcube f = 8.14⋅10-30 m3 = 4⁄3πR3 Volume/atom if the atoms are spheres
Sphere radius = R = 1.25⋅10-10 m
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
The fundamental unit of charge is the "Coulomb", and the electric force
follows the same equations as the gravitational force.
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
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.
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.
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
Magnetic field = B
Magnetic force on a charge = F = Q V × B
Magnetic force on a current = F = 2e-7 I × B
Quantity MKS units CGS units Conversion factor
Mass M kg gram .001
Wire length Z meter cm .01
Radial distance from wire R meter cm .01
Time T second second 1
Force F Newton dyne 100000
Charge Q Coulomb Franklin 3.336e-10
Velocity of a charge V meter/second cm/s .01
Speed of light C 2.999e8 meter/second cm/s 100
Energy E Joule erg e-7
Electric current I Ampere = Coulomb/s Franklin/s 3.336e-10
Electric potential V Volt Statvolt 299.79
Electric field E Volt/meter StatVolt/cm 29979
Magnetic field B Tesla Gauss 10000
Capacitance C Farad cm 1.11e-12
Inductance L Henry s2/cm 9e-11
Electric force constant Ke = 8.988e9 N m2/C2 Ke = 1 dyne cm2 / Franklin2
Magnetic force constant Km = 2e-7 = Ke/C2 Km = 1/C2
Vacuum permittivity ε = 8.854e-12 F/m =1/4/π/Ke
Vacuum permeability μ = 4 π e-7 Vs/A/m =2 π Km
Proton charge Qpro = 1.602e-19 Coulomb Qpro= 4.803e-10 Franklin
Electric field from a charge E = Ke Q / R2 E = Q / R2
Electric force on a charge F = Q E F = Q E
Electric force between charges F = Ke Q Q / R2 F = Q Q / R2
Magnetic field of moving charge B = Km V Q / R2 B = (V/C) Q / R2
Magnetic field around a wire B = Km I / R B = (V/C) I / R
Magnetic force on a charge F = Q V B F = (V/C) Q B
Magnetic force on a wire F = Km B Z F = I B z
Magnetic force between charges F = Km V2 Q1 Q2 / R2 F = (V/C)2 Q Q / R2
Magnetic force between wires F = Km I1 I2 Z / R F = I1 I2 Z / R
Energy of a capacitor E = .5 C V2
Field energy per volume Z = (8 π Ke)-1 (E2 + B2/C2) Z = .5 (E2 + B2/C2)
Speed of light C
Electric field E
Electric field, time derivative Et
Magnetic field B
Magnetic field, time derivative Bt
Charge Q
Charge density q
Current density J
MKS CGS
Ke=8.988e9 Ke=1
Km=2e-7 Km=2/C
∇˙E = 4 π Ke q ∇˙E = 4 π q
∇˙B = 0 ∇˙B = 0
∇×E = -Bt ∇×E = -Bt / C
∇×B = 2 π Km J + Et / C2 ∇×B = 4 π J / C + Et / C
Charge = Q Coulombs
Voltage = V Volts
Energy = E = VQ Joules
Time = T seconds
Current = I = Q/T Amperes
Resistance = R = V/I Ohms
Power = P = QV/T Watts
= IV
= V2/R
= I2R
Ohm's Law: V = IR
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
Voltage = V Volts
Capacitance = C Farads
Total energy = E = ½ C V2 Joules
Effective = Ee = ¼ C V2 Joules
Not all of the energy in a capacitor is harnessable because the voltage diminishes
as the charge diminishes, hence the effective energy is less than the total energy.
A = Plate area
Z = Plate spacing
Ke = Electric force constant = 8.9876e9 N m2 / C2
Q = Max charge on the plate (Coulombs)
Emax= Max electric field = 4 Pi Ke Q / A
V = Voltage between plates = E Z = 4 Pi Ke Q Z / A
En = Energy = .5 Q V = .5 A Z E2 / (4 π Ke)
e = Energy/Volume = E / A Z = .5 E2 / (4 π Ke)
q = Charge/Volume = Q / A / Z
C = Capacitance = Q/V = (4 Pi Ke)
A capacitor can be specified by two parameters:
*) Maximum energy density or maximum electric field
*) Voltage between the plates
Eair = Maximum electric field for air before electical breakdown
Emax = Maximum electric field in the capacitor
Rbohr= Bohr radius
= Characteristic size of atoms
= 5.2918e-11 m
= hbar2 / (ElectronMass*ElectronCharge2*Ke)
Ebohr= Bohr electric field
= Field generated by a proton at a distance of 1 Bohr radius
= 5.142e11 Volt/m
Maximum energy density = .5 * 8.854e-12 Emax2
Emax (MVolt/m) Energy density
(Joule/kg)
Al electrolyte capacitor 15.0 1000
Supercapacitor 90.2 36000
Bohr limit 510000 1.2e12 Capacitor with a Bohr electric field
.png)
N = Number of wire loops
Z = Length
A = Area
Mu = Magnetic constant = 4 π 10-7
I = Current
It = Current change/time
F = Magnetic flux = N B A (Tesla meter2)
Ft = Flux change/time (Tesla meter2 / second)
B = Magnetic field = Mu N I / Z
V = Voltage = Ft = L It = N A Bt = Mu N2 A It / Z
L = Inductance = Ft / It = Mu N2 A / Z (Henrys)
E = Energy = .5 L I2
Hyperphysics: Inductor
White: High conductivity
