Mechanics.

Today the academic discipline of mechanics includes two components: kinematics and dynamics. Kinematics is the "geometry of motion" using only the concepts of position, velocity and acceleration. Dynamics introduces the concepts of mass and force.

The so-called "scientific revolution" of the 16th and 17th centuries put the study of mechanics on a firm mathematical foundation, with models that were capable of making predictions that could be experimentally tested.

Copernicus and Kepler model the planetary system.

The Copernican heliocentric solar system model was drastically simplified (removing all epicycles) and put on a mathematically sound basis with Kepler's three laws of planetary motion:

1. Each planet moves in an elliptical orbit with the sun at one focus of the ellipse.
2. The radius vector of a planet sweeps equal areas in equal times.
3. The square of the orbital period of a planet is equal to the cube of its mean orbital radius.

Galileo finds laws of terrestrial motion.

Galileo studied terrestrial motion of bodies with constant acceleration, which included falling bodies and projectiles. He defined the concept of acceleration, and found three laws of constantly accelerated motion:

1. d  =  (t/2) (vo  +  v)
2. d  =  vot  +  (1/2)at2
3. v2  =  vo2  +  2a •d

Newtonian dynamics.

Newton looked at motion in a more general way. His great accomplishment was to unify the understanding of motion to include both terrestrial (earthly) and celestial (heavenly) motions. He introduced (rather, refined and clarified) two concepts: mass and force. We state his three laws using somewhat modern language.

1. A body at rest remains at rest unless a force acts upon it. An object in a state of uniform motion remains in that state of motion unless an external force acts upon it.
2. When a body is acted upon by an external force, it accelerates. The relation between the force and the acceleration is F  =  ma.
3. When body A exerts a force on body B, then B exerts and equal and oppositely directed force on A.

We can state the content of these laws in a more compact way as equations:

1st and second law: F  =  ma

3rd law: Fab  =   -  Fba

Force and acceleration are vector quantities, so we indicate that fact with boldface symbols. Mass is a scalar quantity.

The compact notation is nice, but we should always do more than memorize equations. Me must have clearly in mind (a) what every symbol means, and (b) any qualifications of limitations on the equations. I.e., we need to know the context in which the equations apply.

Here it's absolutely crucial that we think, as we read the first equation: "The net (vector sum) of all the forces acting on the mass is equal to that mass times the acceleration of that mass." Too often students include forces acting on some other mass or forget to find the vector sum of all forces acting on the body in question. Note that the two forces in the second equation act on different bodies.

These laws seem innocent enough, but contain a powerful amount of information and insight. Philosophers have discussed the implications of F  =  ma at great length. Is it a definition of force? Is it a definition of mass? Is it a law of nature? One might argue that it is all three at once. The third law is just as important, and absolutely essential to apply the first two laws meaningfully. We will see later that the third law figures importantly in the law of conservation of momentum. In like fashion, Galileo's third equation is edging close to the concepts of work and kinetic energy, once it is combined with Newton's F  =  ma.

Newton's gravitational law.

Galileo's work showed that bodies near the earth have constant acceleration directed toward the earth. The climate of thought about heavenly and earthly laws was changing, and people now asked questions such as "What keeps the planets moving?" and "What keeps the moon in its orbit?" Asking such questions supposes that there's an underlying reason similar to the reasons for projectile motion. Of course, no one had reasons for either, but many were beginning to suspect that there were unifying principles and laws that might embrace both earthly and celestial phenomena.

Newton pondered the question of what kept the moon in its orbit, and the answer did not come in a flash of insight as he observed an apple fall from a tree. But he did finally conclude that the moon was moving in a circular orbit because it was continually falling toward the earth. In one of his papers he shows a remarkable picture illustrating this, and nicely relating the moon's orbit to the cannonball trajectories of Galileo.

Imagine an impossibly high mountain on the earth, as shown in the figure. A cannonball fired from the top of the mountain would fall to earth, but if it were fired with higher velocity, it would have a greater range. If one could somehow increase the firing velocity enough, it would fall into a circular orbit completely around the earth. The moon is falling in such a near-circular orbit.

If the firing velocity of the cannonball is still larger, it could move in an elliptical orbit. This diagram illustrated the idea that all of these motions are due to the same cause, an acceleration downward, that is, always directed toward the center of the earth.

