For the Love of Physics (9 page)

The difference is actually simple. Your mass is the same no matter where you are in the universe. That’s right—on the Moon, in outer space, or on the surface of an asteroid. It’s your
weight
that varies. So what is weight, then? Here’s where things get a little tricky. Weight is the result of gravitational attraction. Weight is a force: it is mass times the gravitational acceleration (
F = mg
). So our weight varies depending upon the strength of gravity acting on us, which is why astronauts weigh less on the Moon. The Moon’s gravity is about a sixth as strong as Earth’s, so on the Moon astronauts weigh about one-sixth what they weigh on Earth.

For a given mass, the gravitational attraction of the Earth is about the same no matter where you are on it. So we can get away with saying, “She
weighs a hundred twenty pounds”
*
or “He weighs eighty kilograms,”
*
even though by doing so we are confusing these two categories (mass and weight). I thought long and hard about whether to use the technical physics unit for force (thus weight) in this book instead of kilos and pounds, and decided against it on the grounds that it would be too confusing—no one, not even a physicist whose mass is 80 kilograms would say, “I weigh seven hundred eighty-four newtons” (80 × 9.8 = 784). So instead I’ll ask you to remember the distinction—and we’ll come back to it in just a little while, when we return to the mystery of why a scale goes crazy when we stand on our tiptoes on it.

The fact that gravitational acceleration is effectively the same everywhere on Earth is behind a mystery that you may well have heard of: that objects of different masses fall at the same speed. A famous story about Galileo, which was first told in an early biography, recounts that he performed an experiment from the top of the Leaning Tower of Pisa in which he threw a cannonball and a smaller wooden ball off the tower at the same time. His intent, reputedly, was to disprove an assertion attributed to Aristotle that heavier objects would fall faster than light ones. The account has long been doubted, and it seems pretty clear now that Galileo never did perform this experiment, but it still makes for a good story—such a good story that the commander of the Apollo 15 Moon mission, David Scott, famously dropped a hammer and a falcon feather onto the surface of the Moon at the same time to see if objects of different mass would fall to the ground at the same rate in a vacuum. It’s a wonderful video, which you can access here:
http://video.google.com/videoplay?docid=6926891572259784994#
.

The striking thing to me about this video is just how
slowly
they both drop. Without thinking about it, you might expect them both to drop quickly, at least surely the hammer. But they both fall slowly because the gravitational acceleration on the Moon is about six times less than it is on Earth.

Why was Galileo right that two objects of different mass would land at the same time? The reason is that the gravitational acceleration is the same for all objects. According to
F
=
ma
, the larger the mass, the larger the gravitational force, but the acceleration is the same for all objects. Thus they reach the ground with the same speed. Of course, the object with the larger mass will have more energy and will therefore have a greater impact.

Now it’s important to note here that the feather and the hammer would not land at the same time if you performed this experiment on Earth. This is the result of air drag, which we’ve discounted until now. Air drag is a force that opposes the motion of moving objects. Also wind would have much more effect on the feather than on the hammer.

This brings us to a very important feature of the second law. The word
net
in the equation as given above is vital, as nearly always in nature more than one force is acting on an object; all have to be taken into account. This means that the forces have to be added. Now, it’s not really as simple as this, because forces are what we call vectors, meaning that they have a magnitude as well as a direction, which means that you cannot really make a calculation like 2 + 3 = 5 for determining the net force. Suppose only two forces act on a mass of 4 kilograms; one force of 3 newtons is pointing upward, and another of 2 newtons is pointing downward. The sum of these two forces is then 1 newton in the upward direction and, according to Newton’s second law, the object will be accelerated upward with an acceleration of 0.25 meters per second per second.

The sum of two forces can even be zero. If I place an object of mass
m
on my table, according to Newton’s second law, the gravitational force on the object is then
mg
(mass × gravitational acceleration) newtons in the downward direction. Since the object is not being accelerated, the net force on the object must be zero. That means that there must be another force of
mg
newtons upward. That is the force with which the table pushes upward on the object. A force of
mg
down and one of
mg
up add up to a force of zero!

This brings us to Newton’s third law: “To every action there is always
an equal and opposite reaction.” This means that the force that two objects exert on each other are always equal and are directed in opposite directions. As I like to put it, action equals minus reaction, or, as it’s known more popularly, “For every action there is an equal and opposite reaction.”

Some of the implications of this law are intuitive: a rifle recoils backward against your shoulder when it fires. But consider also that when you push against a wall, it pushes back on you in the opposite direction with the exact same force. The strawberry shortcake you had for your birthday pushed down on the cake plate, which pushed right back at it with an equal amount of force. In fact, odd as the third law is, we are completely surrounded by examples of it in action.

Have you ever turned on the faucet connected to a hose lying on the ground and seen the hose snake all over the place, maybe spraying your little brother if you were lucky? Why does that happen? Because as the water is pushed out of the hose, it also pushes back on the hose, and the result is that the hose is whipped all around. Or surely you’ve blown up a balloon and then let go of it to see it fly crazily around the room. What’s happening is that the balloon is pushing the air out, and the air coming out of the balloon pushes back on the balloon, making it zip around, an airborne version of the snaking garden hose. This is no different from the principle behind jet planes and rockets. They eject gas at a very high speed and that makes them move in the opposite direction.

Now, to truly grasp just how strange and profound an insight this is, consider what Newton’s laws tell us is happening if we throw an apple off the top of a thirty-story building. We know the acceleration will be
g
, about 9.8 meters per second per second. Now, say the apple is about half a kilogram (about 1.1 pounds) in mass. Using the second law,
F
=
ma
, we find that the Earth attracts the apple with a force of 0.5 × 9.8 = 4.9 newtons. So far so good.

