First slide

Simple hormonic motion

Question

A block is suspended by a spring and its time period of small vertical oscillation is found to be T. Now the spring is cut into two segments the ratio of whose length is 1 : 2. The segments are arranged in parallel combination and the same block is suspended from the combination. If the block is now allowed to oscillate vertically, find its new time period.

Moderate

Solution

$\begin{array}{l} k_{1} = (\frac{1 + 2}{1}) k = 3 k and k_{2} = (\frac{1 + 2}{2}) k = \frac{3}{2} k . \\ ∴ k_{e} = k_{1} + k_{2} = \frac{9}{2} k \\ Now, T = 2 π \sqrt{\frac{M}{k}} and T^{'} = 2 π \sqrt{\frac{M}{K_{e}}} \\ ∴ \frac{T^{'}}{T} = \sqrt{\frac{k}{k_{e}}} = \sqrt{\frac{2}{9}} \\ ∴ T^{'} = \frac{\sqrt{2} T}{3} \end{array}$

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