Red: Low conductivity
Electric Thermal Density Electric C/Ct Heat Heat Melt $/kg Young Tensile Poisson Brinell
conduct conduct conduct/ cap cap number hardness
(e7 A/V/m) (W/K/m) (g/cm^3) Density (AK/VW) (J/g/K) (J/cm^3K) (K) (GPa) (GPa) (GPa)
Silver 6.30 429 10.49 .60 147 .235 2.47 1235 590 83 .17 .37 .024
Copper 5.96 401 8.96 .67 147 .385 3.21 1358 6 130 .21 .34 .87
Gold 4.52 318 19.30 .234 142 .129 2.49 1337 24000 78 .124 .44 .24
Aluminum 3.50 237 2.70 1.30 148 .897 2.42 933 2 70 .05 .35 .245
Beryllium 2.5 200 1.85 1.35 125 1.825 3.38 1560 850 287 .448 .032 .6
Magnesium 2.3 156 1.74 1.32 147 1.023 1.78 923 3 45 .22 .29 .26
Iridium 2.12 147 22.56 .094 144 .131 2.96 2917 13000 528 1.32 .26 1.67
Rhodium 2.0 150 12.41 .161 133 .243 3.02 2237 13000 275 .95 .26 1.1
Tungsten 1.89 173 19.25 .098 137 .132 2.54 3695 50 441 1.51 .28 2.57
Molybdenum 1.87 138 10.28 .182 136 .251 2896 24 330 .55 .31 1.5
Cobalt 1.7 100 8.90 .170 .421 1768 30 209 .76 .31 .7
Zinc 1.69 116 7.14 .388 693 2 108 .2 .25 .41
Nickel 1.4 90.9 8.91 .444 1728 15
Ruthenium 1.25 117 12.45 2607 5600
Cadmium 1.25 96.6 8.65 594 2 50 .078 .30 .20
Osmium 1.23 87.6 22.59 .130 3306 12000
Indium 1.19 81.8 7.31 430 750 11 .004 .45 .009
Iron 1.0 80.4 7.87 .449 1811 211 .35 .29 .49
Palladium .95 71.8 1828
Tin .83 66.8 505 22 47 .20 .36 .005
Chromium .79 93.9 .449 2180
Platinum .95 .133 2041
Tantalum .76 .140 3290
Gallium .74 303
Thorium .68
Niobium .55 53.7 2750
Rhenium .52 .137 3459
Vanadium .5 30.7 2183
Uranium .35
Titanium .25 21.9 .523 1941
Scandium .18 15.8 1814
Neodymium .156 1297
Mercury .10 8.30 .140 234
Manganese .062 7.81 1519
Germanium .00019 1211
Diamondiso 10 3320
Diamond e-16 2200 .509
Nanotube 10 3500 Carbon nanotube. Electric conductivity = e-16 laterally
Tube bulk 200 Carbon nanotubes in bulk
Graphene 10 5000
Graphite 2 400 .709 Natural graphite
Al Nitride e-11 180
Brass 1.5 120
Steel 45 Carbon steel
Bronze .65 40
Steel Cr .15 20 Stainless steel (usually 10% chromium)
Quartz (C) 12 Crystalline quartz. Thermal conductivity is anisotropic
Quartz (F) e-16 2 Fused quartz
Granite 2.5
Marble 2.2
Ice 2
Concrete 1.5
Limestone 1.3
Soil 1
Glass e-12 .85
Water e-4 .6
Seawater 1 .6
Brick .5
Plastic .5
Wood .2
Wood (dry) .1
Plexiglass e-14 .18
Rubber e-13 .16
Snow .15
Paper .05
Plastic foam .03
Air 5e-15 .025
Nitrogen .025 1.04
Oxygen .025 .92
Silica aerogel .01
Siemens: Amperes^2 Seconds^3 / kg / meters^2 = 1 Ohm^-1
For most metals,
Electric conductivity / Thermal conductivity ~ 140 J/g/K
Teslas
Field generated by brain 10-12
Wire carrying 1 Amp .00002 1 cm from the wire
Earth magnetic field .0000305 at the equator
Neodymium magnet 1.4
Magnetic resonance imaging machine 8
Large Hadron Collider magnets 8.3
Field for frog levitation 16
Strongest electromagnet 32.2 without using superconductors
Strongest electromagnet 45 using superconductors
Neutron star 1010
Magnetar neutron star 1014
MVolt/meter
Air 3
Glass 12
Polystyrene 20
Rubber 20
Distilled water 68
Vacuum 30 Depends on electrode shape
Diamond 2000
Relative permittivity
Vacuum 1 (Exact)
Air 1.00059
Polyethylene 2.5
Sapphire 10
Concrete 4.5
Glass ~ 6
Rubber 7
Diamond ~ 8
Graphite ~12
Silicon 11.7
Water (0 C) 88
Water (20 C) 80
Water (100 C) 55
TiO2 ~ 150
SrTiO3 310
BaSrTiO3 500
Ba TiO3 ~ 5000
CaCuTiO3 250000
Relative Magnetic Maximum Critical
permeability moment frequency temperature
(kHz) (K)
Metglas 2714A 1000000 100 Rapidly-cooled metal
Iron 200000 2.2 1043
Iron + nickel 100000 Mu-metal or permalloy
Cobalt + iron 18000
Nickel 600 .606 627
Cobalt 250 1.72 1388
Carbon steel 100
Neodymium magnet 1.05
Manganese 1.001
Air 1.000
Superconductor 0
Dysprosium 10.2 88
Gadolinium 7.63 292
EuO 6.8 69
Y3Fe5O12 5.0 560
MnBi 3.52 630
MnAs 3.4 318
NiO + Fe 2.4 858
CrO2 2.03 386
293 K 300 K 500 K
Beryllium 35.6 37.6 99
Magnesium 43.9 45.1 78.6
Aluminum 26.5 27.33 49.9
Copper 16.78 17.25 30.9
Silver 15.87 16.29 28.7
Gauge Diameter Continuous 10 second 1 second 32 ms Resistance
mm current current current current
Ampere Ampere Ampere Ampere Ohm/meter
0 8.3 125 1900 16000 91000 .00032
2 6.5 95 1300 10200 57000 .00051
4 5.2 70 946 6400 36000 .00082
6 4.1 55 668 4000 23000 .00130
12 2.0 20 235 1000 5600 .0052
18 1.02 10 83 250 1400 .021
24 .51 3.5 29 62 348 .084
30 .255 .86 10 15 86 .339
36 .127 .18 4 10 22 1.361
40 .080 1 1.5 8 3.441
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.
