A uniform solid sphere of mass M and radius R has a spherical cavity of radius R /2 . The center of the cavity is located at a distance R /2 from the center of the solid sphere. Let the center of the

Question

A uniform solid sphere of mass M and radius R has a spherical cavity of radius R /2 . The center of the cavity is located at a distance R /2 from the center of the solid sphere. Let the center of the solid sphere be the origin O. A point mass m is placed at an arbitrary point P inside the cavity. Which of the following statements is/are correct?

TestHub · Accepted Answer

Let the center of the solid sphere be the origin O $(0, 0, 0)$ . Let the center of the cavity be O' at $(R /2, 0, 0)$ . The gravitational field $E_{n e t}$ inside the cavity is found by superposition: $E_{n e t} = E_{so l i d} - E_{re m o v e d_s p h ere}$ .

For a point P at position $r$ inside the cavity, $E_{so l i d} = - \frac{GM}{R ^{3}} r$ . The mass of the removed sphere is $M^{'} = M \frac{( R /2 ) ^{3}}{R ^{3}} = M /8$ . The field due to the removed sphere at P is $E_{re m o v e d_s p h ere} = - \frac{G M ^{'}}{( R /2 ) ^{3}} (r - d)$ , where $d$ is the vector from O to O', i.e., $d = (R /2, 0, 0)$ . Substituting these, $E_{n e t} = - \frac{GM}{R ^{3}} r - (- \frac{G ( M /8 )}{( R /2 ) ^{3}} (r - d)) = - \frac{GM}{R ^{3}} r + \frac{GM}{R ^{3}} (r - d) = - \frac{GM}{R ^{3}} d$ . Since $d$ is a constant vector from O to O', $E_{n e t}$ is a constant vector. Specifically, $E_{n e t} = - \frac{GM}{R ^{3}} (\frac{R}{2} \hat{i}) = - \frac{GM}{2 R ^{2}} \hat{i}$ .

Option A: The gravitational force $F = m E_{n e t}$ is constant in magnitude and direction. This is **correct**.

Option C: The gravitational field $E_{n e t}$ is uniform and directed along $- \hat{i}$ , which is towards the center O. This is **correct**.

Option B: The gravitational potential $V (r) = - \int E_{n e t} \cdot d r = - \int (- \frac{GM}{2 R ^{2}} \hat{i}) \cdot (d x \hat{i} + d y \hat{j} + d z \hat{k}) = \frac{GM}{2 R ^{2}} x + C$ . Since $V$ depends on $x$ , it is not uniform. This is **incorrect**.

Option D: The gravitational potential energy $U = mV = m (\frac{GM}{2 R ^{2}} x + C)$ . To minimize $U$ , $x$ must be minimized. The cavity is centered at $(R /2, 0, 0)$ with radius $R /2$ , so its $x$ -coordinates range from $0$ to $R$ . The minimum $x$ value within the cavity is $x = 0$ , which corresponds to the point O (the center of the solid sphere) and lies on the cavity boundary. Thus, potential energy is minimum at the point closest to O. This is **correct**.

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