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Two light identical springs of spring constant k are attached horizontally at the two ends of a uniform horizontal rod AB of length l and mass m. The rod is pivoted at its centre ‘O’ and can rotate freely in horizontal plane. The other ends of the two springs are fixed to rigid supports as shown in figure. The rod is gently pushed through a small angle and released. The frequency of resulting oscillation is:

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$\frac{1}{2 π} \sqrt{\frac{6 k}{m}}$

$\frac{1}{2 π} \sqrt{\frac{2 k}{m}}$

$\frac{1}{2 π} \sqrt{\frac{k}{m}}$

$\frac{1}{2 π} \sqrt{\frac{3 k}{m}}$

answer is A.

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Detailed Solution

The rod's ends' linear displacement is given by,

$x = \frac{l}{2} sinθ$

the net torque due to spring force,

$τ = - 2 k x \times r_{⊥} = - 2 k \times \frac{l}{2} \sin θ \times \frac{l}{2} c o s θ$

$τ = - \frac{k l^{2} \sin 2 θ}{4}$

$τ = - \frac{k l^{2}}{4} \times 2 θ a s a n g l e i s s m a l l$

$τ = - \frac{k l^{2} θ}{2}$

We know

$τ = I α$

$- \frac{k l^{2} θ}{2} = \frac{m l^{2}}{12} α$

$α = - \frac{6 k}{m} θ = - ω^{2} θ$

$ω = \sqrt{\frac{6 k}{m}}$

Thus the linear frequency,

$f = \frac{ω}{2 π} = \frac{1}{2 π} \sqrt{\frac{6 k}{m}}$

Hence the correct answer is $\frac{1}{2 π} \sqrt{\frac{6 k}{m}}$ .

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