A body executes SHM under the influence of one force and has time period $' T_{1}'$ seconds. The same body executes SHM with period $' T_{2}'$ seconds under the influence of another force. When both forces act simultaneously and in the same direction then the time period of the same body is

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\sqrt{T_{1}^{2} + T_{2}^{2}} \sec

\sqrt{\frac{T_{1} + T_{2}}{T_{1} T_{2}}} \sec

\sqrt{\frac{T_{1}^{2} T_{2}^{2}}{(T_{1}^{2} + T_{2}^{2})}} \sec

(T_{1} + T_{2}) \sec

answer is D.

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Detailed Solution

F = mr ω^{2}, T = \frac{2 π}{ω}

$ω = \frac{2 π}{T}$

$F = mr (\frac{4 π^{2}}{T^{2}})$

$F \propto \frac{1}{T_{2}^{2}}$

$F_{1} \propto \frac{1}{T_{1}^{2}}; F_{1} + F_{2} = \frac{1}{T^{2}}$

$\Rightarrow F_{2} \propto \frac{1}{T_{2}^{2}}$

$\frac{1}{T^{2}} = \frac{1}{T_{1}^{2}} + \frac{1}{T_{2}^{2}}$

$T = \sqrt{\frac{T_{1}^{2} T_{2}^{2}}{T_{1}^{2} + T_{2}^{2}}}$

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