The speed of sound waves being likewise independent of the motion of the source.

Material bodies contract when they are moving, and that this foreshortening is only in the direction of the motion.

What happens if a constant force acts on a body for a long time? In relativity, the body keeps picking up, not speed, but momentum, which can continually increase because the mass is increasing. After a while there is practically no acceleration in the sense of a change of velocity, but the momentum continues to increase. ... The inertia is very great when v is nearly as great as c. 

Consider the motion of the molecules in a small tank of gas. When the gas is heated, the speed of the molecules is increased, and therefore the mass is also increased and the gas is heavier. The increase in mass of all this body of gas is equal to the increase in kinetic energy divided by c^2.


If the derivatives of two quantities are equal, the quantities themselves differ at most by a constant, say C.

This theory of equivalence of mass and energy has been beautifully verified by experiments in which matter is annihilated - converted totally to energy: An electron and a positron come together at rest, each with a rest mass m0. When they come together they disintegrate and two gamma rays emerge, each with the measured energy of m0*c^2.



The result is just  as significant in chemistry. For instance, if we were to weigh the carbon dioxide molecule and compare its mass with that of the carbon and the oxygen, we could find out how much energy would be liberated when carbon and oxygen form carbon dioxide. The only trouble here is that the differences in masses are so small that it is technically very difficult to do.



We could use any notation we want; do not laugh at notations; invent them, they are powerful. In fact, mathematics is, to a large extent, invention of better notations.

The energy of a photon is a certain constant, called Planck's constant, times the frequency of the photon: E=hv.

Such a photon also carries a momentum, and the momentum of a photon (or of any other particle, in fact) is h divided by the wavelength.

The number of waves per second, times the wavelength of each, is the distance that the light goes in one second, which, of course, is c.

Thus we see immediately that the energy of a photon must be the momentum times c.

"Space of itself, and time of itself will sink into mere shadows, and only a kind of union between them shall survive."



The external force on the total object is equal to the sum of all the forces on all its constituent particles.

There is a very interesting relationship between rotation in two dimensions and one-dimensional displacement, in which almost every quantity has its analog.

Torque bears the same relationship to rotation as force does to linear movement.

The torque is also often called the moment of the force.

The amount of twist, or torque, is proportional both to the radial distance and to the tangential component of the force.

The formula for the torque can also be written as the magnitude of the force times the length of the lever arm.

Just as external force is the rate of change of a quantity p, which we call the total momentum of a collection of particles, so the external torque is the rate of change of a quantity L which we call the angular momentum of the group of particles.

If we want to know the angular momentum of a particle about an axis, we take only the component of the momentum that is tangential, and multiply it by the radius.

The angular momentum is the magnitude of the momentum times the momentum lever arm.

The angular momentum of the planet going around the sun must remain constant. 

The rate of change of the total angular momentum about any axis is equal to the external torque about that axis!

The law of conservation of angular momentum: if no external torques act upon a system of particles, the angular momentum remains constant.

The moment of inertia is analogous to the mass. A body has inertia for turning which depends, not just on the masses, but on how far away they are from the axis.

The moment of inertia is the inertia against turning, and is the sum of the contributions of all the masses, times their distances squared, from the axis.

There is one important difference between mass and moment of inertia. The mass of an object never changes, but its moment of inertia can be changed.

If the external torque is zero, then the angular momentum, the moment of inertia times omega, remains constant. If we reduce the moment of inertia, we have to increase the angular velocity.



Newton's law has the peculiar property that if it is right on a certain small scale, then it will be right on a larger scale.

In case the object is so large that the nonparallelism of the gravitational forces is significant, one must distinguish between the center of mass and the center of gravity.

The theorem that torque equals the rate of change of angular momentum is true in two general cases: (1) a fixed axis in inertial space, (2) an axis through the center of mass, even though the object may be accelerating.

Theorem of Pappus works like this: if we take any closed area in a plane and generate a solid by moving it through space such that each point is always moved perpendicular to the plane of the area, the resulting solid has a total volume equal to the area of the cross section times the distance that the center of mass moved!

If we want to locate the center of mass of a plane sheet of uniform density, we can remember that the volume generated by spinning it about an axis is the distance that the center of mass goes around, times the area of the sheet.

