Lenny: “What makes things move, George?
George: “Forces do, Lenny.”
Lenny: “What makes things stop moving, George?”
George: “Forces do, Lenny.”
Aristotle lived in a world dominated by friction. To make anything move—a heavy cart with wooden wheels, for example—you had to push it, you had to apply a force to it. The harder you pushed it, the faster it moved; but if you stopped pushing, the cart very quickly came to rest. Aristotle came to some wrong conclusions because he didn’t understand that friction is a force. But still, it’s worth exploring his ideas in modern language. If he had known calculus, Aristotle might have proposed the following law of motion:
The velocity of any object is proportional to the total applied force.
Had he known how to write vector equations, his law would have looked like this:
is of course the applied force, and the response (according to Aristotle) would be the velocity vector, 
. The factor m relating the two is some characteristic quantity describing the resistance of the body to being moved; for a given force, the bigger the m of the object, the smaller its velocity. With a little reflection, the old philosopher might have identified m with the mass of the object. It would have been obvious that heavier things are harder to move than lighter things, so somehow the mass of the object has to be in the equation.
One suspects that Aristotle never went ice skating, or he would have known that it is just as hard to stop a body as to get it moving. Aristotle’s law is just plain wrong, but it is nevertheless worth studying as an example of how equations of motion can determine the future of a system. From now on, let’s call the body a particle.
Consider one-dimensional motion of a particle along the x axis under the influence of a given force. What I mean by a given force is simply that we know what the force is at any time. We can call it F(t) (note that vector notation would be a bit redundant in one dimension). Using the fact that the velocity is the time derivative of position, x, we find that Aristotle’s equation takes the form
Before solving the equation, let’s see how it compares to the deterministic laws of Chapter 1. One obvious difference is that Aristotle’s equation is not stroboscopic—that is neither t nor x is discrete. They do not change in sudden stroboscopic steps; they change continuously. Nevertheless, we can see the similarity if we assume that time is broken up into intervals of size Δt and replace the derivative by 
. Doing so gives
In other words, wherever the particle happens to be at time t, at the next instant its position will have shifted by a definite amount. For example, if the force is constant and positive, then in each incremental step the particle moves forward by an amount 
. This law is obviously deterministic. Knowing that the particle was at a point x(0) at time t = 0 (or x0), one can easily predict where it will be in the future. So by the criteria of Chapter 1, Aristotle did not commit any crime.
Let’s go back to the exact equation of motion:
Equations for unknown functions that involve derivatives are called differential equations. This one is a first-order differential equation because it contains only first derivatives. Equations like this are easy to solve. The trick is to integrate both sides of the equation:
The left side of the equation is the integral of a derivative. That’s where the fundamental theorem of calculus comes in handy. The left side is just x(t) + c.
The right side, on the other hand, is the integral of some specified function and, apart from a constant, is also determined. For example, if F is constant, then the right side is
Note that we included an additive constant. Putting an arbitrary constant on both sides of the equation is redundant. In this case, the equation of motion is satisfied by
How do you fix the constant c? The answer is by the initial condition. For example, if we knew that the particle started at x = 1, at time t = 3 we would plug these values in, obtaining
and solve for c:
Exercise 1: Given a force that varies with time according to F = 2t2, and with the initial condition at time zero, x(0) = π, use Aristotle’s law to find x(t) at all times.
Aristotle’s equations of motion are deterministic, but are they reversible? In Lecture 1, I explained that reversible means that if all the arrows were reversed, the resulting new law of motion would also be deterministic. The analogous procedure to reversing the arrows when time is continuous is very simple. Everywhere you see time in the equations, replace it with minus time. That will have the effect of interchanging the future and the past. Changing t to −t also includes changing the sign of small differences in time. In other words, every Δt must be replaced with −Δt. In fact, you can do it right at the level of the differentials dt. Reversing the arrows means changing the differential dt to −dt. Let’s go back to Aristotle’s equation
and change the sign of time. The result is
The left-hand side of the equation is the force, but the force evaluated at time −t, not at time t. However, if F(t) is a known function, then so is F(−t). In the reversed problem, the force is also a known function of reversed time.
