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Example G.7: Gravitational Potential Energy

Example:

A point mass m = 59 x 102 kg is placed at the center of the base of a uniform hemisphere of mass M = 110 x 108 kg and radius R = 110 m. What is the potential energy of interaction due to gravitational force?

Solution:

Consider an elementary section of thickness dr at a distance r from the center O of the hemisphere as shown in the figure. Let dM be the mass of this elementary section. Then,

Using equation (G.17) for a distributed mass system, the potential V at the point O is given by

The potential energy of interaction due to gravitational force is the product of the potential V and the mass m. Thus,

Now substituting m = 59 x 102 kg, M = 110 x 108 kg and radius R = 110 m in the expression obtained above, the potential energy of interaction U = - 59.03 J.

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