homework and exercises - Can vertical SHM occur in a system of a mass between 2 springs between 2 vertical pillars?

The problem is detailed above. I have worked through problems involving SHM in the horizontal plane, but unsure how to go about it vertically. I know the weight component would need to be incorporated...

Thanks so much for helping!! :)

Answer

Since this is a 1D problem the frequency of small oscillations is simply given by the relation $$\omega^2 = \frac1m U''(x_0),$$ where $U(x)$ is the potential energy of the system and $x_0$ the equilibrium position. This is explicitly given by $$U(x) = mgx + k\left(\sqrt{x^2+d^2}-d\right)^2,$$ and from here one finds $$\omega^2 = \frac{2k}m\left(1-\frac dl + \frac{dx^2}{l^3}\right),$$ where $l^2 = x^2 + d^2$ for convenience. Observe that $d/l = \cos\theta$ and $x/l = \sin\theta$, so that when $x=x_0$, the angle $\theta$ is that of equilibrium, and the above equation is seen to become $$\omega^2 = \frac{2k}m\sin^2\theta.$$