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Springs

A body is in ...

Question

A body is in SHM with period $T$ when oscillated from a freely suspended spring.If this spring is cut in two parts of length ratio 1:3 & again oscillated from the two parts separately,then the periods are $T_{1}$ & $T_{2}$ then $\frac{T_{1}}{T_{2}}$ is given as $\frac{1}{\sqrt{x}}$ . Find $x$ .

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Solution

Time period is inversely proportional to square root of spring constants.
Spring constant is inversely proportional to the length of spring.
$\frac{L_{1}}{L_{2}} = \frac{1}{3} = \frac{k_{2}}{k_{1}}$
So $\frac{T_{1}}{T_{2}} = \sqrt{\frac{k_{2}}{k_{1}}} = \frac{1}{\sqrt{3}}$

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