A uniform rod of mass m and length L is hinged at one of its ends with the ceiling and the other end of the rod is attached with a thread which is attached with horizontal ceiling at point P. If one end of the rod is slightly displaced horizontally and perpendicular to the rod and released, and the time period of small oscillation is 2π√[(2L sinθ)/(xg)], find x.
Physics · Mixed Chapter
JEE Mains

Question

A uniform rod of mass m and length L is hinged at one of its ends with the ceiling and the other end of the rod is attached with a thread which is attached with horizontal ceiling at point P. If one end of the rod is slightly displaced horizontally and perpendicular to the rod and released, and the time period of small oscillation is 2π√[(2L sinθ)/(xg)], find x.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: For a small lateral displacement, the rod and supporting thread constrain the motion to a small rotation about the horizontal support line OP. Step 2: The moment of inertia about that axis is I=(mL²/3)sin²θ, and the centre-of-mass height changes to second order with the angular deflection. Step 3: Linearizing the restoring torque for a small angular displacement gives ω²=3g/(2L sinθ). Step 4: Hence T=2π/ω=2π√(2L sinθ/(3g)); comparing to the given form gives x=3.