We've all dropped balls from our hands or seen them arc gracefully through the air after being
thrown. These motions are simplicity itself and, not surprisingly, they’re governed by only a
few universal rules. We encountered several of those rules in the previous section, but we’re
about to examine our fi rst important type of force—gravity. Like Newton, who reportedly
began his investigations after seeing an apple fall from a tree, we'll start simply by exploring
gravity and its effects on motion in the context of falling objects.
Questions to Think About: What do we mean by “falling,” and why do balls fall? Which falls
faster, a heavy ball or a light ball? Can a ball that’s heading upward still be falling? How does
gravity affect a ball that’s thrown sideways?
Experiments to Do: A few seconds with a baseball will help you see some of the behaviors
that we’ll be exploring. Toss the ball into the air to various heights, catching it in your hand
as it returns. Have a friend time the flight of the ball. As you toss the ball higher, how much
more time does it spend in the air? How does it feel coming back to your hands? Is there any
difference in the impact it makes? Which takes the ball longer: rising from your hand to its
peak height or returning from its peak height back to your hand?
Now drop two different balls—a baseball, say, and a golf ball. If you drop them simultaneously,
without pushing either one up or down, does one ball strike the ground first or do
they arrive together? Now throw one ball horizontally while dropping the second. If they both
leave your hands at the same time and the first one’s initial motion is truly horizontal, which
one reaches the ground first?
Weight and Gravity:
Like everything else around us, a ball has a weight. For example, a golf ball weighs about 0.45 N (0.10 lbf)—but what is weight? Evidently it’s a force, since both the newton (N) and the pound-force (lbf) are units of force. To understand what weight is, howeverand, inparticular, where it comes from—we need to look at gravity.
Gravity is a physical phenomenon that produces attractive forces between every pair
of objects in the universe. In our daily lives, however, the only object massive enough and
near enough to have obvious gravitational effects on us is our planet, Earth. Gravity weakens
with distance; the moon and sun are so far away that we notice their gravities only
through such subtle effects as the ocean tides.
Earth’s gravity exerts a downward force on any object near its surface. That object is
attracted directly toward the center of Earth with a force we call the object’s weight
(Fig. 1.2.1). Remarkably enough, this weight is exactly proportional to the object’s mass—
if one ball has twice the mass of another ball, it also has twice the weight. Such a relationship
between weight and mass is astonishing because weight and mass are very different
attributes: weight is how hard gravity pulls on a ball, and mass is how diffi cult that ball is
to accelerate. Because of this proportionality, a ball that’s heavy is also hard to shake back
and forth!
An object’s weight is also proportional to the local strength of gravity, which is measured
by a downward vector called the acceleration due to gravity—an odd name that I’ll
explain shortly. At the surface of Earth, the acceleration due to gravity is about 9.8 N/kg
(1.0 lbf/lbm). That value means that a mass of 1 kilogram has a weight of 9.8 newtons and
that a mass of 1 pound-mass has a weight of 1 pound-force.
More generally, an object’s weight is equal to its mass times the acceleration due to
gravity, which can be written as a word equation:
weight = mass * acceleration due to gravity (1.2.1)
in symbols:
w = m * g,
and in everyday language:
You can lose weight either by reducing your mass or by going someplace, like
a small planet, where the gravity is weaker.
But why acceleration due to gravity? What acceleration do we mean? To answer that
question, let’s consider what happens to a ball when you drop it.
If the only force on the ball is its weight, the ball accelerates downward; in other
words, it falls. Although a ball moving through Earth’s atmosphere encounters additional
forces due to air resistance, let’s ignore those forces for the time being. Doing so costs us
only a little in terms of accuracy—the effects of air resistance are negligible as long as the
ball is dense and its speed relatively small—and allows us to focus exclusively on the
effects of gravity.
How much does the falling ball accelerate? According to Eq. 1.1.1, the ball’s acceleration
is equal to the net force exerted on it divided by its mass. Because the ball is falling,
however, the only force on it is its own weight. That weight, according to Eq. 1.2.1, is equal
to the ball’s mass times the acceleration due to gravity. Using a little algebra, we get
falling ball’s acceleration = ball’s weight/ball’s mass
= ball’s mass * acceleration due to gravity/ball’s mass
= acceleration due to gravity.
As you can see, the falling ball’s acceleration is equal to the acceleration due to gravity.
Thus, acceleration due to gravity really is an acceleration after all: it’s the acceleration of a
freely falling object. Moreover, the units of acceleration due to gravity can be transformed
easily from those relating weight to mass, 9.8 N/kg (1.0 lbf/lbm), into those describing the
acceleration of free fall, 9.8 m/s2 (32 ft/s2).
Thus a ball falling near Earth’s surface experiences a downward acceleration of 9.8 m/s2
(32 ft/s2), regardless of its mass. This downward acceleration is substantially more than that
of an elevator starting its descent. When you drop a ball, it picks up speed very quickly in
the downward direction.
Because all falling objects at Earth’s surface accelerate downward at exactly the same
rate, a billiard ball and a bowling ball dropped simultaneously from the same height will
reach the ground together. (Remember that we’re not considering air resistance yet.)
Although the bowling ball weighs more than the billiard ball, it also has more mass; so
while the bowling ball experiences a larger downward force, its larger mass ensures that its
downward acceleration is equal to that of the lighter and less massive billiard ball.
Check Your Understanding #1: Weight and Mass
Out in deep space, far from any celestial object that exerts significant gravity, would an astronaut
weigh anything? Would that astronaut have a mass?
Answer:
The astronaut would have zero weight but would still have a normal mass.
Why: Weight is a measure of the force exerted on the astronaut by gravity. Far from Earth or any
other large object, the astronaut would experience virtually no gravitational force and would have zero
weight. But mass is a measure of inertia and doesn’t depend at all on gravity. Even in deep space, it
would be much harder to accelerate a school bus than to accelerate a baseball because the school
bus has more mass than the baseball.
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