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
Isotope Protons Neutrons Natural fraction
Hydrogen-1 1 0 .9998
Hydrogen-2 1 1 .0002
Helium-3 2 1 .000002
Helium-4 2 2 .999998
Lithium-6 3 3 .05
Lithium-7 3 4 .95
Beryllium-9 4 5 1
Boron-10 5 5 .20
Boron-11 5 6 .80
Carbon-12 6 6 .989
Carbon-13 6 7 .011
Teaching simulation for isotopes at phet.colorado.edu
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".
Time = T
Half life = Th
Original mass = M
Mass remaining after time "T" = m = M exp(-T/Th)
Time Mass of element remaining (kg)
0 1
2 1/2
4 1/4
6 1/8
8 1/16
From the point of view of nucleons: Neutron -> Proton + electron + antineutrino
From the point of view of quarks: Down quark -> Up quark + electron + antineutrino
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
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
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.
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.
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 =
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
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
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
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
Deuterium + Tritium -> Helium-4 + Neutron + 17.590 MeV
but the neutrons it produces are a nuisance to the reactor.
Energy/Mass Power/Mass
MJoule/kg Watts/kg
Battery, Lithium ion .8 1200
Battery, Lithium polymer 1.0 1000
Battery, Lithium titanate .4 4000
Battery, Lithium sulfur 1.8 800
Battery, Lithium air 6.1 200
Battery, Aluminum air 4.6 130 Not rechargeable
Capacitor, Aluminum, high power .01 100000
Capacitor, Aluminum, high energy .1 10000
Gasoline combustion motor - 8000
Electric motor - 8000
Electric generator - 200
Flywheel .02 200
Rocket, H2O2 2.7 1000000
Rocket, NH3 + H2O2 6
Rocket, Kerosene + H2O2 8.1 2000000
Rocket, Methane + Oxygen 11.1 2500000
Rocket, Kerosene + Oxygen 10.3 5000000
Rocket, Hydrogen + Oxygen 13.2 1700000
Rocket, Al+NH4NO3 (solid fuel) 6.9 9000000
Nuclear alpha, Plutonium-241 200000 40
Nuclear beta, Tungsten-185 20000 40
Nuclear fission electric 10000000 40
Human, cycling sprint .000188 17.4
Eagle 42
Hummingbird 300
Fusion bomb, D + Li-6 22000000 Large
Energy/Mass Mass fraction
MJoule/kg
Antimatter 90,000,000,000 1
Fusion, D + Li6 268,000,000 .00298
Fusion bomb 25,000,000 .000278 Maximum practical yield of a bomb
Fission, U-235 83,000,000 .000918
Fission bomb 6,000,000 .000067 Maximum practical yield of a bomb
Fission, fast neutron 28,000,000 Fast neutrons, unenriched fuel
Fission, slow neutron 500,000 Slow neutrons, unenriched fuel
Nuclear battery, Co60 4,300,000 Half life 5.3 year
Nuclear battery, Pu238 2,260,000 Half life 88 year
Nuclear battery, Pu241 1,960,000 Half life 14.4 year
Nuclear battery, Sr90 590,000 Half life 29 year
Hydrogen 141.8
Methane 55.5 1 carbon. Natural gas
Ethane 51.9 2 carbons
Propane 50.4 3 carbons
Butane 49.5 4 carbons
Octane 47.8 8 carbons
Kerosene 46 12 carbons
Diesel 46 16 carbons
Oil 46 36 carbons
Fat 37 20 carbons. 9 Calories/gram
Pure carbon 32.8
Coal 32 Similar to pure carbon
Ethanol 29 7 Calories/gram
Wood 22
Sugar 17 4 Calories/gram
Protein 17 4 Calories/gram
Plastic explosive 8.0 HMX
Smokeless powder 5.2 Modern gunpowder
TNT 4.7
Black powder 2.6 Medieval gunpowder
Phosphocreatine .137 Recharges ATP
ATP .057 Adenosine triphosphate
Aluminum capacitor .010
Spring .0003
Battery, aluminum-air 4.68
Battery, Li-S 1.44
Battery, Li-ion .8
Battery, Li-polymer .6
Battery, Alkaline .4
Battery, Lead acid .15
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.
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 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.
Mass of 10 bowling pins / Mass of bowling ball = 2.18
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
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
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.
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
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
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
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.
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
_overhead_photo_in_August_2014.JPG)
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.
