A body executes $SHM$ under the action of force $F_1$ with frequency $n_1$ . If the force is changed to $F_2$ , it executes SHM with a frequency $n_2$ . If both the forces act simultaneously in the same direction on the body, then find the new frequency of oscillations.

We have,

            $F_1= -m{a}_1$ and $F_2= -m{a}_2$

                 $= -m{{\omega }_1}^{2}y$         $= -m{{\omega }_2}^{2}y$

$\therefore$                         $F = F_1+F_1 = m({{\omega }_1}^2+{{\omega }_2}^2)(-y)$       …(i)

If  $\omega$ is the new frequency, then

                             $F=-m{\omega }^{2}y$    …(ii)

On comparing equations (i) and (ii),

                     ${\omega }^2 = {{\omega }_1}^2+{{\omega }_2}^2$

or                 $n^2 = {n_1}^2+{n_2}^2$

and              $\frac{1}{T^2} = \frac{1}{{T_1}^2} + \frac{1}{{T_2}^2}$