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GATE 2026 ME – Question 42

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

A 5 kg mass is suspended at the free end of an overhanging massless beam, having a pin support and a roller support, as shown in the figure below. Young's modulus of the material of the beam is 200 GPa and area moment of inertia of the beam is $10^{-8}$ m$^4$. The natural frequency of the beam in rad/s is

A beam on a pin support at the left and a roller support 1 m from it. The beam overhangs a further 2 m beyond the roller, and a 5 kg mass hangs from the free end.
  1. 10
  2. 10/3
  3. 5
  4. 20/3

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Correct answer: (A) 10

Explanation

Treat the span between the supports as 1 m ($L$) and the overhang as 2 m ($a$). A load $P$ at the tip deflects it by $\frac{Pa^2(L + a)}{3EI} = \frac{P \times 4 \times 3}{3EI} = \frac{4P}{EI}$. With $EI = 200 \times 10^9 \times 10^{-8} = 2000$ N m$^2$, the stiffness at the tip is $k = \frac{EI}{4} = 500$ N/m. The natural frequency is $\omega = \sqrt{\frac{500}{5}} = 10$ rad/s.