On the right-hand side of the equation we’ve replaced dt with −dt, thereby changing the sign of the whole expression. In fact, one can shift the minus sign to the left-hand side of the equation:
The implication is simple: The reversed equation of motion is exactly like the original, but with a different rule for the force as a function of time. The conclusion is clear: If Aristotle’s equations of motion are deterministic into the future, they are also deterministic into the past. The problem with Aristotle’s equations is not that they are inconsistent; they are just the wrong equations.
It is interesting that Aristotle’s equations do have an application—not as fundamental laws, but as approximations. Frictional forces do exist, and in many cases they are so important that Aristotle’s intuition—things stop if you stop pushing—is almost correct. Frictional forces are not fundamental. They are a consequence of a body interacting with a huge number of other tiny bodies—atoms and molecules—that are too small and too numerous to keep track of. So we average over all the hidden degrees of freedom. The result is frictional forces. When frictional forces are very strong such as in a stone moving through mud—then Aristotle’s equation is a very good approximation, but with a qualification. It’s not the mass that determines the proportionality between force and velocity. It’s the so-called viscous drag coefficient. But that may be more than you want to know.
Aristotle’s mistake was to think that a net “applied” force is needed to keep an object moving. The right idea is that one force—the applied force—is needed to overcome another force—the force of friction. An isolated object moving in free space, with no forces acting on it, requires nothing to keep it moving. In fact, it needs a force to stop it. This is the law of inertia. What forces do is change the state of motion of a body. If the body is initially at rest, it takes a force to start it moving. If it’s moving, it takes a force to stop it. If it is moving in a particular direction, it takes a force to change the direction of motion. All of these examples involve a change in the velocity of an object, and therefore an acceleration.
From experience we know that some objects have more inertia than others; it requires a larger force to change their velocities. Obvious examples of objects possessing large and small inertia are locomotives and Ping-Pong balls, respectively. The quantitative measure of an object’s inertia is its mass.
Newton’s law of motion involves three quantities: acceleration, mass, and force. Acceleration we studied in Lecture 2. By monitoring the position of an object as it moves, a clever observer—with a bit of mathematics—can determine its acceleration. Mass is a new concept that is actually defined in terms of force and acceleration. But so far we haven’t defined force. It sounds like we are in a logical circle in which force is defined by the ability to change the motion of a given mass, and mass is defined by the resistance to that change. To break that circle, let’s take a closer look at how force is defined and measured in practice.
There are very sophisticated devices that can measure force to great accuracy, but it will suit our purposes best to imagine a very old-fashioned device, namely, a spring balance. It consists of a spring and a ruler to measure how much the spring is stretched from its natural equilibrium length (see Figure 1).
Figure 1: A spring balance.
The spring has two hooks, one to attach to the massive body whose mass is being measured, and one to pull on. In fact, while you are at it, make several such identical devices.
Let’s define a unit of force by pulling on one hook, while holding the other hook fixed to some object A, until the pointer registers one “tick” on the ruler. Thus we are applying a unit of force to A.
To define two units of force, we could pull just hard enough to stretch the spring to two ticks. But this assumes that the spring behaves the same way between one tick and two ticks of stretching. This will lead us back to a vicious circle of reasoning that we don’t want to get into. Instead, we define two units of force by attaching two spring balances to A and pulling both of them with a single unit of force (see Figure 2).
In other words, we pull both hooks so that each pointer records a single tick. Three units of force would be defined by using three springs, and so on.
When we do this experiment in free space, we discover the interesting fact that object A accelerates along the direction in which we pull the hook. More exactly, the acceleration is proportional to the force—twice as big for two units of force, three times as big for three units, and so on.
Figure 2: Twice the force.
Let us do something to change the inertia of A. In particular, we will double the inertia by hooking together two identical versions of object A (see Figure 3).
Figure 3: Twice the mass.
What we find is that when we apply a single unit of force (by pulling the whole thing with a single spring stretched to one tick) the acceleration is only half what it was originally. The inertia (mass) is now twice as big as before.
The experiment can obviously be generalized; hook up three masses, and the acceleration is only a third as big, and so on.
We can do many more experiments in which we hook any number of springs to any number of A’s. The observations are summarized by a single formula, Newton’s second law of motion, which tells us that force equals mass times acceleration,
|
(1) |
This equation can also be written in the form
|
(2) |
In other words, force equals mass times the rate of change of velocity: no force—no change in velocity.
