First slide

Simple hormonic motion

Question

Two springs are made to oscillate simple harmonically due to the same mass individually. The time periods obtained are $T_{1}$ and $T_{2}$ . If both the springs are connected in series and then made to oscillate by the same mass, the resulting time period will be

Moderate

A

$T_{1} + T_{2}$
B

$\frac{T_{1} T_{2}}{T_{1} + T_{2}}$
C

$\sqrt{T_{1}^{2} + T_{2}^{2}}$
D

$\frac{T_{1} + T_{2}}{2}$

Solution

Let spring constant of two springs are $K_{1}$ and $K_{2}$ , respectively, then
$T_{1} = 2 π \sqrt{\frac{m}{k_{1}}}$ and $T_{2} = 2 π \sqrt{\frac{m}{k_{2}}}$
$k_{1} = \frac{4 π^{2} m}{T_{1}^{2}}$ and $k_{2} = \frac{4 π^{2} m}{T_{2}^{2}}$
so when two springs are connected in series $T = 2 π \sqrt{\frac{m}{k_{e q}}}$ where $k_{e q} = \frac{k_{1} k_{2}}{k_{1} + k_{2}}$
Hence $T = \sqrt{T_{1}^{2} + T_{2}^{2}}$

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