*Michael Fowler, Physics Dept., U.Va. *

We are now ready to move on to Newton’s Laws of Motion, which for the first time presented a completely coherent analysis of motion, making clear that the motion in the heavens could be understood in the same terms as motion of ordinary objects here on earth.

The crucial Second Law, as we shall see below, links the *acceleration*
of a body with the force acting on the body.
To understand what it says, it is necessary to be completely clear what
is meant by acceleration, so let us briefly review.

*Speed* is just how fast something’s moving, so is fully specified by a
positive number and suitable units, such as 55 mph or 10 meters per second.

*Velocity*, on the other hand, means to a scientist more than
speed---it also *includes* a specification of the *direction* of the
motion, so 55 mph *to the northwest* is a velocity. Usually wind *velocities* are given in a
weather forecast, since the direction of the wind affects future temperature
changes in a direct way. The standard
way of representing a velocity in physics is with an arrow pointing in the
appropriate direction, its length representing the speed in suitable units. These arrows are called “*vectors*”.

(WARNING: Notice, though, that for a moving object such as a projectile, *both*
its position at a given time (compared with where it started) *and *its
velocity at that time can be represented by vectors, so you must be clear what
your arrow represents!)

*Acceleration*: as we have stated, acceleration is defined as* rate
of change of velocity*.

It is* not* defined as rate of change of speed. *A body can have nonzero acceleration while
moving at constant speed!*

Consider *speed. Of course, its velocity is changing
constantly, because velocity includes direction. *

Let us look at how its *velocity* changes over a period of one second. (Actually, in the diagram below we exaggerate
how far it would move in one second, the distance would in fact be one-five
thousandth of the distance around the circle, impossible to draw.)

Here we show the cannonball (greatly exaggerated in size!) at two points in its orbit, and the velocity vectors at those points. On the right, we show the two velocity vectors again, but we put their ends together so that we can see the difference between them, which is the small dashed vector.

In other words, the small dashed vector is the velocity that has to be added
to the first velocity to get the second velocity: it is the *change* in
velocity on going around that bit of the orbit.

Now, if we think of the two points in the orbit as corresponding to positions of the cannonball one second apart, the small dashed vector will represent the change in velocity in one second, and that is—by definition—the acceleration. The acceleration is the rate of change of velocity, and that is how much the velocity changes in one second (for motions that change reasonably smoothly over the one-second period, which is certainly the case here. To find the rate of change of velocity of a fly’s wing at some instant, we obviously would have to measure its velocity change over some shorter interval, maybe a thousandth of a second).

So we see that, with our definition of acceleration as the rate of change of velocity, which is a vector, a body moving at a steady speed around a circle is accelerating towards the center all the time, although it never gets any closer to it. If this thought makes you uncomfortable, it is because you are still thinking that acceleration must mean a change of speed, and just changing direction doesn’t count.

It is possible to find an explicit expression for the magnitude of the
acceleration towards the center (sometimes called the *centripetal *acceleration)
for a body moving on a circular path at speed *v*. Look again at the diagram above showing two
values of the velocity of the cannonball one second apart. As is explained above, the magnitude *a*
of the acceleration is the length of the small dashed vector on the right,
where the other two sides of this long narrow triangle have lengths equal to
the speed *v* of the cannonball. We’ll
call this the “*vav*” triangle, because those are the lengths of its sides. What about the *angle* between the two long
sides? That is just the angle the
velocity vector turns through in one second as the cannonball moves around its
orbit. Now look over at the circle
diagram on the left showing the cannonball’s path. Label the cannonball’s position at the
beginning of the second *A*, and at the end of the second *B*, so the
length *AB* is how far the cannonball travels in one second, that is, *v*. (It’s true that the part of the path *AB*
is slightly curved, but we can safely ignore that very tiny effect.) Call the center of the circle *C*. Draw the triangle *ACB*. (The reader should
sketch the figure and actually draw these triangles!) The two long sides *AC* and *BC*
have lengths equal to the radius of the circular orbit. We could call this long thin triangle an “*rvr*”
triangle, since those are the lengths of its sides.

