A body executes sample harmonic motion under the action of a force $F_1$ with a time period $(4 / 5)\ sec$. If the force is changed to $F_ 2$ it executes $SHM$ with time period $(3 / 5)\ sec$. If both the forces $F_1$ and $F_2$ act simultaneously in the same direction on the body, it's time period (in $second$) is
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$F_{1}=m \omega^{2} x$
$T=\frac{2 \pi}{\omega}$
$F=m a=4 m \pi^{2} x \frac{1}{T^{2}}$
$\therefore F_{1}=\frac{4 m \pi^{2}}{T_{1}^{2}} x$
$F_{2}=\frac{4 m \pi^{2}}{T_{2}^{2}} x$
$F_{1}+F_{2}=\frac{4 m \pi^{2} x}{T^{2}}+\frac{4 m \pi^{2} x}{T_{2}^{2}}$
$\frac{1}{T^{2}}=\frac{1}{T_{1}^{2}}+\frac{1}{T_{2}^{2}}$
$\therefore T=\frac{12}{25}$
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