Question

Find the potential energy of the gravitational interaction of a mass point 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

• A.
• B.
• C.
• D. -1

Answer 1

The correct option is A

To find potential due to rod at the site of m.

Consider an element dx at a distance x from the center of the rod.Mass of the element = $\frac{m}{l}dx$

$dV=-\frac{G\frac{m}{l}dx}{(a-x)}$

Potential due to the rod is

$V=-\frac{GM}{l}{\int }_{\frac{-l}{2}}^{\frac{l}{2}}\frac{dx}{a-x}=-\frac{GM}{l}[-ln(a-x){]}_{\frac{-l}{2}}^{\frac{l}{2}}$

= $-\frac{GM}{l}ln\left(\frac{a+\frac{l}{2}}{a-\frac{l}{2}}\right)$

Potential energy of system

= $V_m=-\frac{GMm}{l}ln\left(\frac{a+\frac{1}{2}}{a-\frac{1}{2}}\right)$