Two springs of force constants $k_{1}$ and $K_{2}$ are connected to a mass m as shown in figure (a) and (b). what is the ratio of the time periods of vertical oscillation in cases (a) and (b) if $k_{1} = k_{2}$ ?

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Solution

The correct option is B
In case (a), the springs are connected in parallel. Therefore,
$k^{'} = k_{1} + k_{2} = 2 k$
$∴ T_{a} = 2 π \sqrt{\frac{m}{k^{'}}} = 2 π \sqrt{\frac{m}{2 k}}$
In case (b), the springs are connected in series. Therefore
$T_{b} = 2 π \sqrt{\frac{2 m}{k}}$
$∴ \frac{T_{a}}{T_{b}} = \frac{1}{2}$
Hence the correct choice is (b)

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