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 \(4 \pi^2\) )
\(\frac{1}{\mathrm{~T}_3^2}=\frac{1}{\mathrm{~T}_1^2}+\frac{1}{\mathrm{~T}_2^2}\)
\(\mathrm{T}_3^2=\sqrt{\frac{\mathrm{T}_1^2 \mathrm{~T}_2^2}{\mathrm{~T}_1^2+\mathrm{T}_{2 \ell}^2}}\)