Under the influence of force ' $F_1$ ' the body oscillates with a period ' $\mathrm{T}_1$ ' and due to…
Under the influence of force ' $F_1$ ' the body oscillates with a period ' $\mathrm{T}_1$ ' and due to another force ' $\mathrm{F}_2$ ' body oscillates with period ' $\mathrm{T}_2$ '. If both forces acts simultaneously then the resultant period is
(consider displacement is same in all three cases)
$\mathrm{F}=\mathrm{kx}$ and $\mathrm{k}=\mathrm{m} \omega^2$
$\therefore \quad F-m \omega^2 x$
$\therefore \quad \omega^2=\frac{F}{m x}$
For First Force, $\omega_1^2=\frac{\mathrm{F}_1}{\mathrm{mx}}$
For Second Force, $\omega_2^2=\frac{\mathrm{F}_2}{\mathrm{mx}}$
$\therefore \quad$ Resultant Force will be
$\omega_3^2=\frac{\mathrm{F}_1+\mathrm{F}_2}{\mathrm{mx}}$
$\omega_3^2=\omega_1^2+\omega_2^2$
$\frac{\omega_3^2}{4 \pi^2}=\frac{\omega_1^2}{4 \pi^2}+\frac{\omega_2^2}{4 \pi^2}$ ....(dividing by $4 \pi^2$ )
$\frac{1}{\mathrm{~T}_3^2}=\frac{1}{\mathrm{~T}_1^2}+\frac{1}{\mathrm{~T}_2^2}$
$\mathrm{T}_3^2=\sqrt{\frac{\mathrm{T}_1^2 \mathrm{~T}_2^2}{\mathrm{~T}_1^2+\mathrm{T}_{2 \ell}^2}}$