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:
Reynolds Soccer Golf Baseball Tennis
number
40000 .49 .48 .49 .6
45000 .50 .35 .50
50000 .50 .30 .50
60000 .50 .24 .50
90000 .50 .25 .50
110000 .50 .25 .32
240000 .49 .26
300000 .46
330000 .39
350000 .20
375000 .09
400000 .07
500000 .07
800000 .10
1000000 .12 .35
2000000 .15
4000000 .18 .30
Data
Drag 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
1672 Newton is the first to note the Magnus effect while observing tennis players
at Cambridge College.
1742 Robins, a British mathematician and ballistics researcher, explains deviations
in musket ball trajectories in terms of the Magnus effect.
1852 The German physicist Magnus describes the Magnus effect.
For a spinning tennis ball,
Velocity = V = 55 m/s Swift groundstroke
Radius = R = .067 m
Area = Area = .0141 m2
Mass = M = .058 kg
Spin number = S = W R / V = .25 Heavy topspin
Spin rate = W = V / R = 205 Hz
Air density = Dair = 1.22 kg/m3
Ball density= Dball
Drag coef = Cdrag = .5 For a sphere
Spin coef = Cspin = 1 For a sphere and for S < .25
Drag force = Fdrag = ½ Cdrag Dair Area V2 = 13.0 Newtons
Spin force = Fspin = ½ Cspin Dair Area V2 S = 6.5 Newtons
Drag accel = Adrag = 224 m/s2
Spin accel = Aspin = 112 m/s2
Gravity = Fgrav = M g
For a rolling ball the spin number is S=1.
V2 S C R-1 Dair/Dball = .0383
Velocity = V
Fluid density = D = 1.22 kg/m2 (Air at sea level)
Cross-sectional area = A
Drag coefficient = C
Drag force = F = ½ C A D V2
Drag power = P = ½ C A D V3 = F V
Drag parameter = K = C A
"Terminal velocity" occurs when the drag force equals the
gravitational force.
M g = ½ C D A V2
Suppose we want to estimate the parachute size required for a soft landing.
Let a "soft landing" be the speed reached if you jump from a height of 2
meters, which is Vt = 6 m/s. If a skydiver has a mass of 100 kg
then the area of the parachute required for this velocity is 46
meters2, which corresponds to a parachute radius of 3.8 meters.
Drag coefficient
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
1672 Newton is the first to note the Magnus effect while observing tennis players
at Cambridge College.
1742 Robins, a British mathematician and ballistics researcher, explains deviations
in musket ball trajectories in terms of the Magnus effect.
1852 The German physicist Magnus describes the Magnus effect.
For a spinning tennis ball,
Velocity = V = 55 m/s Swift groundstroke
Radius = R = .067 m
Area = Area = .0141 m2
Mass = M = .058 kg
Spin number = S = W R / V = .25 Heavy topspin
Spin rate = W = V / R = 205 Hz
Air density = Dair = 1.22 kg/m3
Ball density= Dball
Drag coef = Cdrag = .5 For a sphere
Spin coef = Cspin = 1 For a sphere and for S < .25
Drag force = Fdrag = ½ Cdrag Dair Area V2 = 13.0 Newtons
Spin force = Fspin = ½ Cspin Dair Area V2 S = 6.5 Newtons
Drag accel = Adrag = 224 m/s2
Spin accel = Aspin = 112 m/s2
Gravity = Fgrav = M g
For a rolling ball the spin number is S=1.
V2 S C R-1 Dair/Dball = .0383
Force of the wheel normal to ground = Fnormal
Rolling friction coefficient = Croll
Rolling friction force = Froll = Croll Fnormal
Typical car tires have a rolling drag coefficient of .01 and specialized tires
can achieve lower values.
Croll
Railroad .00035 Steel wheels on steel rails
Steel ball bearings on steel .00125
Racing bicycle tires .0025 8 bars of pressure
Typical bicycle tires .004
18-wheeler truck tires .005
Best car tires .0075
Typical car tires .01
Car tires on sand .3
Wheel diameter = D
Wheel sinkage depth = Z
Rolling coefficient = Croll ≈ (Z/D)½
Car mass = M = 1200 kg
Gravity constant = g = 9.8 m/s2
Tire rolling drag coeff = Cr =.0075
Rolling drag force = Fr = Cr M g = 88 Newtons
Air drag coefficient = Ca = .25
Air density = D = 1.22 kg/meter3
Air drag cross-section = A = 2.0 m2
Car velocity = V = 17 m/s (City speed. 38 mph)
Air drag force = Fa = ½CaADV2 = 88 Newtons
Total drag force = F = Fr + Fa = 176 Newtons
Drag speed = Vd = 17 m/s Speed for which air drag equals rolling drag
Car electrical efficiency = Q = .80
Battery energy = E = 60 MJoules
Work done from drag = EQ = F X = Cr M g [1 + (V/Vd)2] X
Range = X = EQ/(CrMg)/[1+(V/Vd)2] = 272 km
The range is determined by equating the work from drag with the energy delivered
by the battery. E Q = F X.
Drag speed = Vd = [Cr M g / (½ Ca D A)]½ = 4.01 [Cr M /(Ca A)]½ = 17.0 meters/second
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.
