Physicists like to study seemingly unrelated phenomena to show that a relationship can be found if the phenomena are examined closely enough. This search for unification has been going on for centuries. In 1665, the 23-year-old Isaac Newton made a basic contribution to physics when he showed that the force that holds the Moon in its orbit is the same force that makes an apple fall. We take this knowledge so much for granted now that it is not easy for us to comprehend the ancient belief that the motions of earthbound bodies and heavenly bodies were different in kind and were governed by different laws.
Newton concluded not only that Earth attracts both apples and the Moon but also that every body in the universe attracts every other body; this tendency of bodies to move toward each other is called gravitation. Newton’s conclusion takes a little getting used to, because the familiar attraction of Earth for earth-bound bodies is so great that it overwhelms the attraction that earthbound bodies have for each other. For example, Earth attracts an apple with a force magnitude of about 0.8 N. You also attract a nearby apple (and it attracts you), but the force of attraction has less magnitude than the weight of a speck of dust.
Newton proposed a force law that we call Newton’s law of gravitation: Every particle attracts any other particle with a gravitational force of magnitude
Here m1 and m2 are the masses of the particles, r is the distance between them, and G is the gravitational constant, with a value that is now known to be
In Fig. 13-2a, 
is the gravitational force acting on particle 1 (mass m1) due to particle 2 (mass m2). The force is directed toward particle 2 and is said to be an attractive force because particle 1 is attracted toward particle 2. The magnitude of the force is given by Eq. 13-1.
We can describe as being in the positive direction of an r axis extending radially from particle 1 through particle 2 (Fig. 13-2b). We can also describe by using a radial unit vector 
(a dimensionless vector of magnitude 1) that is directed away from particle 1 along the r axis (Fig. 13-2c). From Eq. 13-1, the force on particle 1 is then
The gravitational force on particle 2 due to particle 1 has the same magnitude as the force on particle 1 but the opposite direction. These two forces form a third-law force pair, and we can speak of the gravitational force between the two particles as having a magnitude given by Eq. 13-1. This force between two particles is not altered by other objects, even if they are located between the particles. Put another way, no object can shield either particle from the gravitational force due to the other particle.
Fig. 13-2 (a) The gravitational force on particle 1 due to particle 2 is an attractive force because particle 1 is attracted to particle 2. (b) Force is directed along a radial coordinate axis r extending from particle 1 through particle 2. (c) is in the direction of a unit vector along the r axis.
The strength of the gravitational force—that is, how strongly two particles with given masses at a given separation attract each other—depends on the value of the gravitational constant G. If G—by some miracle—were suddenly multiplied by a factor of 10, you would be crushed to the floor by Earth’s attraction. If G were divided by this factor, Earth’s attraction would be so weak that you could jump over a building.
Although Newton’s law of gravitation applies strictly to particles, we can also apply it to real objects as long as the sizes of the objects are small relative to the distance between them. The Moon and Earth are far enough apart so that, to a good approximation, we can treat them both as particles—but what about an apple and Earth? From the point of view of the apple, the broad and level Earth, stretching out to the horizon beneath the apple, certainly does not look like a particle.
Newton solved the apple–Earth problem by proving an important theorem called the shell theorem:
A uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell’s mass were concentrated at its center.
Earth can be thought of as a nest of such shells, one within another and each shell attracting a particle outside Earth’s surface as if the mass of that shell were at the center of the shell. Thus, from the apple’s point of view, Earth does behave like a particle, one that is located at the center of Earth and has a mass equal to that of Earth.
Suppose that, as in Fig. 13-3, Earth pulls down on an apple with a force of magnitude 0.80 N. The apple must then pull up on Earth with a force of magnitude 0.80 N, which we take to act at the center of Earth. Although the forces are matched in magnitude, they produce different accelerations when the apple is released. The acceleration of the apple is about 9.8 m/s2, the familiar acceleration of a falling body near Earth’s surface. The acceleration of Earth, however, measured in a reference frame attached to the center of mass of the apple–Earth system, is only about 1 × 10−25 m/s2.
Fig. 13-3 The apple pulls up on Earth just as hard as Earth pulls down on the apple.
CHECKPOINT 1 A particle is to be placed, in turn, outside four objects, each of mass m: (1) a large uniform solid sphere, (2) a large uniform spherical shell, (3) a small uniform solid sphere, and (4) a small uniform shell. In each situation, the distance between the particle and the center of the object is d. Rank the objects according to the magnitude of the gravitational force they exert on the particle, greatest first.
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