A uniform spring whose unstretched length is l has a force constant k. The spring is cut into two pieces of unstretched lengths l1 and l2 , where l1 = nl2 and n is being an integer. Now. a mass m is made to oscillate with first spring. The time period of its oscillation would be

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A uniform spring whose unstretched length is l has a force constant k. The spring is cut into two pieces of unstretched lengths l₁ and l₂ , where l₁ = nl₂ and n is being an integer. Now. a mass m is made to oscillate with first spring. The time period of its oscillation would be

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$T = 2 π \sqrt{\frac{m}{k (n + 1)}}$

$T = 2 π \sqrt{\frac{m}{nk}}$

$T = 2 π \sqrt{\frac{mn}{k (n + 1)}}$

$T = 2 π \sqrt{\frac{m (n + 1)}{nk}}$

answer is A.

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$k \propto \frac{1}{l}$

$∵ \frac{l_{1}}{l_{2}} = n$

$\Rightarrow l_{1} = (\frac{n}{n + 1}) l$ or $k' = (\frac{n + 1}{n}) k$

$∴ T = 2 π \sqrt{\frac{mn}{k (n + 1)}}$

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