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
Speed Power Force Force Force Mass Drag Drag Area Drag Roll Year 100kph
(total) (fluid) (roll) (data) (specs) coef coef time
m/s kWatt kN kN kN ton m2 m2 m2 s
eSkate 5.3 .11 .021 .013 .008 .08 .76 1.0 .01
eScooter Zoomair 7.2 .25 .035 .027 .008 .08 .85 1.0 .01
Bike 10 .30 .035 .030 .005 .10 .49 1.0 .005
Bike 18 1.78 .103 .098 .005 .10 .50 1.0 .005
eBike 250 Watt 8.9 .25 .028 .023 .005 .10 .48 1.0 .005
eBike 750 Watt 10.6 .75 .071 .066 .005 .10 .96 1.0 .005
eBike 1 kWatt 12.5 1.0 .080 .075 .005 .10 .79 1.0 .005
eBike 1.5 kWatt 15.3 1.5 .098 .093 .005 .10 .65 1.0 .005
eBike 3 kWatt 16.7 3.0 .180 .175 .005 .10 1.03 1.0 .005
eBike Stealth H 22.2 5.2 .234 .228 .006 .12 .76 1.0 .005
eBike Wolverine 29.2 7.0 .240 .234 .006 .12 .45 1.0 .005
Bike, record 22.9 3.66 .160 .155 .005 .10 .48 1.0 .005
Bike, steamline 38.7 3.66 .095 .090 .005 .10 .099 .11 .005
Loremo 27.8 45 1.62 1.58 .035 .47 3.39 .25 1.25 .20 .0075 2009
Mitsubishi MiEV 36.1 47 1.30 1.22 .081 1.08 1.53 .35 .0075 2011
Aptera 2 38.1 82 2.15 2.09 .062 .82 2.36 .19 1.27 .15 .0075 2011
Nissan Leaf SL 41.7 80 1.92 1.81 .114 1.52 1.71 .72 2.50 .29 .0075 2012 10.1
Volkswagen XL1 43.9* 55 1.25 1.19 .060 .80 1.01 .28 1.47 .19 .0075 2013 11.9
Chevrolet Volt 45.3 210 4.64 4.52 .121 1.61 3.61 .62 2.21 .28 .0075 2014 7.3
Saab 900 58.3 137 2.35 2.25 .100 1.34 1.09 .66 1.94 .34 .0075 1995 7.7
Tesla S P85 249+ 69.2* 568 8.21 8.06 .150 2.00 2.76 .58 2.40 .24 .0075 2012 3.0
BMW i8 69.4* 260 3.75 3.63 .116 1.54 1.24 .55 2.11 .26 .0075 2015 4.4
Nissan GTR 87.2 357 4.09 3.96 .130 1.74 .85 .56 2.09 .27 .0075 2008 3.4
Lamborghini Dia 90.3 362 4.01 3.89 .118 1.58 .78 .57 1.85 .31 .0075 1995
Porsche 918 94.4 661 7.00 6.88 .124 1.66 1.27 .29 .0075
LaFerrari 96.9 708 7.31 7.19 .119 1.58 1.26 .0075
Lamborghini SV 97.2 559 5.75 5.62 .130 1.73 .98 .0075
Bugatti Veyron 119.7 883 7.38 7.24 .142 1.89 .83 .74 .0075 2005
Hummer H2 242 .218 2.90 2.46 4.32 .57 .0075 2003
Formula 1 .053 .702 .9 .0075 2017
Bus (2 decks) 138 12.6 .005 2012
Subway (R160) 24.7 448 38.6 .0004 2006
Airbus A380 320 435200 1360 1360 21.8
F-22 Raptor 740 231000 312 312 .93
Blackbird SR71 1100 332000 302 302 .41
Skydive, min 40 30 .75 .75 0 .075 .77 1.0 0
Skydive, max 124 101 .75 .75 0 .075 .080 1.0 0
Sub, human power 4.1 1.78 .434 .051
Blue Whale 13.9 3750 270 2.74
Sub, nuke 17.4 30000 1724 11.2 Virginia Class
*: The top speed is electronically limited
Drag (data) Drag parameter obtained from the power and top speed
Data (specs) Drag parameter from Wikipedia
Force (total) Total drag force = Fluid drag force + Roll drag force
Force (fluid) Fluid drag force
Force (roll) Roll drag force
Area Cross section from Wikipedia
Drag coef Drag coefficient from Wikipedia
Roll coef Roll coefficient. Assume .0075 for cars and .005 for bikes.
100kph time Time to accelerate to 100 kph
For the skydiver, the minimum speed is for a maximum cross section (spread
eagled) and the maximum speed is for a minimum cross section (dive).
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
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.
Place Points Place Points
1 25 6 8
2 18 7 6
3 15 8 4
4 12 9 2
5 10 10 1
Power Max Range Motor Battery Battery
speed cost cost energy
kWatt mph miles $ $ MJoule
.75 30 10 160 40 .5
1.5 35 20 240 60 1.2
3 45 40 570 100 1.8
6 55 80 1150 200 3.6
The bikes have one electric wheel and one conventional wheel except for 6 kWatt bike,
which has 2 electric wheels with 3 kWatt each.
Speed Power License
mph kWatt required?
Connecticut 30 1.5 Yes
California 28 .75 No
Massachusetts 25 .75 Yes
Oregon 20 1.0 No
Washington 20 1.0 No
Pennsylvania 20 .75 No
Delaware 20 .75 No
Maryland 20 .5 No
DC 20 ? No
Wing drag force = F→
Wing lift force = F↑
Wing lift-to-drag coef.= Qw = F↑ / F→
Wing length = L
Wing width = W
Wing lift-to-drag coef.= Qw ~ L/W = Wing aspect ratio.
