Problem 1.214

There is a uniform sphere of mass \(M\) and radius \(R .\) Find the strength \(\mathbf{G}\) and the potential \(\varphi\) of the gravitational field of this sphere as a function of the distance \(r\) from its centre (with \(r<R\) and \(r>R\) ). Draw the approximate plots of the functions \(G(r)\) and \(\varphi(r)\)

\begin{array}{l} \mathbf{G}=\left\{\begin{array}{l} -\left(\gamma M / R^{3}\right) \mathbf{r} \text { for } r \leqslant R \\ -\left(\gamma M / r^{3}\right) \mathbf{r} \text { for } r \geqslant R ; \end{array}\right. \\ \varphi=\left\{\begin{array}{ll} -3 / 2\left(1-r^{2} / 3 R^{2}\right) & \gamma M / R \end{array}\right. \text { for } r \leqslant R \\ -\gamma M / r & \text { for } r \geqslant R . \text { See Fig. } 10 . \end{array}

Universal Gravitation