Question

A massless rod rigidly fixed at $O$. A string carrying a mass $m$ at one end is attached to point $A$ on the rod so that $OA=a$. At another point, $B(OB=b)$ of the rod, a horizontal spring of force constant $k$ is attached as shown. Find the period of small vertical oscillations of mass $m$ around its equilibrium position.

Answer 1

At Equilibrium torque balance,

$k{x}_{0}b=mga$

When block of mass m displace x.

$mg-T=mA$ where A is acceleration

$mg-k(x_0+(x\frac{b}{a})\frac{b}{a}=mA$

$mg-k{x}_0\frac{b}{a}-kx(\frac{b}{a}{)}^2=mA$

$-k(\frac{b}{a}{)}^{2}x=mA$

$\omega =\sqrt{k}\frac{b}{a}$

$T=\frac{2\pi }{\omega }=\frac{2\pi a}{b\sqrt{k}}$