GRAVITATIONAL POTENTIAL ENERGY
When we assumed that the earth’s gravitational force on a body of mass m doesn’t depend on the body’s height. This led to the expression U = mgy. But Eq. (13.2), Fg = GmEm/r^{2} , shows that the gravitational force exerted by the earth (mass mE) does in general depend on the distance r from the body to the earth’s center. For problems in which a body can be far from the earth’s surface, we need a more general expression for gravitational potential energy
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To find this expression, we follow the same steps as in Section 7.1. We consider a body of mass m outside the earth, and first compute the work W_{grav} done by the gravitational force when the body moves directly away from or toward the center of the earth from r = r1 to r = r2, as in Fig. 13.10. This work is given by
where Fr is the radial component of the gravitational force F S —that is, the component in the direction outward from the center of the earth. Because F S points directly inward toward the center of the earth, Fr is negative. It differs from Eq. (13.2), the magnitude of the gravitational force, by a minus sign:
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Substituting Eq. (13.7) into Eq. (13.6), we see that Wgrav is given by
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The path doesn’t have to be a straight line; it could also be a curve like the one in Fig. 13.10. By an argument similar to that in Section 7.1, this work depends on only the initial and final values of r, not on the path taken. This also proves that the gravitational force is always conservative.
We now define the corresponding potential energy U so that Wgrav = U1 - U2, as in Eq. (7.3). Comparing this with Eq. (13.8), we see that the appropriate definition for gravitational potential energy is
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Figure 13.11 shows how the gravitational potential energy depends on the distance r between the body of mass m and the center of the earth. When the body moves away from the earth, r increases, the gravitational force does negative work, and U increases (i.e., becomes less negative). When the body “falls” toward earth, r decreases, the gravitational work is positive, and the potential energy decreases (i.e., becomes more negative)
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You may be troubled by Eq. (13.9) because it states that gravitational potential energy is always negative. But in fact you’ve seen negative values of U before. In using the formula U = mgy in Section 7.1, we found that U was negative whenever the body of mass m was at a value of y below the arbitrary height we chose to be y = 0—that is, whenever the body and the earth were closer together than some arbitrary distance. (See, for instance, Example 7.2 in Section 7.1.) In defining U by Eq. (13.9), we have chosen U to be zero when the body of mass m is infinitely far from the earth 1r = ∞2. As the body moves toward the earth, gravitational potential energy decreases and so becomes negative.
If we wanted, we could make U = 0 at the earth’s surface, where r = RE, by adding the quantity GmEm>RE to Eq. (13.9). This would make U positive when r 7 RE. We won’t do this for two reasons: One, it would complicate the expression for U; two, the added term would not affect the difference in potential energy between any two points, which is the only physically significant quantity. If the earth’s gravitational force on a body is the only force that does work, then the total mechanical energy of the system of the earth and body is constant, or conserved. In the following example we’ll use this principle to calculate escape speed, the speed required for a body to escape completely from a planet.