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

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Solution

The correct option is A
The two springs are connected in parallel. Therefore, the force constant of the combination is
$K^{^{'}} = k_{1} + k_{2}$
This can be shown as follows: Let x be the extension in each spring. Since the total force is mg,
we get, $m g = - (k_{1} + k_{2}) x = - k^{^{'}} x . T h u s k^{^{'}} = k_{1} + k_{2}$ . Therefore
$T = 2 π \sqrt{\frac{m}{k^{^{'}}}} = 2 π \sqrt{\frac{m}{(k_{1} + k_{2})}}$
Hence the correct choice is (a).

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Two springs of force constants k1 and k2 are connected as shown figure. The time period of vertical oscillations of mass m is given by

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