Aristotle had considered motion toward the earth to be "natural", requiring no force. For Aristotle, straight line motion would be "unnatural" and require a force. Newton is turning this on its head, saying that straight line motion requires no force, while all other paths do require a force. He further says that that force is always in the direction of the acceleration, and related to the acceleration by F  =  ma. Bodies fall toward the earth because the earth exerts force on them. This force we call the gravitational force. Newton worked out the law for this force, his law of gravitation. It says that the size of the gravitational force a body exerts on another body is proportional to the product of the masses of the two interacting bodies, divided by the square of their distance of separation.

Fg = G M m / R2

M and m are the masses of the two bodies, and R is the distance of their separation. G is a "universal gravitational constant. The force each body experiences is directed toward the other body. It's size is Fg. The forces each exerts on the other have the same size, and they are oppositley directed, satisfying Newton's third law.

G is the universal gravitational constant.

G = 6.67300 × 10-11 m3 kg-1 s-2

The force on each body of an interacting pair has the same size, and the force on each is directed toward the other body. They are therefore equal size and opposite direction, as required by Newton's third law.

Quite a number of people at the time suspected that there was such a gravitational force, and even supposed it might have an inverse-square dependence on distance (1/r2). Robert Hooke, Edmund Halley, and Christopher Wren were all thinking along these lines. But it was Newton who "sweated the details" and worked this into a comprehensive and unifying system of mechanics, and checking the predictions against astronomical data. He also invented new mathematical methods, calculus, to aid him in working out the theory and the calculations. One reason Newton's Principia took so long to bring to publication was that the predictions for the moon were not coming out well. It turned out that the data Newton had been supplied were not very accurate, and when he obtained better data, the predictions of his theory fit the data.

Still, many critics complained that Newton still hadn't explained gravity, but had only given a law for it. Newton's response was classic "I feign [invent] no hypotheses." He meant that he was not going to speculate on a cause by inventing fictitious "reasons", for it was enough to have a correct set of laws. Other critics were bothered by the very notion that bodies could exert forces on each other with no medium intervening between them. They called this force an "occult force", for it seemed mystical or magical. It was, in fact, occult, for "occult" means "hidden".

The French philosopher Rene Descartes had a competing theory. It explained all forces between planets as due to an intervening medium, an "ether" substance, and he even postulated properties of this ether to account for both gravitational and magnetic force interactions. You seldom see it mentioned in textbooks, for it died an early death. It was a philosopher's grandiose speculative theory of everything, but lacking the detail and discipline to make any firm and testable predictions. If I described it in detail here, you'd think it a rather crackpot theory. Even the philosopher Voltaire, who was also French, could not swallow Descartes' theory, prefering Newton's instead. Descartes was trying to do physics in the same way Aristotle had, with much speculation and very little contact with the real world of hard data. But other natural philosophers had progressed beyond that method, and Descartes' notions were not at all influential. Christian Huygens chided Descartes, saying that if Descartes had only read and understood the works of his contemporaries he wouldn't have made such mistakes.

Newton also believed that space was filled with an "ether" substance, but he didn't develop a detailed theory of it, nor did he make any use of that idea in formulating his mechanics! We will see the ether idea revived in the 19th century, to the point where it becaume the object of laboratory experiments and much theorizing, only to die again in the early 20th century when Albert Einstin developed relativity theory, and, like Newton, Einstein did not use the ether idea. It played no role in his theory. This fact showed that the ether idea simply wasn't necessary to understand the physical phenomena, and never had been necessary. But more of that later.

Circular motion.

As an example of the power of the Newtonian mechanics, we consider the problem of circular motion.

The acceleration of a body is the change in its velocity divided by the time duraton of that change, a = ΔV/Δt. Since velocity is a vector, any change in its size or direction, or both, requires acceleration. A body moving in a curved path is accelerating, even if its speed is constant in size. Therefore, from Newton's law, we know that the net force on it is non-zero.

The diagram shows a a body moving with constant speed, V on a curved path of radius R. Two positions are shown, during which the body has moved a distance S along an arc. At the beginning of the time interval the body's velocity is V1. At the end of the interval it is V2. We give them distinguishing subscripts because they have different directions, even though they have the same size. During that time the body has moved through angle α. At the right we show a vector diagram of the relation between these velocities and their vector difference, Δ V.

Now consider the limiting case as the time interval gets very small, approaching zero. The angle approaches zero also. The diagrams have two similar, very skinny triangles. We can write:

Δ S / R = Δ V / V

So:

V Δ S = R Δ V

and:

V (Δ S / Δ t ) = R (Δ V / Δ t ).