The important point to realize now is that the “*vav*” triangle and the
“*rvr*” triangle are *similar*, because since the velocity vector is
always perpendicular to the radius line from the center of the circle to the
point where the cannonball is in orbit, *the angle the velocity vector
rotates by in one second is the same as the angle the radius line turns through
in one second*. Therefore, the two
triangles are similar, and their corresponding sides are in the same ratios,
that is, *a*/*v* = *v*/*r*.
It follows immediately that *the magnitude of
the acceleration a for an object moving at steady speed v in a circle of radius
r is v ^{2}/r directed towards the center of the circle. *

This result is true for all circular motions, even those where the moving
body goes round a large part of the circle in one second. To establish it in a case like that, recall
that the acceleration is the rate of change of velocity, and we would have to
pick a smaller time interval than one second, so that the body didn’t move far
around the circle in the time chosen. If,
for example, we looked at two velocity vectors one-hundredth of a second apart,
and they were pretty close, then the acceleration would be given by the
difference vector between them *multiplied by one-hundred*, since
acceleration is defined as what the velocity change in one second would be if
it continued to change at that rate. (In
the circular motion situation, the acceleration is of course changing all the
time. To see why it is sometimes
necessary to pick small time intervals, consider what would happen if the body
goes around the circle completely in one second. Then, if you pick two times one second apart,
you would conclude the velocity isn’t changing at all, so there is no acceleration.)

**W**e’ve stated before that a ball thrown vertically upwards has
constant *downward* acceleration of 10 meters per second in each second, even
when it’s at the very top and isn’t moving at all. The key point here is that acceleration is
rate of change of velocity. You can’t
tell what the rate of change of something is unless you know its value at more
than one time. For example, speed on a straight
road is rate of change of distance from some given point. You can’t get a speeding ticket just for
being at a particular point at a certain time—the cop has to prove that a short
time later you were at a point well removed from the first point, say, three
meters away after one-tenth of a second.
That would establish that your speed was thirty meters per second, which
is illegal in a 55 m.p.h. zone. In just the same way that speed is rate of
change of position, acceleration is rate of change of velocity. Thus to find acceleration, you need to know
velocity at two different times. The
ball thrown vertically upwards does have zero velocity at the top of its path,
but that is only at a *single instant* of time. One second later it is dropping at ten meters
per second. One millionth of a second
after it reached the top, it is falling at one hundred-thousandth of a meter
per second. Both of these facts
correspond to a downward acceleration, or rate of change of velocity, of 10
meters per second in each second*. It
would only have zero acceleration if it stayed at rest at the top for some
finite period of time*, so that you could say that its velocity remained the
same—zero—for, say, a thousandth of a second, and during that period the rate
of change of velocity, the acceleration, would then of course be zero. Part of the problem is that the speed is very
small near the top, and also that our eyes tend to lock on to a moving object
to see it better, so there is the illusion that it comes to rest and stays
there, even if not for long.

Galileo’s analysis of projectile motion was based on two concepts:

1. *Naturally accelerated motion*,
describing the *vertical* component of motion, in which the body picks up
speed at a uniform rate.

2. *Natural horizontal motion*,
which is motion at a *steady speed in a straight line*, and happens to a
ball rolling across a smooth table, for example, when frictional forces from
surface or air can be ignored.

*different aspects of the same thing*. He did this by introducing the idea of motion
being affected by a *force*, then expressing this idea in a quantitative
way. Galileo, of course, had been well
aware that motion is affected by external forces. Indeed, his definition of natural horizontal
motion explicitly states that it applies to the situation where such forces can
be neglected. He knew that friction
would ultimately slow the ball down, and—very important—a force pushing it from
behind would cause it to accelerate. What
he didn’t say, though, and *must* be
the result of a vertical force on the body, without which the natural vertical
motion would *also* be at a constant speed, just like natural horizontal
motion. This vertical force is of course
just the force of gravity.

