Two springs of force constants $k_{1}$ and $k_{2}$ are connected as shown in the figure. The time period of vertical oscillations of mass m is given by

$2 π \sqrt{\frac{m}{(k_{1} + k_{2})}}$

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$2 π \sqrt{\frac{m (k_{1} + k_{2})}{k_{1} k_{2}}}$

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$2 π \sqrt{\frac{m}{(k_{1} - k_{2})}}$

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$2 π \sqrt{\frac{m (k_{1} - k_{2})}{k_{1} k_{2}}}$

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Solution

The correct option is A
$2 π \sqrt{\frac{m}{(k_{1} + k_{2})}}$

The two springs are connected in parallel. Therefore, the force constant of the combination, $k^{'} = k_{1} + k_{2}$
Now, time period, $T = 2 π \sqrt{\frac{m}{k^{'}}} = 2 π \sqrt{\frac{m}{(k_{1} + k_{2})}}$
Hence, the correct choice is (a).

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