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As a final note, let’s show that when we are close to the earth’s surface, Eq. (13.9) reduces to the familiar U = mgy from Chapter 7. We first rewrite Eq. (13.8) as

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If the body stays close to the earth, then in the denominator we may replace r1 and r2 by RE, the earth’s radius, so,

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According to Eq. (13.4), g = GmE/RE 2 , so,

 

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If we replace the r’s by y’s, this is just Eq. (7.1) for the work done by a constant gravitational force. In Section 7.1 we used this equation to derive Eq. (7.2), U = mgy, so we may consider Eq. (7.2) for gravitational potential energy to be a special case of the more general Eq. (13.9).


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