The Feynman Lectures on Physics, Vol. I: The New Millennium Edition: Mainly Mechanics, Radiation, and Heat (Volume 1)

by Richard P. Feynman

This result, which has now been established for both translation and rotation of axes, has certain consequences: first, no one can claim his particular axes are unique, but of course they can be more convenient for certain particular problems. For example, it is handy to have gravity along one axis, but this is not physically necessary. Second, it means that any piece of equipment which is completely self-contained, with all the force-generating equipment completely inside the apparatus, would work the same when turned at an angle.

Not only Newton’s laws, but also the other laws of physics, so far as we know today, have the two properties which we call invariance (or symmetry) under translation of axes and rotation of axes.

The fact that a physical relationship can be expressed as a vector equation assures us the relationship is unchanged by a mere rotation of the coordinate system. That is the reason why vectors are so useful in physics.

Vector algebra

Thus velocity is a vector because it is the difference of two vectors. It is also the right definition of velocity because its components are , , and .

Newton’s laws in vector notation

Scalar product of vectors

The quantity is called the scalar product of two vectors, and

What good is the dot product?

By a unit vector we mean one whose dot product with itself is equal to unity.

Characteristics of Force

What is a force?

One of the most important characteristics of force is that it has a material origin, and this is not just a definition.

If you insist upon a precise definition of force, you will never get it! First, because Newton’s Second Law is not exact, and second, because in order to understand physical laws you must understand that they are all some kind of approximation.

One may be dissatisfied with the approximate view of nature that physics tries to obtain (the attempt is always to increase the accuracy of the approximation), and may prefer a mathematical definition; but mathematical definitions can never work in the real world. A mathematical definition will be good for mathematics, in which all the logic can be followed out completely, but the physical world is complex, as we have indicated in a number of examples, such as those of the ocean waves and a glass of wine.

when one dissolves in the other? The forces on a single thing already involve approximation, and if we have a system of discourse about the real world, then that system, at least for the present day, must involve approximations of some kind.

Friction

Molecular forces

Fundamental forces.

Fields

It would be trivial, just another way of writing the same thing, if the laws of force were simple, but the laws of force are so complicated that it turns out that the fields have a reality that is almost independent of the objects which create them.

Pseudo forces

One very important feature of pseudo forces is that they are always proportional to the masses; the same is true of gravity. The possibility exists, therefore, that gravity itself is a pseudo force. Is it not possible that perhaps gravitation is due simply to the fact that we do not have the right coordinate system? After all, we can always get a force proportional to the mass if we imagine that a body is accelerating. For instance, a man shut up in a box that is standing still on the earth finds himself held to the floor of the box with a certain force that is proportional to his mass. But if there were no earth at all and the box were standing still, the man inside would float in space. On the other hand, if there were no earth at all and something were pulling the box along with an acceleration , then the man in the box, analyzing physics, would find a pseudo force which would pull him to the floor, just as gravity does.

Einstein found that gravity could be considered a pseudo force only at one point at a time, and was led by his considerations to suggest that the geometry of the world is more complicated than ordinary Euclidean geometry.

Nuclear forces

Work and Potential Energy (A)

Energy of a falling body

For clarity we shall start with the simplest possible example, and then develop harder and harder examples.

We thus have a marvelous theorem: the rate of change of kinetic energy of an object is equal to the power expended by the forces acting on it.

Work done by gravity

the work done in going around any path in a gravitational field is zero.

Summation of energy

Gravitational field of large objects

Work and Potential Energy (conclusion)

Work

Physical work is expressed as , called “the line integral of dot ,” which means that if the force, for instance, is in one direction and the object on which the force is working is displaced in a certain direction, then only the component of force in the direction of the displacement does any work.

Constrained motion

In motion with a fixed frictionless constraint, no work is done by the constraint because the forces of constraint are always at right angles to the motion.

the work done by the net force is equal to the sum of the works done by all the parts

Conservative forces

if only conservative forces act, the kinetic energy plus the potential energy remains constant:

We find that although forces and velocities “dissolve” and disappear when we consider the more advanced forces between nuclear matter and between molecules and so on, the energy concept remains. Therefore we find curves of potential energy in quantum mechanics books, but very rarely do we ever see a curve for the force between two molecules, because by that time people who are doing analyses are thinking in terms of energy rather than of force.

Nonconservative forces

As a matter of fact, all the fundamental forces in nature appear to be conservative.

Potentials and fields

When the potential is constant, there is no field, or when the potential energy is constant there is no force, because if we move an object from one place to another anywhere inside the sphere the work done by the force is exactly zero. Why? Because the work done in moving the object from one place to the other is equal to minus the change in the potential energy (or, the corresponding field integral is the change of the potential). But the potential energy is the same at any two points inside, so there is zero change in potential energy, and therefore no work is done in going between any two points inside the shell. The only way the work can be zero for all directions of displacement is that there is no force at all.

It is easy to show from Eq. (12.11) that the force on a particle due to magnetic fields is always at right angles to its velocity, and also at right angles to the field. Since the force due to magnetism on a moving charge is at right angles to the velocity, no work is done by the magnetism on the moving charge because the motion is at right angles to the force. Therefore, in calculating theorems of kinetic energy in electric and magnetic fields we can disregard the contribution from the magnetic field, since it does not change the kinetic energy.) We suppose that there is only an electric field.

The Special Theory of Relativity

The principle of relativity

The principle of relativity was first stated by Newton, in one of his corollaries to the laws of motion: “The motions of bodies included in a given space are the same among themselves, whether that space is at rest or moves uniformly forward in a straight line.”