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GATE 2020 ME (ME1) – Question 41

Vibrations · Free vibration of single degree of freedom systems, damping · 2 marks · Multiple choice

A rigid massless rod of length L is connected through a frictionless revolute joint to a disc pulley of mass m and radius r = L/4. Its other end is hinged frictionlessly to a wall. A 2k spring is attached at the rod midpoint. An inextensible rope passes over half the disc and is secured to a k spring at C as shown. There is no slip; the rope is always taut and the system is in static equilibrium in the shown configuration. Neglect gravity. Find its natural frequency for small-amplitude vibration.

Diagram for GATE 2020 ME (ME1) question 41
  1. $\sqrt{3/2}\sqrt{k/m}$
  2. $(3/\sqrt2)\sqrt{k/m}$
  3. $\sqrt3\sqrt{k/m}$
  4. $\sqrt{k/m}$

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Show answer and explanation

Correct answer: (C) $\sqrt3\sqrt{k/m}$

Explanation

Let pulley-centre displacement be y. No slip at the fixed rope gives rotation y/r; disc inertia is mr²/2, so effective translating mass is m+I/r²=3m/2. The rod’s midpoint moves y/2 and the free rope end 2y. Effective stiffness is $2k(1/2)^2+k(2)^2=9k/2$. Therefore $\omega_n=\sqrt{(9k/2)/(3m/2)}=\mathbf{\sqrt{3k/m}}$, **C**.