Width = ½ Area / Length
Qw Aspect Wing Wing Wing
ratio length width area
meter meter meter2
U-2 23 10.6 High-altitude spy plane
Albatros 20 1.7 Largest bird
Gossamer 20 10.4 14.6 1.4 41.3 Gossamer albatross, human-powered aircraft
Hang glider 15
Tern 12
Herring Gull 10
Airbus A380 7.5 7.5 36.3 11.6 845
Concorde 7.1 .7 11.4 15.7 358.2
Boeing 747 7 7.9 23.3 11.3 525
Cessna 150 7 2.6 4.5 1.7 15
Sparrow 4
Human wingsuit 2.5 1 1.0 1.0 2
Flying lemur ? Most capable gliding mammal. 2 kg max
Flying squirrel 2.0
Air density = D = 1.22 kg/meter2
Velocity = V
Cross-sectional area = A
Drag coefficient = C
Drag force = F = ½ C D A V2
Human mass = M = 80 kg
Gravity = g = 10 meter/second2
Gravity force = F↓ = 800 Newton
Chute drag coefficient = C = 1 Dimensionless
Air density = D =1.22 kg/meter2
Parachute area = A = 100 meter2
Drag force = F↑ = ½ C D A V2 = F↓
Terminal velocity = V = 3.6 meter/second
Drag force = F = ½ C D A V2
Drag power = P = F V = ½ C D A V3
Drag coef Drag area Power Max speed
dimensionless meter2 Watt meter/second
Bike 1 .5 400 11
Car .4 3 300000 74
Cw
F-4 Phantom .021 (subsonic)
Cessna 310 .027
Airbus A380 .027
Boeing 747 .031
F-4 Phantom .044 (supersonic)
Drag force = F→
Lift force = F↑ = Fgrav
Wing lift/drag ratio = Qw = F↑ / F→
Horizontal speed = V→
Vertical descent speed = V↓
Glide ratio = G = V→ / V↑
Gravitational force = Fgrav
Drag power = Pdrag = F→ V→
Power from gravity = Pgrav = Fgrav V↓
If the glider descends at constant velocity,
Pdrag = Pgrav
The goal of a glider is to maximize the glide ratio
V→ / V↓ = (Pdrag / F→) / (Pgrav / Fgrav)
= Fgrav / F→
= Qw
The glide ratio is equal to the lift coefficient. Qw = G
Air density = D
Wing area = A
Wing drag coefficient = Cw
Wing drag = F→ = ½ Cw D A V2
Wing lift = F↑
Wing lift/drag ratio = Qw = F↑ / F→
Aircraft speed = V
Aircraft mass = M
Gravity = g = 9.8 meters/second2
Gravity force = Fgrav= M g
Engine force = Feng = V F→
Drag power = P→ = F→ V = ½ Cw D A V3
Agility (Power/mass) = p = P→ / M = V g / Qw
For flight at constant velocity,
Feng = F→ Horizontal force balance
F↑ = Fgrav Vertical force balance
F↑ = F→ Qw Definition of the wing lift/drag coefficient
Fgrav= Fdrag Qw → M g = Qw ½ Cw D A V2
Cruising speed = V = M½ g½ Qw-½ (½ Cw D A)-½ ~ M1/6
Agility (Power/mass) = p = M½ g3/2 Qw-3/2 (½ Cw D A)-½ ~ M1/6
Aircraft energy/mass = e ~ M0
Flight time = T = e/p ~ M-1/6
Range = X = V T ~ M0
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
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Speed Mass Takeoff Ceiling Thrust Range Cost Number Year Stealth
Mach ton ton km kN km M$
SR-71 Blackbird 3.3 30.6 78.0 25.9 302 5400 32 1966
MiG-25 Foxbat 2.83 20.0 36.7 20.7 200.2 1730 1186 1970
MiG-31 Foxhound 2.83 21.8 46.2 20.6 304 1450 519 1981
F-22A Raptor 2.51 19.7 38.0 19.8 312 2960 150 195 2005 *
F-15 Eagle 2.5 12.7 30.8 20.0 211.4 4000 28 192 1976
F-14 Tomcat 2.34 19.8 33.7 15.2 268 2960 712 1974
MiG-29 Fulcrum 2.25 11.0 20.0 18.0 162.8 1430 29 1600 1982
Su-35 2.25 18.4 34.5 18.0 284 3600 40 48 1988
F-4 Phantom II 2.23 13.8 28.0 18.3 1500 5195 1958
Chengdu J-10 2.2 9.8 19.3 18.0 130 1850 28 400 2005
F-16 Falcon 2.0 8.6 19.2 15.2 127 1200 15 957 1978
Chengdu J-7 2.0 5.3 9.1 17.5 64.7 850 2400 1966
Dassault Rafale 1.8 10.3 24.5 15.2 151.2 3700 79 152 2001
Euro Typhoon 1.75 11.0 23.5 19.8 180 2900 90 478 2003
F-35A Lightning 1.61 13.2 31.8 15.2 191 2220 85 77 2006 *
B-52 .99 83.2 220 15.0 608 14080 84 744 1952
B-2 Bomber .95 71.7 170.6 15.2 308 11100 740 21 1997 *
A-10C Warthog .83 11.3 23.0 13.7 80.6 1200 19 291 1972
Drone RQ-180 ~15 18.3 ~2200 2015 *
Drone X-47B .95 6.4 20.2 12.2 3890 2 2011 * Carrier
Drone Avenger .70 8.3 15.2 17.8 2900 12 3 2009 *
Drone RQ-4 .60 6.8 14.6 18.3 34 22800 131 42 1998
Drone Reaper .34 2.2 4.8 15.2 5.0 1852 17 163 2007
Drone RQ-170 15 20 2007 *
India HAL AMCA 2.5 14.0 36.0 18.0 250 2800 ? 0 2023 *
India HAL FGFA 2.3 18.0 35.0 20.0 352 3500 ? 0 >2020 *
Mitsubishi F-3 2.25 9.7 ? ? 98.1 3200 ? 1 2024 *
Chengdu J-20 2.0 19.4 36.3 ? 359.8 ? 110 4 2018 *
Sukhoi PAK FA 2.0 18.0 35.0 20.0 334 3500 50 6 2018 *
Shenyang J-31 1.8 17.6 25.0 ? 200 4000 ? 0 2018 *
Mach 1 = 295 m/s
5th generation fighters: F-22, F-35, X-2, HAL AMCA, J-20, J-31, Sukhoi PAK FA
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
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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
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
Speed Mass Range Year
mach 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 17000 2 Test flights
USA X-41 8 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.
