A rod of mass M and length L is hinged at its one end and carries a block of mass m at its lower end. A spring of force constant k 1 is installed at distance a from the hinge and another of force constant k 2 at a distance b as shown in the figure. If the whole arrangement rests on a smooth horizontal table top, find the frequency of vibration.

Question

A rod of mass M and length L is hinged at its one end and carries a block of mass m at its lower end. A spring of force constant k 1 is installed at distance a from the hinge and another of force constant k 2 at a distance b as shown in the figure. If the whole arrangement rests on a smooth horizontal table top, find the frequency of vibration., 1 2 π k 1 a 2 + k 2 b 2 L 2 ( m + M 3 ), 1 2 π k 1 a 2 + k 2 b 2 L 2 ( m − M 3 ), 1 2 π k 1 a 2 − k 2 b 2 L 2 ( m + M 3 ), 1 4 π k 1 a 2 + k 2 b 2 L 2 ( m − M 3 )

MARKS App · Accepted Answer

For small angular displacement θ, net torque towards mean position is τ = k 1 a θ a + k 2 b θ b or I α = k 1 a 2 + k 2 b 2 θ or mL 2 + 1 3 ML 2 α = k 1 a 2 + k 2 b 2 θ or α = k 1 a 2 + k 2 b 2 θ L 2 m + M 3 ∴ ω 2 = k 1 a 2 + k 2 b 2 L 2 m + M 3 Hence, frequency f = 2 π = 1 2 π k 1 a 2 + k 2 b 2 L 2 m + M 3

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