Forces On Bodies Problems Involving Multiple Bodies

• 2018: A uniform rod of mass M and length L is hinged at one end and is free to rotate in a vertical plane. A particle of mass m is attached to the other end of the rod. The particle is released from rest at a height h above the horizontal. The value of h such that the rod will just be able to rotate about the hinge without the particle leaving the rod is:

(A) $\frac{L}{2}$ (B) $\frac{3L}{4}$ (C) $\frac{L}{3}$ (D) $\frac{2L}{3}$

The answer is (C) $\frac{L}{3}$

Let’s analyze the question. The rod is free to rotate in a vertical plane. The particle is released from rest at a height h above the horizontal. We need to find the value of h such that the rod will just be able to rotate about the hinge without the particle leaving the rod.

The particle will leave the rod when the tension in the string is zero. The tension in the string is equal to the weight of the particle minus the centripetal force. The centripetal force is equal to mv^2/r