Question

The potential energy of gravitational interaction of a point mass \( m \) and a thin uniform rod of mass \( M \) and length \( L \), if they are located along a straight line at a distance \( a \) from each other, is

• A. \( U = -\frac{GMm}{L} \ln \left( \frac{a + L}{a} \right) \)
• B. \( U = -\frac{GMm}{L} \ln \left( \frac{a}{a + L} \right) \)
• C. \( U = \frac{GMm}{L} \ln \left( \frac{a + L}{a} \right) \)
• D. \( U = \frac{GMm}{L} \ln \left( \frac{a}{a + L} \right) \)

Hint:

The potential energy of the gravitational interaction between a point mass and a uniform rod is derived from integrating over the length of the rod.

Answer 1

The Correct Option is D

The potential energy of the gravitational interaction between a point mass and a uniform rod is calculated by integrating the gravitational potential energy contributions from each infinitesimal segment of the rod. The formula for the potential energy is: \[ U = -\frac{GMm}{L} \ln \left( \frac{a}{a + L} \right) \] Hence, the correct answer is (d).