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A massless rod is hinged at a fixed point O. A string attached to the rod at point A (where OA = a) supports a mass m hanging vertically. A horizontal spring of force constant k is connected to the rod at point B (where OB = b). Find the time period of small vertical oscillations of the mass m about its equilibrium position.

1. 2*pi*sqrt(m*a² / (k*b²))
2. 2*pi*sqrt(m*b² / (k*a²))
3. 2*pi*sqrt(m / k)
4. 2*pi*sqrt(m*a / (k*b))

Correct answer: 2*pi*sqrt(m*a² / (k*b²))

Solution

Taking torque about O for small displacement theta: the spring (at distance b) provides restoring torque k*b²*theta, and the moment of inertia is m*a², giving SHM with period 2*pi*sqrt(m*a²/(k*b²)).

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