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GATE 2021 AE – Question 36

Structures · Structural dynamics: free and forced vibration of SDOF systems and free vibration of 2-DOF systems · 2 marks · Multiple choice

A massless bent rod, pivoted as shown, carries mass m at coordinates (L,H) and a spring k at horizontal distance L/2 from the pivot. Its small-oscillation frequency in Hz is

source diagram and its labels are reproduced in the attached image.
  1. $\frac1{2\pi}\sqrt{\frac{kL^2}{4m(L^2+H^2)}}$
  2. $\frac1{2\pi}\sqrt{\frac{kL^2}{m(L^2+H^2)}}$
  3. $\frac1{2\pi}\sqrt{\frac{4kL^2}{m(L^2+H^2)}}$
  4. $\frac1{2\pi}\sqrt{\frac{k(L^2+H^2)}{4mL^2}}$

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Correct answer: (A) $\frac1{2\pi}\sqrt{\frac{kL^2}{4m(L^2+H^2)}}$

Explanation

Use the energy method for a small rotation $\theta$ about the pivot.

**Kinetic energy.** The mass $m$ sits at $(L,H)$ from the pivot, at a distance $\sqrt{L^2+H^2}$, so its speed is $\sqrt{L^2+H^2}\,\dot\theta$ and the equivalent moment of inertia is
$$I=m\,(L^2+H^2).$$

**Potential energy.** The spring is attached at a horizontal distance $L/2$ from the pivot, so its extension is $\dfrac L2\theta$ and the restoring moment is
$$M=k\left(\frac L2\theta\right)\frac L2=\frac{kL^2}{4}\theta .$$

**Natural frequency.**
$$\omega_n=\sqrt{\frac{kL^2/4}{m(L^2+H^2)}}=\sqrt{\frac{kL^2}{4m(L^2+H^2)}},\qquad f=\frac{\omega_n}{2\pi}.$$

Answer: option **A**.