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Universal Gravitation

Problem 1.217

Find the proper potential energy of gravitational interaction of matter forming (a) a thin uniform spherical layer of mass mm and radius RR; (b) a uniform sphere of mass mm and radius RR (make use of the answer to Problem 1.214 ).

Reveal Answer
1.217. (a) Let us subdivide the spherical layer into small elements, each of mass δm.\delta m . In this case the energy of interaction of each element with all others is δU=−γmδm/R\delta U=-\gamma m \delta m / R. Summing over all elements and taking into account that each pair of interacting elements appears twice in the result, we obtain U=−vm2/2R;U=-v m^{2} / 2 R ; (b) U=−3γm2/5RU=-3 \gamma m^{2} / 5 R.
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