Question: (Gravitational attraction of a large ball) Let \(B\) be a very large metal ball with the radius \(R\) (imagine the Earth!) that has mass density depending only on the distance from its center. Let \(M\) be the total mass of \(B\). Let a point object \(P\) with unit mass is located at the distance \(D\) from the center of \(B\) and assume that \(D\) is greater than \(R\) : in other words, \(P\) is outside the ball \(B\) (imagine a satellite in the sky). Note that by Newton's law of gravitation, for two point objects \(P_{1}\) and \(P_{2}\) having mass \(m_{1}\) and \(m_{2}\) respectively, there is the gravitational force attracts the two point objects with the magnitude of force equal to \(G \frac{m_{1} m_{2}}{d^{2}}\), where \(d\) denotes the distance between the two object and \(G\) is the gravitational constant. In the following you are asked to prove that the gravitational force \(\vec{F}\) of \(B\) on \(P\) has the magnitude

Note that the ball \(B\) is large so you cannot consider it as a point even approximately! (This exercise is one of the fundamental computations in Astrophysics, where the stars or planets are often considered as point masses in the description of their dynamics. To verify this result, Isaac Newton had to postpone publishing his Principia.)

1. (3 marks). It is often convenient to use a potential function of the force \(\vec{F}\), i.e. a function \(\varphi\) such that \(\vec{F}=-\nabla \varphi .\) Consider a shell of \(B\), i.e. the sphere \(S_{\rho}\) of radius \(\rho(0 \leq \rho \leq R)\) whose center is the same as the ball \(B\). Suppose this shell \(S_{\rho}\) has an infinitesimal width \(d \rho\). On this surface \(S_{\rho}\) the mass density is constant. Let \(f(\rho)\) denote this density (it is the function of only \(\rho)\). Thus the total mass $d M$ of \(S_{\rho}\) is
   \[d M=f(\rho) \cdot \text { the surface area of } S_{\rho} \cdot d \rho\]
   The infinitesimal gravitational potential \(d \varphi\) at \(P\) of the gravitational force of the shell \(S_{\rho}\) can be computed by
   \[d \varphi=G f(\rho) d \rho \iint_{S_{\rho}} \frac{1}{\delta} d S\]
   where \(\delta\) denotes the distance of a point on the shell \(S_{\rho}\) to the point \(P\). Show that
   \[d \varphi=\frac{G}{D} d M\]
2. ( 1 mark). Use (a) to compute the potential \(\varphi\) at \(P\) of the gravitational force of the ball \(B\). Here one can use the superposition principle of the gravitational potential that \(\varphi=\int d \varphi\),
   i.e. the potential \(\varphi\) of the ball is the sum of the potentials of all the infinitesimal shells.
3. ( 1 mark ). Use (b) to deduce \((*)\).

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