Therefore the point Newton is making is that the essential difference
between Galileo’s natural steady speed horizontal motion and the natural
accelerated vertical motion is that vertically, there is always the force of
gravity acting, and without that—for example far into space—the natural motion
(that is, with no forces acting) in *any* direction would be at a steady
speed in a straight line.

(Actually, it took *Never at Rest*, by Richard Westfall, and
I have summarized some of the points here.)

To put it in his own words (although actually he wrote it in Latin, this is from an 1803 translation):

*Law 1*

*Every body perseveres in its state of rest, or of uniform motion in a
right line, unless it is compelled to change that state by forces impressed
thereon. *

He immediately adds, tying this in precisely with Galileo’s work:

*Projectiles persevere in their motions, so far as they are not retarded
by the resistance of the air, or impelled downwards by the force of gravity. *

Notice that here “persevere in their motions” must mean in *steady speed
straight line* motions, because he is adding the gravitational acceleration
on to this.

This is sometimes called “The Law of Inertia”: in the absence of an external force, a body in motion will continue to move at constant speed and direction, that is, at constant velocity.

So *any* acceleration, or change in speed (or direction of motion), of
a body* signals that it is being acted on by some force*.

*for a
given body*, the acceleration produced is proportional to the strength of
the external force, so doubling the external force will cause the body to pick
up speed twice as fast.

*Law 2*

*The alteration of motion is ever proportional to the motive force
impressed; and is made in the direction of the right line in which that force
is impressed. *

Another rather obvious point he doesn’t bother to make is that for a given *force*,
such as, for example, the hardest you can push, applied to two different
objects, say a wooden ball and a lead ball of the same size, with the lead ball
weighing seven times as much as the wooden ball, then the lead ball will only
pick up speed at one-seventh the rate the wooden one will.

Now let us consider the significance of this law for falling bodies. Neglecting air resistance, bodies of all masses accelerate downwards at the same rate. This was Galileo’s discovery.

Let us put this well established fact together with *proportional*
to the external force, but *inversely proportional* to the *mass* of
the body the force acts on.

Consider two falling bodies, one having *twice* the mass of the other. Since their acceleration is the same, the
body having twice the mass must be experiencing a gravitational force which is
twice as strong. Of course, we are well
aware of this, all it’s saying is that two bricks weigh twice as much as one
brick. Any weight measuring device, such
as a bathroom scales, is just measuring the force of gravity. However, this proportionality of mass and
weight is not a completely trivial point.
Masses can be measured against each other *without using gravity at
all*, for example far into space, by comparing their relative accelerations
when subject to a standard force, a push.
If one object accelerates at half the rate of another when subject to
our standard push, we conclude it has twice the mass. Thinking of the mass in this way as a measure
of resistance to having velocity changed by an outside force, *inertia*. (Note that this is a bit different from
everyday speech, where we think of inertia as being displayed by something that
stays at rest. For

To return to the concept of mass, it is really just a measure of the *amount
of stuff*. For a uniform material,
such as water, or a uniform solid, the mass is the volume multiplied by the
density—the density being defined as the mass of a unit of volume, so water,
for example, has a density of one gram per cubic centimeter, or sixty-two
pounds per cubic foot.

Hence, from Galileo’s discovery of the uniform acceleration of all falling
bodies, we conclude that the *weight* of a body, which is the gravitational
attraction it feels towards the earth, is directly proportional to its mass,
the amount of stuff it’s made of.

All the statements above about force, mass and acceleration are statements about proportionality. We have said that for a body being accelerated by a force acting on it the acceleration is proportional to the (total) external force acting on the body, and, for a given force, inversely proportional to the mass of the body.

If we denote the force, mass and acceleration by *F*, *m* and *a*
respectively (bearing in mind that really *F* and *a* are vectors
pointing in the same direction) we could write this:

*F* is
proportional to *ma*

To make any progress in applying

*The unit of force is that force which
causes a unit mass (one kilogram) to accelerate with unit acceleration (one
meter per second per second). *

This unit of force is named, appropriately, the ** newton**.

