All Problems Universal Gravitation
Problem 1.211
A particle of mass \(m\) is located outside a uniform sphere of mass \(M\) at a distance \(r\) from its centre. Find: (a) the potential energy of gravitational interaction of the particle and the sphere; (b) the gravitational force which the sphere exerts on the particle.
1.211. (a) First let us consider a thin spherical layer of radius \(\rho\) and mass \(\delta M .\) The energy of interaction of the particle with an elementary belt \(\delta S\) of that layer is equal to (Fig. 8\()\) \[ d U=-\gamma(m \delta M / 2 l) \sin \theta d \theta \] According to the cosine theorem in the triangle \(O A P l^{2}=\rho^{2}+\) \(+r^{2}-2 \rho r \cos \theta,\) Having determined the differential of this expression, we can reduce Eq. ( \(^{*}\) ) to the form that is convenient for integration. After integrating over the whole layer we obtain \(\delta U=\) \(=-\gamma m \delta M / r\). And finally, integrating over all layers of the sphere, we obtain \(U=-\gamma m M / r\) (b) \(\vec{F}_{r}=-\partial U / \partial r=-\gamma m M / r^{2}\).