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 cons

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 k₁is installed at distance a from the hinge and another of force constant k₂at a distance b as shown in the figure. If the whole arrangement rests on a smooth horizontal table top, find the frequency of vibration.

TestHub · 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 $({mL}^{2} + \frac{1}{3} {ML}^{2}) α = (k_{1} a^{2} + k_{2} b^{2}) θ$

or $α = \frac{(k_{1} a^{2} + k_{2} b^{2}) θ}{L^{2} (m + \frac{M}{3})}$

∴ $ω^{2} = \frac{k_{1} a^{2} + k_{2} b^{2}}{L^{2} (m + \frac{M}{3})}$

Hence, frequency

$f = \frac{2}{π} = \frac{1}{2 π} \sqrt{\frac{k_{1} a^{2} + k_{2} b^{2}}{L^{2} (m + \frac{M}{3})}}$

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