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GATE 2021 ME (ME2) – Question 44

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

Consider the system shown in the figure. A rope goes over a pulley. A mass, m, is hanging from the rope. A spring of stiffness, k, is attached at one end of the rope. Assume rope is inextensible, massless and there is no slip between pulley and rope. The pulley radius is r and its mass moment of inertia is J. Assume that the mass is vibrating harmonically about its static equilibrium position. The natural frequency of the system is

Mass m suspended by a rope over a pulley of radius r and inertia J, with the other rope end attached to spring k. See the attached source image.
  1. $\sqrt{kr^2/(J-mr^2)}$
  2. $\sqrt{kr^2/(J+mr^2)}$
  3. $\sqrt{k/m}$
  4. $\sqrt{kr^2/J}$

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Correct answer: (B) $\sqrt{kr^2/(J+mr^2)}$

Explanation

With mass displacement x, pulley angular speed is ẋ/r. Kinetic energy is $\tfrac12(m+J/r^2)\dot x^2$, and incremental spring energy is kx²/2.

Thus $\omega_n=\sqrt{k/(m+J/r^2)}=\sqrt{kr^2/(J+mr^2)}$, **B**. Gravity only sets the static equilibrium offset.