The center of mass of any uniform triangular area is where the three medians, the lines from the vertices through the centers of the opposite sides, all meet. That point is 1/3 of the way along each median. Clue: Slice the triangle up into a lot of little pieces, each parallel to a base. Note that the median line bisects every piece, and therefore the center of mass must lie on this line.

To summarize, the moment of inertia of an object about a given axis, which we shall call the z-axis, has the following properties: (1)... (2)If the object is made of a number of parts, each of whose moment of inertia is known, the total moment of inertia is the sum of the moments of inertia of the pieces. (3)The moment of inertia about any given axis is equal to the moment of inertia about a parallel axis through the CM plus the total mass times the square of the distance from the axis to the CM. This theorem is called the parallel-axis theorem. (4)If the object is a plane figure, the moment of inertia about an axis perpendicular to the plane is equal to the sum of the moments of inertia about any two mutually perpendicular axes lying in the plane and intersecting at the perpendicular axis. (I.e., if one has a plane figure and a set of coordinate axes with origin in the plane and z-axis perpendicular to the plane, then the moment of inertia of this figure about the z-axis is equal to the sum of the moments of inertia about the x- and y-axes.)

What is the kinetic energy of a rigid body, rotating about a certain axis with an angular velocity w? We can immediately guess the correct answer by using our analogies. The moment of inertia corresponds to the mass, angular velocity corresponds to velocity, and so the kinetic energy ought to be 0.5*I*w^2, and indeed it is.

When we are rotating, there is centrifugal force on the weights. They are trying to fly out, so when we are going around we have to pull the weights in against the centrifugal force. So, the work we do against the centrifugal force ought to agree with the difference in rotational energy, and of course it does. That is where the extra kinetic energy comes from. 

Among the forces that are developed in a rotating system, centrifugal force is not the entire story, there is another force. This other force is called Coriolis force, and it has the very strange property that when we move something in a rotating system, it seems to be pushed sidewise.



Although a torque is a twist on a plane, and it has no a priori vector character, mathematically it does behave like a vector.

The external torque on a system is the rate of change of the total angular momentum.

The law of conservation of angular momentum: if the total external torque is zero, then the total vector angular momentum of the system is a constant. If there is no torque on a given system, its angular momentum cannot change.

What about angular velocity? Is it a vector? We have already discussed turning a solid object about a fixed axis, but for a moment suppose that we are turning it simultaneously about two axes. It might be turning about an axis inside a box, while the box is turning about some other axis. The net result of such combined motions is that the object simply turns about some new axis! The wonderful thing about this new axis is that it can be figured out this way. If the rate of turning in the xy-plane is written as a vector in the z-direction whose length is equal to the rate of rotation in the plane, and if another vector is drawn in the y-direction, say, which is the rate of rotation in the zx-plane, then if we add these together as a vector, the magnitude of the result tells us how fast the object is turning, and the direction tells us in what plane, by the rule of the parallelogram. That is to say, simply, angular velocity is a vector, where we draw the magnitudes of the rotations in the three planes as projections at right angles to those planes.

Cycloid - the path followed by a pebble that is stuck in the tread of an automobile tire.

The slower the gyroscope spins, the more obvious the nutation is. 

When the motion settles down, the axis of the gyro is a little bit lower than it was at the start.

The angular momentum of a rigid body is not necessarily in the same direction as the angular velocity.

Any rigid body, even an irregular one like a potato, possesses three mutually perpendicular axes through the CM, such that the moment of inertia about one of these axes has the greatest possible value for any axis through the CM, the moment of inertia about another of the axes has the minimum possible value, and the moment of inertia about the third axis is intermediate between these two (or equal to one of them). These axes are called the principal axes of the body, and they have the important property that if the body is rotating about one of them, its angular momentum is in the same direction as the angular velocity. For a body having axes of symmetry, the principal axes are along the symmetry axes.



A linear differential equation with constant coefficients is a differential equation consisting of a sum of several terms, each term being a derivative of the dependent variable with respect to the independent variable, and multiplied by some constant.

One of the most important properties of linear differential equations: if we multiply a solution of the equation by any constant, it is again a solution.