But now consider what the third law demands: if the Earth attracts the apple with a force of 4.9 newtons, then the apple will attract the Earth
with a force of 4.9 newtons. Thus, as the apple falls to Earth, the Earth falls to the apple. This seems ridiculous, right? But hold on. Since the mass of the Earth is so much greater than that of the apple, the numbers get pretty wild. Since we know that the mass of the Earth is about 6 × 10
²⁴
kilograms, we can calculate how far it falls up toward the apple: about 10
^–22
meters, about one ten-millionth of the size of a proton, a distance so small it cannot even be measured; in fact, it’s meaningless.

This whole idea, that the force between two bodies is both equal and in opposite directions, is at play everywhere in our lives, and it’s the key to why your scale goes berserk when you lift yourself up onto your toes on it. This brings us back to the issue of just what weight is, and lets us understand it more precisely.

When you stand on a bathroom scale, gravity is pulling down on you with force
mg
(where
m
is your mass) and the scale is pushing up on you with the same force so that the net force on you is zero. This force pushing up against you is what the scale actually measures, and this is what registers as your weight. Remember, weight is not the same thing as mass. For your mass to change, you’d have to go on a diet (or, of course, you might do the opposite, and eat more), but your weight can change much more readily.

Let’s say that your mass (
m
) is 55 kilograms (that’s about 120 pounds). When you stand on a scale in your bathroom, you push down on the scale with a force
mg
, and the scale will push back on you with the same force,
mg.
The net force on you is zero. The force with which the scale pushes back on you is what you will read on the scale. Since your scale may indicate your weight in pounds, it will read 120 pounds.

Let’s now weigh you in an elevator. While the elevator stands still (or while the elevator is moving at constant speed), you are not being accelerated (neither is the elevator) and the scale will indicate that you weigh 120 pounds, as was the case when you weighed yourself in your bathroom. We enter the elevator (the elevator is at rest), you go on the scale, and it reads 120 pounds. Now I press the button for the top floor, and the
elevator briefly accelerates upward to get up to speed. Let’s assume that this acceleration is 2 meters per second per second and that it is constant. During the brief time that the elevator accelerates, the net force on you cannot be zero. According to Newton’s second, the net force
F
_net
on you must be F
_net
= ma
_net
. Since the net acceleration is 2 meters per second per second, the net force on you is
m
× 2 upward. Since the force of gravity on you is
mg
down, there must be a force of
mg
+
m
2, which can also be written as
m
(
g
+ 2), on you in upward direction. Where does this force come from? It must come from the scale (where else?). The scale is exerting a force
m
(
g
+ 2) on you upward. But remember that the weight that the scale indicates is the force with which it pushes upward on you. Thus the scale tells you that your weight is about 144 pounds (remember,
g
is about 10 meters per second per second). You have gained quite a bit of weight!

According to Newton’s third, if the scale exerts a force of
m
(
g
+ 2) on you upward, then you must exert the same force on the scale downward. You may now reason that if the scale pushes on you with the same force that you push on the scale, that then the net force on you is zero, thus you cannot be accelerated. If you reason this way, you make a very common mistake. There are only two forces acting on you:
mg
down due to gravity and
m
(
g
+ 2) up due to the scale, and thus a net force of 2
m
is exerted on you in an upward direction, which will accelerate you at 2 meters per second per second.

The moment the elevator stops accelerating, your weight goes back to normal. Thus it’s only during the short time of the upward acceleration that your weight goes up.

You should now be able to figure out on your own that if the elevator is being accelerated downward, you lose weight. During the time that the acceleration downward is 2 meters per second per second, the scale will register that your weight is
m
(
g
– 2), which is about 96 pounds. Since an elevator that goes up must come to a halt, it must be briefly accelerated downward before it comes to a stop. Thus near the end of your elevator
ride up you will see that you lost weight, which you may enjoy! However, shortly after that, when the elevator has come to a stop, your weight will again go back to normal (120 pounds).

Suppose now, someone who really, really dislikes you cuts the cable and you start zooming down the elevator shaft, going down with an acceleration of
g.
I realize you probably wouldn’t be thinking about physics at that point, but it would make for a (briefly) interesting experience. Your weight will become
m
(
g
–
g
) = 0; you are weightless. Because the scale is falling downward at the same acceleration as you, it no longer exerts a force on you upward. If you looked down at the scale it would register zero. In truth, you would be floating, and everything in the elevator would be floating. If you had a glass of water you could turn it over and the water would not fall out, though of course this is one experiment I urge you not to try!

This explains why astronauts float in spaceships. When a space module, or the space shuttle, is in orbit, it is actually in a state of free fall, just like the free fall of the elevator. What exactly is free fall? The answer might surprise you. Free fall is when the force acting upon you is exclusively gravitational, and no other forces act on you. In orbit, the astronauts, the spaceship, and everything inside it are all falling toward Earth in free fall. The reason why the astronauts don’t go splat is because the Earth is curved and the astronauts, the spaceship, and everything inside it are moving so fast that as they fall toward Earth, the surface of the planet curves away from them, and they will never hit the Earth’s surface.

Thus the astronauts in the shuttle are weightless. If you were in the shuttle, you would think that there is no gravity; after all, nothing in the shuttle has any weight. It’s often said that the shuttle in orbit is a zero-gravity environment, since that’s the way you perceive it. However, if there were no gravity, the shuttle would not stay in orbit.