More on Gravitational potential energy
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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
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,
According to Eq. (13.4), g = Gm_{E}/R_{E} ^{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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