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.
Rotor radius = R
Air density = D = 1.22 kg/meter3 at sea level
Rotor tip speed = V
Rotor width param= Cr
Rotor lift force = F↑ = D Cr R2 V2
Rotor drag force = F→
Rotor lift/drag = Qr = F↑ / F→
Rotor power = P = F→ V = F↑ V / Qr
Rotor force/power= Z = F↑/ P
= Qr / V
= R F↑-½ D½ Cr½ Qr
= R F↑-½ D½ qr
Rotor quality = qr = Qr Cr½
The physical parameters of a propeller are {Qr,Cr,qr}, with typical values of
Qr = 5.5
Cr = .045
qr = 1.17
Most propellers have 2 blades and some have 3. If there are 4 or more blades then
qr declines.
K ≈ Aspect ratio
Cr ≈ K-½
Qr ≈ K
qr ≈ K½
Aircraft mass = M
Gravity = g
Aircraft force = F↑ = M g
Rotor radius = R ~ M1/3
Hovering force/power = Z = qr D½ R F↑-½ ~ M-1/6
Hovering power/mass = p = g / Z ~ M1/6
Aircraft energy/mass = e ~ M0
Hovering time = T = e / p ~ M-1/6
Rotor radius = R
Air density = D = 1.22 kg/meter3
Aircraft speed = U
Rotor speed coef. = s
Rotor tip speed = V = s U
Rotor lift force = F↑
Rotor drag force = F↓
Rotor lift/drag = Qr = F↑ / F↓
Rotor power = P = F↓ V = F↑ V / Q
Rotor force/power = Z = Q / V
Typically, Q ~ 5.5 and s ~ 3.
Mass = M
Gravity = g
Rotor quality = q
Hover force = F = M g
Hover power = P
Force/Power ratio = Z = F/P
Power/Mass ratio = p = P/M = g/Z
Air density = Dair= 1.22
Seawater density = Dwater= 1025
Gravity = g = 9.8 meters/second2
Wing drag coef. = Cw = .03
Wing Lift/drag = Qw = 7
Rotor lift/drag = Qr = 5.5
Rotor width param = Cr = .045
Rotor quality = qr = 1.17 = Qr Cr½
Rotor force/power = Zr
Rotor agility = pr = g/Zr
Wing agility = pw
Aircraft mass = M
Gravity = g
Air density = D = 1.22 kg/meter3
Aircraft speed = U
Rotor speed coef. = s
Rotor tip speed = V = s U
Aircraft lift force = F = M g
Rotor lift force = F↑
Wing lift/drag = Qw = F / F↑
Rotor drag force = F→
Rotor lift/drag = Qr = F↑ / F→
Rotor power = P = F→ V = F↑ V / Qr = F V / (Qr Qw)
Aircraft force/power = Z = F / P = [Qr Qw / s] / U
There is a tradeoff between Qr and s.
Hovering time = 25 minutes
Cruise speed = 100 meters/second
Range = 155 km
Hovering power = 40 kWatts
Vehicle mass = 320 kg
Battery energy/mass= .8 MJoules/kg
Battery power/mass = 1200 Watts/kg
Battery cost/MJoule= 100 $/MJoule
Battery mass = 80 kg
Battery energy = 64 MJoules
Battery power = 96 kWatts
Battery cost = 6400 $
Drone mass = M = 1.0 kg
Battery mass = m = .5 kg
Battery energy/mass= e = E/m = .8 MJoules/kg
Battery energy = E = .4 MJoules
Hover power/mass = p = P/M = 94 Watts/kg (Hover power for a 1 kg drone with a 1/4 meter radius rotor)
Hover power = P = p M = 94 Watts
Flight time = T = E/P = 3990 seconds = 66 minutes
The flight time is
T = (e/p)⋅(m/M)
Mass = M = 1 kg
Gravity constant = g = 9.8 meters/second
Rotor radius = R = .25 meters
Rotor quality = q = 1.3
Hover power/mass = p = M½ g3/2 q-1 R-1 = 94 Watts
The rotor radius scales as M1/3 and the hover power/mass scales as M1/6.
If we scale the above vehicle from 1 kg up to 300 kg (the mass of a 1-person vehicle)
the hovering power/mass is 240 Watts/kg and the total power is 73 kWatts, or 98 horsepower.
