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When we first developed the concept of gravitational potential e we assumed that the gravitational force on a body is constant in  This led to the expression U. But we now know that the force on a body of mass m at any point outside the earth is given met(12-2), where is the mass of the earth and r is the from the earth’s center. For problems in which r changes enough force can’t be considered constant, we need a more general potential energy.To find this expression, we follow the same basic sequence 7-2. We consider a body of mass m outside the earth, and first done by the gravitational force when the body moves directly center of the earth from r = r, to r = r2 as in Fig. 12-8.


where F, is the radial component of the gravitational force F. that the direction outward from the center of the earth. Because F toward the center of the earth, F, is negative. It differs from Eq. of the gravitational force,

The path doesn’t have to be a straight line; it could also be a 12-8. By an argument siinilar to that in Section 7-2, this work and final values of r, not on the path taken. This also pro, force is always conservative.
We now define the corresponding potential energy U so Eq. (7-3). Comparing this with Eq. (12-8), we see that the gravitational potential energy depends on the distance r and the center of the earth. When the body moves away the  ravitational force does negative work, and U increases When the body “falls” toward earth, r decreases, the gravand the potential energy decreases (i.e., becomes more _ Eq. (12-9) because it states that gravitational potential in fact you’ve seen negative values of U before. In using 7-2, we found that U was negative whenever the body below the arbitrary height we chose to be y :0 O-that is, earth were closer together than some certain arbitrary 7-2 in Section 7-2.) In defining U by Eq. (12-9), we the body of mass m is infinitely far away from the earth earth, gravitational potential energy decreases and anted, we could make U = 0 at the surface of the earth the quantity to Eq. (12-9).

This would make the price of making the expression for U more not affect the difference in U between any two points, quantity; that is why we omit this term and energy to confuse the expressions for gravitational force, Eq energy, Eq. (12-9). The force F, is proportional to is proportional to 1/r..•• e can now use general energy relations for problems in gravitational force has to be included. If the only force that does work, the total mechanical energy conserved. In the following example we’ll use this principle,the speed required for a body to escape completely from figure.


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