Newton's Law of Gravitation
The gravitational attraction that’s most familiar to you is your weight, the force that attracts you toward the earth. By studying the motions of the moon and planets, Newton discovered a fundamental law of gravitation that describes the gravitational attraction between any two bodies. Newton published this law in 1687 along with his three laws of motion. In modern language, it says
“Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them.”
Figure 13.1 depicts this law, which we can express as an equation:
The gravitational constant G in Eq. (13.1) is a fundamental physical constant that has the same value for any two particles. We’ll see shortly what the value of G is and how this value is measured.
Equation (13.1) tells us that the gravitational force between two particles decreases with increasing distance r: If the distance is doubled, the force is only one-fourth as great, and so on. Although many of the stars in the night sky are far more massive than the sun, they are so far away that their gravitational force on the earth is negligibly small.
Caution: Don’t confuse g and G The symbols g and G are similar, but they represent two very different gravitational quantities. Lowercase g is the acceleration due to gravity, which relates the weight w of a body to its mass m: w = mg. The value of g is different at different locations on the earth’s surface and on the surfaces of other planets. By contrast, capital G relates the gravitational force between any two bodies to their masses and the distance between them. We call G a universal constant because it has the same value for any two bodies, no matter where in space they are located. We’ll soon see how the values of g and G are related.
Gravitational forces always act along the line joining the two particles and form an action–reaction pair. Even when the masses of the particles are different, the two interaction forces have equal magnitude (Fig. 13.1). The attractive force that your body exerts on the earth has the same magnitude as the force that the earth exerts on you. When you fall from a diving board into a swimming pool, the entire earth rises up to meet you! (You don’t notice this because the earth’s mass is greater than yours by a factor of about 10^{23.} Hence the earth’s acceleration
is only 10^{-23} as great as yours.)
Gravitation and spherically symmetric Bodies
We have stated the law of gravitation in terms of the interaction between two particles. It turns out that the gravitational interaction of any two bodies having spherically symmetric mass distributions (such as solid spheres or spherical shells) is the same as though we concentrated all the mass of each at its center, as in Fig. 13.2. Thus, if we model the earth as a spherically symmetric body with mass mE, the force it exerts on a particle or on a spherically symmetric body with mass m, at a distance r between centers, is provided that the body lies outside the earth. A force of the same magnitude is exerted on the earth by the body. (We will prove these statements in Section 13.6.)
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At points inside the earth the situation is different. If we could drill a hole to the center of the earth and measure the gravitational force on a body at various depths, we would find that toward the center of the earth the force decreases, rather than increasing as 1/r^{2} . As the body enters the interior of the earth (or other spherical body), some of the earth’s mass is on the side of the body opposite from the center and pulls in the opposite direction. Exactly at the center, the earth’s gravitational force on the body is zero.
Spherically symmetric bodies are an important case because moons, planets, and stars all tend to be spherical. Since all particles in a body gravitationally attract each other, the particles tend to move to minimize the distance between them. As a result, the body naturally tends to assume a spherical shape, just as a lump of clay forms into a sphere if you squeeze it with equal forces on all sides. This effect is greatly reduced in celestial bodies of low mass, since the gravitational attraction is less, and these bodies tend not to be spherical (Fig. 13.3).
Determining the value of G
To determine the value of the gravitational constant G, we have to measure the gravitational force between two bodies of known masses m1 and m2 at a known distance r. The force is extremely small for bodies that are small enough to be brought into the laboratory, but it can be measured with an instrument called a torsion balance, which Sir Henry Cavendish used in 1798 to determine G.
Figure 13.4 shows a modern version of the Cavendish torsion balance. A light, rigid rod shaped like an inverted T is supported by a very thin, vertical quartz fiber. Two small spheres, each of mass m1, are mounted at the ends of the horizontal arms of the T. When we bring two large spheres, each of mass m2, to the positions shown, the attractive gravitational forces twist the T through a small angle. To measure this angle, we shine a beam of light on a mirror fastened to the T. The reflected beam strikes a scale, and as the T twists, the reflected beam moves along the scale.
After calibrating the Cavendish balance, we can measure gravitational forces and thus determine G. The presently accepted value is
To three significant figures, G = 6.67 * 10^{-11} N . m^{2}/kg^{2} . Because 1 N = 1 gm/s ^{2} ,
the units of G can also be expressed as m^{3}/(kg.s^{ 2} ). Gravitational forces combine vectorially. If each of two masses exerts a force on a third, the total force on the third mass is the vector sum of the individual forces of the first two. Example 13.3 makes use of this property, which is often called superposition of forces (see Section 4.1).