Max Mass Max Bombs Max Engine Range # Year Bombs * #
speed mass alt Built
kph ton ton ton km kWatt km MTon
UK Avro Lancaster 454 16.6 32.7 10.0 6.5 4x 954 4073 7377 1942 74
USA B-17 Flying Fortress 462 16.4 29.7 3.6 10.5 4x 895 3219 12731 1938 46
USA B-29 Superfortress 574 33.8 60.6 9.0 9.7 4x1640 5230 3970 1944 36
UK Handley Page Halifax 454 17.7 24.7 5.9 7.3 4x1205 3000 6176 1940 36.4
Soviet Petlyakov Pe-2 580 5.9 8.9 1.6 8.8 2x 903 1160 11427 1941 18
UK Short Stirling 454 21.3 31.8 6.4 5.0 4x1025 3750 2371 1939 15.2
Soviet Tupolev Tu-2 528 7.6 11.8 3.8 9.0 2x1380 2020 2257 1942 8.6
Germany Heinkel He 177 565 16.8 32.0 7.2 8.0 2x2133 1540 1169 1942 8.4
Germany Fokke-Wulf Condor 360 17.0 24.5 5.4 6.0 4x 895 3560 276 1937 1.5
Japan Mitsubishi Ki-67 537 8.6 13.8 1.6 9.5 2x1417 3800 767 1942 1.2
Japan Yokosuka P1Y Ginga 547 7.3 13.5 1.0 9.4 2x1361 5370 1102 1944 1.1
Japan Mitsubishi G4M 428 6.7 12.9 1.0 8.5 2x1141 2852 2435 1941 2.4
UK Total 126
USA Total 82
Soviet Total 27
Germany Total 10
Japan Total 5
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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
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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
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
MJoules Rocket Shock Density Boil
/kg km/s km/s g/cm3 Kelvin
Beryllium+ O2 23.2 5.3
Aluminum + O2 15.5
Magnesium+ O2 14.8
Hydrogen + O2 13.2 4.56 .07 20
Kerosene + O3 12.9
Octanitrocubane 11.2 10.6 1.95
Methane + O2 11.1 3.80 .42 112 CH4
Octane + O2 10.4 .70 399 C8H18
Kerosene + O2 10.3 3.52 .80 410 C12H26
Dinitrodiazeno. 9.2 10.0 1.98
C6H6N12O12 9.1 1.96 China Lake compound
Kerosene + H2O2 8.1 3.2
Kerosene + N2O4 8.0 2.62
HMX (Octogen) 8.0 3.05 9.1 1.86
RDX (Hexagen) 7.5 2.5 8.7 1.78
Al + NH4NO3 6.9
Nitroglycerine 7.2 8.1 1.59 Unstable
PLX 6.5 1.14 95% CH3NO2 + 5% C2H4(NH2)2
Composition 4 6.3 8.04 1.59 91% RDX. "Plastic explosive"
Kerosene + N2O 6.18
Dynamite 5.9 7.2 1.48 75% Nitroglycerine + stabilizer
PETN 5.8 8.35 1.77
Smokeless powder 5.2 6.4 1.4 Used after 1884. Nitrocellulose
TNT 4.7 6.9 1.65 Trinitrotoluene
Al + Fe2O3 4.0 Thermite
H2O2 2.7 1.59 1.45 423 Hydrogen peroxide
Black powder 2.6 .6 1.65 Used before 1884
Al + NH4ClO4 2.6
NH4ClO4 2.5
N2O 1.86 1.76
N2H4 1.6 2.2 1.02 387 Hydrazine
Bombardier beetle .4 Hydroquinone + H2O2 + protein catalyst
N2O4 .10 1.45 294
Rocket: Rocket exhaust speed
Shock: Shock speed
MJoules Shock Density
/kg km/s g/cm3
Octanitrocubane 11.2 10.6 1.95
Dinitrodiazeno. 9.2 10.0 1.98
C6H6N12O12 9.1 1.96 China Lake compound
HMX (Octogen) 8.0 9.1 1.86
RDX (Hexagen) 7.5 8.7 1.78
PLX 6.5 1.14 95% CH3NO2 + 5% C2H4(NH2)2
Composition 4 6.3 8.04 1.59 91% RDX. "Plastic explosive"
Dynamite 5.9 7.2 1.48 75% Nitroglycerine + stabilizer
PETN 5.8 8.35 1.77
Exhaust Energy Density of fuel + oxidizer
speed /mass
km/s MJ/kg g/cm3
Hydrogen H2 4.46 13.2 .32
Methane CH4 3.80 11.1 .83
Ethane C2H6 3.58 10.5 .9
Kerosene C12H26 3.52 10.3 1.03
Hydrazine N2H4 3.46 1.07
Liquid hydrogen is usually not used for the ground stage of rockets because of
its low density.
Energy/Mass Energy/Mass Rocket Rocket Boil Density
with kerosene as monopropellant with kerosene as monopropellant Kelvin g/cm3
MJoule/kg MJoule/kg km/s km/s
O3 12.9 2.97 161
O2 10.3 0 3.52 0 110 1.14
H2O2 8.1 2.7 3.2 1.6 423 1.45
N2O4 8.00 .10 2.62 294 1.44
N2O 6.18 1.86 1.76 185
N2H4 - 1.58 2.2 387 1.02
MJoules Rocket Density
/kg km/s g/cm3
C6H6N12O12 9.1 1.96 China Lake compound
HMX (Octogen) 8.0 3.05 1.86
RDX (Hexagen) 7.5 2.5 1.78
Al + NH4ClO4 2.6
NH4ClO4 2.5
NH3OHNO3 2.5 1.84 Hydrxyammonium nitrate
Al + NH4NO3 6.9
NH4NO3 1.4 2.0 1.12 Ammonium nitrate
~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
Black powder = .75 KNO3 + .19 Carbon + .06 Sulfur
