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GATE 2025 AE – Question 57

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

A uniform rigid bar of mass 3 kg is hinged at point F, and supported by a spring of stiffness k=100 N/m, as shown in the figure. The natural frequency of free vibration of the system is ____rad/s (answer in integer).

A horizontal bar of length 1 m hinged to a wall at F at its left end. The right end G is held by a vertical spring k = 100 N/m attached to the ceiling.

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Correct answer: 10

Explanation

For small rotations $\theta$ about the hinge, the moment of inertia is $I_F = \frac{mL^2}{3} = \frac{3 \times 1^2}{3} = 1$ kg-m$^2$ and the spring gives the restoring moment $kL^2\theta = 100\theta$. So $\omega_n = \sqrt{\frac{100}{1}} = 10$ rad/s. The weight only moves the equilibrium position.