If we now agree to measure forces in newtons, the statement of proportionality above can be written as a simple equation:

*F* = *ma *

which is the usual statement of

If a mass is now observed to accelerate, it is a trivial matter to find the
total force acting on it. The force will
be in the direction of the acceleration, and its magnitude will be the product
of the mass and acceleration, measured in newtons. For example, a 3 kilogram falling body,
accelerating downwards at 10 meters per second per second, is being acted on by
a force *ma* equal to 30 newtons, which is, of course, its weight.

Having established that a force—the action of another body—was necessary to
cause a body to change its state of motion, Newton made one further crucial
observation: such forces *always* arise as a *mutual interaction* of
two bodies, and the other body also feels the force, but in the opposite
direction.

*Law 3*

*To every action there is always opposed an equal and opposite reaction**:
or the mutual actions of two bodies upon each other are always equal, and
directed to contrary parts. *

*Whatever draws or presses another is as much drawn or pressed by that
other. If you press a stone with your
finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the
horse (if I may so say) will be equally drawn back towards the stone: for the
distended rope, by the same endeavour to relax or unbend itself, will draw the
horse as much towards the stone, as it does the stone towards the horse, and
will obstruct the progress of the one as much as it advances that of the other. If a body impinge upon another, and by its
force change the motion of the other, that body also (because of the equality
of the mutual pressure) will undergo an equal change, in its own motion,
towards the contrary part. The changes
made by these actions are equal, not in the velocities but in the motions of
bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed,
the changes of the velocities made towards contrary parts are reciprocally
proportional to the bodies. This law
takes place also in attractions. *

All this maybe sounds kind of obvious.
Anyone who’s had a dog on a leash, especially a big dog, is well aware
that tension in a rope pulls both ways. If
you push against a wall, the wall is pushing you back. If that’s difficult to visualize, imagine
what would happen if the wall suddenly evaporated. *external* force acts on
it, was one of the big forward steps in our understanding of how the Universe
works.

The Second Law states that if a body is accelerating, there must be an external force acting on it. It’s not always obvious what this external force is even in the most trivial everyday occurrences. Suppose you’re standing still, then begin to walk. What was the external force that caused you to accelerate? Think about that for a while. Here’s a clue: it’s very hard to start walking if you’re wearing smooth-bottomed shoes and standing on smooth ice. You tend to skid around in the same place. If you understand that, you also know what external force operates when a car accelerates.

The reason the external force causing the acceleration may not be
immediately evident is that it may not be what’s doing the work. Consider the following scenario: you are
standing on level ground, on rollerskates, facing a wall with your palms
pressed against it. You push against the
wall, and roll away backwards. You
accelerated. Clearly, you did the work
that caused the acceleration. But from *would* be generated directly
by the agent doing the work, you.

Now imagine two people on roller skates, standing close facing each other,
palms raised and pushing the other person away.
According to Newton’s discussion above following his Third Law, the two
bodies involved will undergo equal changes of motion, but to contrary parts,
that is, in opposite directions. That
sounds reasonable. They obviously both
move off backwards. Notice, however,
that *reciprocally proportional to
the bodies*”.

Roller skates actually provide a pretty good example of the necessity of generating an external force if you want to accelerate. If you keep the skates pointing strictly forwards, and only the wheels are in contact with the ground, it’s difficult to get going. The way you start is to turn the skates some, so that there is some sideways push on the wheels. Since the wheels can’t turn sideways, you are thus able to push against the ground, and therefore it is pushing you—you’ve managed to generate the necessary external force to accelerate you. Note that if the wheels were to be replaced by ball bearings somehow, you wouldn’t get anywhere, unless you provided some other way for the ground to push you, such as a ski pole, or maybe twisting your foot so that some fixed part of the skate contacted the ground.

We have now reached the last sentence in *This
law also takes place in attractions*”.
This of course is central to

Let us now put together what we know about the gravitational force:

1. The gravitational force on a body (its weight, at the Earth’s surface) is proportional to its mass.

2. If a body *A* attracts a body
*B* with a gravitational force of a given strength, then *B* attracts
*A* with a force of equal strength in the opposite direction.

