Our modern understanding is the following: chemical energy has two parts, kinetic energy of the electrons inside the atoms, so part of it is kinetic, and electrical energy of interaction of the electrons and the protons - the rest of it, therefore, is electrical.



Galileo's first experiments on motion were done by using his pulse to count off equal intervals of time.

We cannot "see" objects smaller than the wavelength of visible light (about 5*10^(-7) meter).



The set of numbers which appears in such a diagram is known as Pascal's triangle. The numbers are also known as the binomial coefficients, because they also appear in the expansion of (a+b)^n. The number of "ways" to any point on the diagram is just the number of different "paths" (sequences of heads and tails) which can be taken from the starting point. The total number of possible sequences is 2^n (since there are 2 outcomes for each toss).

In its general form the problem of the "random walk" is related to the motion of atoms (or other particles) in a gas - called Brownian motion - and also to the combination of errors in measurements.

A probability distribution.  This case is more representative of something like the thermal motion of a molecule in a gas.



Every object in the universe attracts every other object with a force which for any two bodies is proportional to the mass of each and varies inversely as the square of the distance between them.

An object responds to a force by accelerating in the direction of the force by an amount that is inversely proportional to the mass of the object.

To find something out, it is better to perform some careful experiments than to carry on deep philosophical arguments.

Kepler's three laws: I.Each planet moves around the sun in an ellipse, with the sun at one focus. II.The radius vector from the sun to the planet sweeps out equal areas in equal intervals of time. III.The squares of the periods of any two planets are proportional to the cubes of the semimajor axes of their respective orbits.

The net result is that we get two tidal bulges.

Another topic deserving discussion is Einstein's modification of Newton's law of gravitation. Einstein advanced arguments which suggest that we cannot send signals faster than the speed of light, so Newton's law of gravitation must be wrong. By correcting it to take the delays into account, we have a new law, called Einstein's law of gravitation. One feature of this new law which is quite easy to understand is this: In the Einstein relativity theory, anything which has energy has mass - mass in the sense that it is attracted gravitationally. Even light, which has an energy, has a "mass." When a light beam, which has energy in it, comes past the sun there is an attraction on it by the sun.



The quantity ds/dt is called the "derivative of s with respect to t", and the complicated process of finding it is called finding a derivative, or differentiating. The ds's and dt's which appear separately are called differentials.

One can get formulas for integrals by taking the formulas for derivatives and running them backwards, because they are related to each other inversely.

Acceleration is defined as the time rate of change of velocity.

Acceleration is the derivative of the velocity with respect to the time.

The velocity is equal to the integral of the acceleration. Distance is the integral of the velocity, so distance can be found by twice integrating the acceleration.



The time-rate-of-change of a quantity called momentum is proportional to the force.

We carefully distinguish velocity, which has both magnitude and direction, from speed, which we choose to mean the magnitude of the velocity, but which does not include the direction.

The component of the force in the x-, y-, or z-direction is equal to the mass times the rate of change of the corresponding component of velocity.

According to Newton's Second Law, force is the time rate of change of the momentum.

The law of conservation of momentum: the total momentum of the two particles does not change because of any mutual interactions between them.

The sum of all external forces equals the rate of change of the total momentum of all the particles inside.

Relativity principle: the laws of physics will look the same whether we are standing still or moving with a uniform speed in a straight line.

Symmetrical objects: if the objects are alike there is no reason for right or left to be preferred and so the bodies would do something that is symmetrical.

This is again a symmetrical situation, with no preference between right and left, so we assume that they stand still.

One professor has given this definition of symmetry: a thing is symmetrical if one can subject it to a certain operation and it appears exactly the same after the operation.

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 three numbers which describe the quantity in a given coordinate system are called the components of the vector in the direction of the coordinate axes of that system.

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



It is always important to ask, "What does it mean?"

In order to understand physical laws you must understand that they are all some kind of approximation. Any simple idea is approximate.

It is important to realize that each of these empirical laws has its limitations, beyond which it does not really work.

The mechanism of power loss is that as the slider snaps over the bumps, the bumps deform and then generate waves and atomic motions and, after a while, heat, in the two bodies.

The coefficient of friction (mu) is only roughly a constant, and varies from place to place along the plane. The tables that list purported values of mu for "steel on steel," "copper on copper," and the like, are all false, because they ignore the factors, which really determine mu.

