All Problems Universal Gravitation
Problem 1.217
Find the proper potential energy of gravitational interaction of matter forming (a) a thin uniform spherical layer of mass \(m\) and radius \(R\); (b) a uniform sphere of mass \(m\) and radius \(R\) (make use of the answer to Problem 1.214 ).
1.217. (a) Let us subdivide the spherical layer into small elements, each of mass \(\delta m .\) In this case the energy of interaction of each element with all others is \(\delta U=-\gamma m \delta m / R\). Summing over all elements and taking into account that each pair of interacting elements appears twice in the result, we obtain \(U=-v m^{2} / 2 R ;\) (b) \(U=-3 \gamma m^{2} / 5 R\).