All the springs in fig. (a), (b) and (c) are identical, each having force constant K. Mass attached to each system is ' $m$ '. If $T_a, T_b$ and $T_c$ are the time periods of oscillations of the three systems respectively, then

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Aicharya · Accepted Answer

$\begin{aligned} T_a & =2 \pi \sqrt{\frac{m}{K}} \\ T_b & =2 \pi \sqrt{\frac{m}{\frac{K}{2}}} \\ \therefore \quad T_b & =2 \pi \sqrt{\frac{2 m}{K}}=\sqrt{2} T_a \Rightarrow T_a=\frac{T_b}{\sqrt{2}}...(i) \\ T_c & =2 \pi \sqrt{\frac{m}{2 K}}=\frac{T_a}{\sqrt{2}} \Rightarrow T_a=\sqrt{2} T_c ...(ii)\end{aligned}$
From (i) and (ii), $\frac{\mathrm{T}_{\mathrm{b}}}{\sqrt{2}}=\sqrt{2} \mathrm{~T}_{\mathrm{c}} \Rightarrow \mathrm{~T}_{\mathrm{b}}=2 \mathrm{~T}_{\mathrm{c}}$

All the springs in fig. (a), (b) and (c) are identical, each having force constant K. Mass attached to each…

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