The physics of it is the following. If we have a weight on a spring, and pull it down twice as far, the force is twice as much, the resulting acceleration is twice as great, the velocity it acquires in a given time is twice as great, the distance covered in a given time is twice as great; but it has to cover twice as great a distance in order to get back to the origin because it is pulled down twice as far. So it takes the same time to get back to the origin, irrespective of the initial displacement. In other words, the motion has the same time pattern, no matter how "strong" it is.

If we had a heavier mass, it would take longer to oscillate back and forth on a spring. That is because it has more inertia, and so, while the forces are the same, it takes longer to get the mass moving. Or, if the spring is stronger, it will move more quickly, and that is right: the period is less if the spring is stronger.

The period of oscillation of a mass on a spring does not depend in any way on how it has been started, how far down we pull it.

The period is determined, but the amplitude of the oscillation is not determined by the equation of motion.

The amplitude is determined, in fact, by how we let go of it, by what we call the initial conditions or starting conditions.

The angular frequency is the number of radians by which the phase changes in a second. 

The amplitude of oscillation measures the maximum displacement attained by the mass.

When a particle is moving in a circle, the horizontal component of its motion has an acceleration which is proportional to the horizontal displacement from the center.

The energy is dependent on the square of the amplitude; if we have twice the amplitude, we get an oscillation which has four times the energy.



The negative powers are the reciprocals of the positive powers.



The complex number F that we have so defined is not a real physical force, because no force in physics is really complex; actual forces have no imaginary part, only a real part. We shall, however, speak of the "force" F0*e^(iwt), but of course the actual force is the real part of that expression.

We must emphasize that this separation into a real part and an imaginary part is not valid in general, but is valid only for equations which are linear, that is, for equations in which x appears in every term only in the first power or the zeroth power.

Differentiation is now as easy as multiplication! This idea of using exponentials in linear differential equations is almost as great as the invention of logarithms, in which multiplication is replaced by addition. Here differentiation is replaced by multiplication.

If we think of the charge q on a capacitor as being analogous to the displacement x of a mechanical system, we see that the current, I=dq/dt, is analogous to velocity, 1/C is analogous to a spring constant k. The self-inductance is analogous to the mass in a mechanical oscillating circuit.

The difficulties of science are to a large extent the difficulties of notations, the units, and all the other artificialities which are invented by man, not by nature.



If A is represented by a complex number, then the mean of A^2 is equal to 0.5*A0^2.



When we have no force acting, and suddenly turn one on, we do not immediately get the steady solution that we solved for with the sine wave solution, but for a while there is a transient which sooner or later dies out, if we wait long enough. The "forced" solution does not die out, since it keeps on being driven by the force.

The principle of superposition for linear systems means the following: if we have a complicated force which can be broken up in any convenient manner into a sum of separate pieces, each of which is in some way simple, in the sense that for each special piece into which we have divided the force we can solve the equation, then the answer is available for the whole force, because we may simply add the pieces of the solution back together, in the same manner as the total force is compounded out of pieces.

Finally, we make some remarks on why linear systems are so important. The answer is simple: because we can solve them! So most of the time we solve linear problems. Second (and most important), it turns out that the fundamental laws of physics are often linear. The Maxwell equations for the laws of electricity are linear, for example. The great laws of quantum mechanics turn out, so far as we know, to be linear equations. ... We mention another situation where linear equations are found. When displacements are small, many functions can be approximated linearly.

Mathematical analysis is not the grand thing it is said to be; it solves only the simplest possible equations. As soon as the equations get a little more complicated, just a shade - they cannot be solved analytically. But the numerical method can take care of any equation of physical interest.

The unit of capacitance, C, is the farad; a charge of one coulomb on each plate of a one-farad capacitor yields a voltage difference of one volt.

An analog computer is a device which imitates the problem that we want to solve by making another problem, which has the same equation, but in another circumstance of nature, and which is easier to build, to measure, to adjust, and to destroy!

Kirchhoff's laws for electrical circuits: (1)At any junction, the sum of the currents into a junction is zero. That is, all the current which comes in must come back out. (2)If we carry a charge around any loop, and back to where it started, the net work done is zero.