Physics - SHM Question with Solution | TestHub

$$\text{ Let the mass }m\text{ be displaced towards right }$$ by a distance $x$. Then the spring 1 will be extended and the spring 2 will be compressed by an amount of $x$ each. The restoring force on mass $m$ due to spring 1 is
$$F_1=-kx$$ And the restoring force due to spring 2 is $$F_2=-(2k)x$$ $\therefore$ Total force on the mass $m$ is $$\begin{array}{rl}& F=F_1+F_2=-kx-2kx=-3kx\\ \Rightarrow & m\frac{d^{2}x}{d{t}^2}=-3kx\Rightarrow \frac{d^{2}x}{d{t}^2}+\frac{3k}{m}x=0\\ \Rightarrow & \frac{d^{2}x}{d{t}^2}+{\omega }^{2}x=0\text{ where }{\omega }^2=\frac{3k}{m}\Rightarrow \omega =\sqrt{\frac{3k}{m}}\\ \Rightarrow & f=\frac{\omega }{2\pi }=\frac{1}{2\pi }\sqrt{\frac{3k}{m}}.\end{array}$$