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Consider the system shown in the figure. A rope goes over a pulley. A mass, $\text{m}$, is hanging from the rope. A spring of stiffness, $\text{k}$, 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 $\text{r}$ and its mass moment of inertia is $\text{J}$. Assume that the mass is vibrating harmonically about its static equilibrium position. The natural frequency of the system is

  1. $\sqrt{\frac{kr^{2}}{J - mr^{2}}}$
  1. $\sqrt{\frac{kr^{2}}{J + mr^{2}}}$
  1. $\sqrt{k/m}$
  1. $\sqrt{\frac{kr^{2}}{J}}$
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