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Universal Gravitation

Problem 1.211

A particle of mass mm is located outside a uniform sphere of mass MM at a distance rr 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.

Reveal Answer
1.211. (a) First let us consider a thin spherical layer of radius ρ\rho and mass δM.\delta M . The energy of interaction of the particle with an elementary belt δS\delta S of that layer is equal to (Fig. 8)) dU=−γ(mδM/2l)sin⁡θdθ d U=-\gamma(m \delta M / 2 l) \sin \theta d \theta According to the cosine theorem in the triangle OAPl2=ρ2+O A P l^{2}=\rho^{2}+ +r2−2ρrcos⁡θ,+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 δU=\delta U= =−γmδM/r=-\gamma m \delta M / r. And finally, integrating over all layers of the sphere, we obtain U=−γmM/rU=-\gamma m M / r (b) F⃗r=−∂U/∂r=−γmM/r2\vec{F}_{r}=-\partial U / \partial r=-\gamma m M / r^{2}.
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