In the given figure an L-shaped bar of mass M is pivoted at one of its end, so that it can freely rotate in a vertical plane. Find the frequency of oscillation, if it is slightly disturbed from its equilibrium position.
A
12π√gL
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B
12π√3g√104L
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C
12π√3g4L
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D
12π√3g√54L
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Solution
The correct option is B12π√3g√104L We can consider AB and BC as individual rods of masses M/2 and length L each. Let C1 and C2 be the COM of AB and BC respectivley. Then COM of L-shaped rod is xcom=M2L2M=L4;Ycom=M2L2M=L4
Hence, distance of COM from point of suspension is d=√(3L4)2+(L4)2=L4√10 Moment of inertia about A is IA=(M2)L23+(M2)L212+M2[L2+(L2)2]=ML23 Frequency of physical pendulum is f=12π√MgdIA f=12π
⎷MgL√104ML23=12π√3g√104L f=12π(3g√104L)12