Note that these equations are vector equations. Both force and acceleration are vectors because they not only have magnitude but also direction.
A mathematician might be content to say that the length of a line segment is 3. But a physicist or engineer—or even an ordinary person—would want to know, “Three what?” Three inches, three centimeters, or three light years?
Similarly, it conveys no information to say that the mass of an object is 7 or 12. To give the numbers meaning, we must indicate what units we are using. Let’s begin with length.
Somewhere in Paris rests the defining platinum meter stick. It is kept in a sealed container at a fixed temperature and away from other conditions that might affect its length.1 From here on, we will adopt that meter stick as our unit of length.
Thus we write
[x] = [length] = meters.
Despite its appearance, this is not an equation in the usual sense. The way to read it is x has units of length and is measured in meters.
Similarly, t has units of time and is measured in seconds. The definition of a second could be given by the amount of time it takes a certain pendulum to make a single swing:
[t] = [time] = seconds.
The units meters and seconds are abbreviated as m and s, respectively.
Once we have units for length and time, we can construct units for velocity and acceleration. To compute the velocity of an object, we divide a distance by a time. The result has units of length per time, or—in our units—meters per second.
Similarly, acceleration is the rate of change of velocity, and its units are velocity per unit time, or length per unit time per unit time:
The unit of mass that we will use is the kilogram; it is defined as the mass of a certain lump of platinum, that is also kept somewhere in France. Thus
[m] = [mass] = kilogram = kg.
Now let’s consider the unit of force. One might define it in terms of some particular spring made of a specific metal, stretched a distance of 0.01 meter, or something like that. But in fact, we have no need for a new unit of force. We already have one—namely the force that it takes to accelerate one kilogram by one meter per second per second. Even better is to use Newton’s law F = ma. Evidently, force has units of mass times acceleration,
There is a name for this unit of force. One kilogram meter per second squared is called a Newton, abbreviated N. Newton, himself, being English, probably favored the British unit, namely the pound. There are about 4.4 N to a pound.
The simplest of all examples is a particle with no forces acting on it. The equation of motion is Eq. (2), but with the force set to zero:
or, using the dot notation for time derivative,
We can drop the factor of mass and write the equation in component form as
The solution is simple: The components of velocity are constant and can just be set equal to their initial values,
vx(t) = vx(0). |
(3) |
The same goes for the other two components of velocity. This, incidently, is often referred to as Newton’s first law of motion:
Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.
Equations (1) and (2) are called Newton’s second law of motion,
The relationship between an object’s mass m, its acceleration a, and the applied force F is
F = ma.
But, as we have seen, the first law is simply a special case of the second law when the force is zero.
Recalling that velocity is the derivative of position, we can express Eq. (3) in the form
This is the simplest possible differential equation, whose solution (for all components) is
or, in vector notation,
A more complicated motion results from the application of a constant force. Let’s first carry it out for just the Z direction. Dividing by m, the equation of motion is
Exercise 2: Integrate this equation. Hint: Use definite integrals.
From this result we deduce
or
This is probably the second simplest differential equation. It is easy to solve:
|
(4) |
Exercise 3: Show by differentiation that this satisfies the equation of motion.
This simple case may be familiar. If z represents the height above the surface of the Earth, and 
is replaced with the acceleration due to gravity, 
, then Eq. (4) is the equation describing the motion of an object falling from height z0 with an initial velocity vz(0):
|
(5) |
Let’s consider the case of the simple harmonic oscillator. This system is best thought of as a particle that moves along the x axis, subject to a force that pulls it toward the origin. The force law is
Fx = −kx.
The negative sign indicates that at whatever the value of x, the force pulls it back toward x = 0. Thus, when x is positive, the force is negative, and vice versa. The equation of motion can be written in the form
or, by defining, 
,
|
(6) |
Exercise 4: Show by differentiation that the general solution to Eq. (6) is given in terms of two constants A and B by
Determine the initial position and velocity at time t = 0 in terms of A and B.
The harmonic oscillator is an enormously important system that occurs in contexts ranging from the motion of a pendulum to the oscillations of the electric and magnetic fields in a light wave. It is profitable to study it thoroughly.
1. There is a more modern definition of the meter in terms of the wavelength of light emitted by atoms jumping from one quantum level to another. For our purposes the Paris meter stick will do just fine.