But

V = Δ S / Δ t, and V = Δ V / Δ t,

so we can write:

V2 = R a,

which becomes

a = V2/R .

This is the size of the acceleration vector. The direction of the acceleration vector is inward toward the center of the arc at any instant, and is therefore perpendicular to the velocity vector at that instant, which is always tangent to the curve.

We can associate this acceleration with the inward (radial) component of whatever net force happens to be acting on the body (of mass m). It has size Fradial = m a = mV2/R, and is a vector directed radially inward.

In this way Newton answered an important and difficult question: "How can a gravitational force from the sun, acting on a planet, be the cause of constant speed circular motion of that planet?" The general idea was not new, but Newton put it on a sound mathematical footing. Newton's gravitational law expressed the size of the gravitational force between any two bodies. So, considering the gravitational force of the sun on the earth, given by Newton's law:

Fg = G M m / R2

since this is the only force acting on the earth, and is radial (toward the sun), we use our previously derived result, equating:

Fg = G M m / R2 = m V2 / R

So,

v2 = G M / R.

For a circular orbit, its circumference is related to its radius by c = 2 π R

The time for one revolution (the period, T) of a circular orbit is related to its speed by v = c / T

Combining the last three equations gives:

T2 = [(2π)2 / (GM)] R3

Observe the important idea here. We have used Newton's laws (much as Newton did) to derive Kepler's third law. But we have done more: we have evaluated an expression for the size of the constant of proportionality.

Work, kinetic energy, impulse, momentum.

Further progress was slow. The history of formulation of concepts of energy and momentum is complex, with many persons struggling with the problem. People recognized that the quantity mv2 (called "vis viva", meaning "living force") was a key to characterizing motion. Leibniz is generally given credit for the use this name with this definition in terms of mass and velocity. Another useful quantity arose, the action equivalent to what we would now express as the product of energy and time. The word is still seen in statements of Newton's third law using the terms "action" and "reaction". This language is best avoided by the modern student.

To see how these concepts arise from the raw materials of Galileo and Newton's mechanics, let's play with the equations a bit.

Suppose a force acts on a body of mass m during which time the mass is displaced by amount d. Further suppose that this is the only force acting on the body. If we start with Galileo's third equation, v2  =  vo2  +  2a • d and use Newton's F  =  ma solved for a, we can eliminate a to get v2  −  vo2  =  2(F/m) • d. Multiply both sides by m and divide both sides by 2 to get:

(m/2)v2  −  (m/2)vo2  =  F • d.

In modern textbooks this is called the "work − kinetic energy" theorem, where the (m/2)v2 is kinetic energy and the F • d is the work done by force F in displacing the body by amount d.

Suppose a force acts on a body of mass m for a time Δt. Using Newton's F  =  ma and Galileo's definition of acceleration, a  =  (v  −  vo)/Δt we get F  =  m(v  −  vo)/Δt, which may be rearranged to:

FΔt  =  m(v  − vo) = mv  −  mvo = Δ(mv) .

The quantity mv is momentum. The quantity FΔt is called the impulse exerted by the force acting on the mass for time Δt. The whole equation appears in modern textbooks as one example of the "law of conservation of momentum." Of course, this last mathematical exercise is nothing but a rearrangement of Newton's own form of his second law:

F =  Δ(mv)/Δt

Or, "The net force on a body equals the rate of change of its momentum." Newton's version used calculus, and expressed the right side as the time derivative of the momentum.

You may say "What's the big deal about momentum and kinetic energy? Both depend on velocity and mass. How are they different. Or are they?" They are very different. Kinetic energy is a scalar, but momentum is a vector. Momentum is proportional to the vector velocity. Kinetic energy is proportional to the square of the speed. Obviously they also have different units and dimensions. Once one has a clear definition of a concept, that concept becomes important in thinking about problems clearly. These two equations are getting very close to the conservation laws of energy and momentum, which tell us something profound about what happens in the universe, how it happens, and why some things can't happen.

In physics courses, typically a chapter is devoted to energy, and another to momentum. Some problems of mechanics yield to the application of one or the other of these concepts. But some problems, especially those where two bodies interact, require the simultaneous application of both concepts to arrive at answers.

—Donald E. Simanek, Feb, 2005.

Go to the next chapter, The assymetric Atwood machine. An example of linear and rotational kinematics.
Return to the contents page of A Brief Course in Classical Mechanics.