3. The gravitational attraction between two bodies decreases with distance, being proportional to the inverse square of the distance between them. That is, if the distance is doubled, the gravitational attraction falls to a quarter of what it was.

One interesting point here—think about how the earth is gravitationally
attracting you. Actually, all the
different parts of the earth are attracting you! *all* the earth’s mass were concentrated in *one point at the
center*. So, when we’re talking about
the gravitational attraction between you and the earth, and we talk about the
distance of separation, we mean the distance between you and the *center*
of the earth, which is just less than four thousand miles (6300 kilometers).

Let’s denote the gravitational attractive force between two bodies *A*
and *B* (as mentioned in item 2 above) by *F*. The forces on the two bodies are really equal
and opposite vectors, each pointing to the other body, so our letter *F*
means the *length* of these vectors, the strength of the force of
attraction.

Now, item 1 tells us that the gravitational attraction between the earth and
a mass *m* is proportional to *m*.
This is an immediate consequence of the experimental fact that falling
bodies accelerate at the same rate, usually written* g* (approximately 10
meters per second per second), and the definition of force from

*F* is
proportional to mass *m*

for the earth’s gravitational attraction on a body (often written weight *W*
= *mg*), and *any *gravitational attraction on the body.

From the symmetry of the force (item 2 above) and the proportionality to the
mass (item 1), it follows that the gravitational force between two bodies must
be proportional to *both* masses. So,
if we double both masses, say, the gravitational attraction between them
increases by a factor of four. We see
that if the force is proportional to both masses, let’s call them *M* and *m*,
it is actually proportional to the *product* *Mm* of the masses. From item 3 above, the force is also
proportional to 1/*r*^{2}, where *r* is the distance between
the bodies, so for the gravitational attractive force between two bodies

*F* is
proportional to *Mm*/*r*^{2}

This must mean that by measuring the gravitational force on something, we
should be able to figure out the mass of the Earth! But there’s a catch—all we
know is that the force is *proportional* to the Earth’s mass. From that we could find, for instance, the
ratio of the mass of the Earth to the mass of Jupiter, by comparing how fast
the Moon is “falling” around the Earth to how fast Jupiter’s moons are falling
around Jupiter. For that matter, we
could find the ratio of the Earth’s mass to the Sun’s mass by seeing how fast
the planets swing around the Sun. Still,
knowing all these ratios doesn’t tell us the Earth’s mass in tons. It does tell us that if we find that out, we
can then find the masses of the other planets, at least those that have moons,
and the mass of the Sun.

So how do we measure the mass of the Earth? The only way is to compare the
Earth’s gravitational attraction with that of something we already know the
mass of. We don’t know the masses of any
of the heavenly bodies. What this really
means is that we have to take a known mass, such as a lead ball, and measure
how strongly it attracts a smaller lead ball, say, and compare that force with
the earth’s attraction for the smaller lead ball. This is very difficult to accomplish because
the forces are so small, but it was done successfully in 1798, just over a
century after

In other words, Cavendish took two lead weights *M* and *m*, a few
kilograms each, and actually *detected the tiny gravitational attraction between
them* (of order of magnitude millionths of a newton)! This was a
sufficiently tough experiment that even now, two hundred years later, it’s not
easy to give a lecture demonstration of the effect.

Making this measurement amounts to finding the constant of proportionality
in the statement about *F* above, so that we can sharpen it up from a
statement about proportionality to an actual useable equation,

*F* = *GMm*/*r*^{2}

where the constant *G* is what Cavendish measured, and found to be 6.67
x 10^{-11} in the appropriate units, where the masses are in kilograms,
the distance in meters and the force in newtons. (Notice here that we can’t get rid of the
constant of proportionality *G*, as we did in the equation *F* = *ma*,

From *universal* gravitational attraction, the *same* constant *G*
determines the gravitational attraction between *any two masses* in the
universe. ** This means we can now
find the mass of the earth**. We
just consider a one kilogram mass at the earth’s surface. We know it feels a force of approximately 10
newtons, and is a distance of about 6300 km, or 6,300,000 meters, from the
center of the earth. So we know