Many people believe that the friction to be overcome to get something started (static friction) exceeds the force required to keep it sliding (sliding friction), but with dry metals it is very hard to show any difference. The opinion probably arises from experiences where small bits of oil or lubricant are present, or where blocks, for example, are supported by springs or other flexible supports so that they appear to bind.

The apparent friction is much reduced if the lower surface vibrates very fast.

Apparent decreases of the friction at high speed are often due to vibrations.

At the present time it is impossible even to estimate the coefficient of friction between two substances.

Molecular forces have never been satisfactorily explained on a basis of classical physics; it takes quantum mechanics to understand them fully.

If the displacement is not too great the force is proportional to the displacement. This principle is known as Hooke's law, or the law of elasticity, which says that the force in a body which tries to restore the body to its original condition when it is distorted is proportional to the distortion. This law, of course, holds true only if the distortion is relatively small; when it gets too large the body will be torn apart or crushed, depending on the kind of distortion. The amount of force for which Hooke's law is valid depends upon the material.

If many charges are present, we first work out the total electric field produced at point R by all the charges, and then, knowing the charge that is placed at R, we find the force on it.

The principle of superposition of fields: the total field due to all the sources is the sum of the fields due to each source.

Although the principle of superposition applies exactly for electrical forces, it is not exact for gravity if the field is too strong, and Newton's equation C=C1+C2+C3+... is only approximate, according to Einstein's gravitational theory.

One very important feature of pseudo forces is that they are always proportional to the masses.

Einstein put forward the hypothesis that accelerations give an imitation of gravitation, that the forces of acceleration (the pseudo forces) cannot be distinguished from those of gravity; it is not possible to tell how much of a given force is gravity and how much is pseudo force.

As time goes on, the changes in kinetic energy and in the quantity mgh are equal and opposite, so that the sum of the two quantities remains constant. Q.E.D.

We have shown, from Newton's second law of motion, that energy is conserved for constant forces when we add the potential energy to the kinetic energy.

The force acting on an object times the velocity of the object (vector "dot" product) is the power being delivered to the object by that force.

The rate of change of kinetic energy of an object is equal to the power expended by the forces acting on it.

If an object is moving in any way under the influence of a force, moving in some kind of curved path, then the change in K.E. when it goes from one point to another along the curve is equal to the integral of the component of the force along the curve times the differential displacement ds, the integral being carried out from one point to the other. This integral is called the work done by the force on the object.

First we shall find the gravitational force on a mass that is produced by a plane sheet of material, infinite in extent. ... The force is independent of distance! If we are close, most of the matter is pulling at an unfavorable angle; if we are far away, more of the matter is situated more favorably to exert a pull toward the plane. At any distance, the matter which is most effective lies in a certain cone. When we are farther away the force is smaller by the inverse square, but in the same cone, in the same angle, there is much more matter, larger by just the square of the distance! This analysis can be made rigorous by just noticing that the differential contribution in any given cone is in fact independent of the distance, because of the reciprocal variation of the strength of the force from a given mass, and the amount of mass included in the cone, with changing distance.

The potential energy is easier to work with than is the field because we do not have to worry about angles, we merely add the potential energies of all the pieces of mass.

If the potential energy is the same no matter where an object is placed inside the sphere, there can be no force on it. So there is no force inside.

The important thing to learn and to remember is the relationship, not the proof.

If the force 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.

There are two kinds of muscles in the human body and in other animals: one kind, called striated or skeletal muscle, is the type of muscle we have in our arms, for example, which is under voluntary control; the other kind, called smooth muscle, is like the muscle in the intestines or, in the clam, the greater adductor muscle that closes the shell.

To go away from the earth, we need 2^(0.5) times the velocity we need to just go around the earth near its surface. We need, in other words, twice as much energy (because energy goes as the square of the velocity) to leave the earth as we do to go around it.

The idea of force is not particularly suitable for quantum mechanics; there the idea of energy is most natural. 

All the fundamental forces in nature appear to be conservative.

Inside the shell the potential turns out to be a constant! When the potential is constant, there is no field, or when the potential energy is constant there is no force.

The force in the x-direction is minus the partial derivative of the potential energy with respect to x.

"Grad" or "gradient" is not a quantity but an operator that makes a vector from a scalar.