JEE Main · 2024 · 31 Jan 2024 · Shift 2 · Elasticity & SHM

Two springs of stiffness k, mounted side by side, act in parallel and are equivalent to a single spring of stiffness 2k. This 2k spring is in series with another spring of stiffness k below it:

$k_{series}=\frac{2k\cdot k}{2k+k}=\frac{2k}{3}$

This series branch acts in parallel with the remaining spring of stiffness k that directly supports the mass, so

$k_{eq}=k+\frac{2k}{3}=\frac{5k}{3}$

The angular frequency of oscillation is

$\omega =\sqrt{\frac{k_{eq}}{M}}=\sqrt{\frac{5k}{3M}}$

and the time period is

$T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{3M}{5k}}=\pi \sqrt{\frac{12M}{5k}}$

Comparing with $\pi \sqrt{\frac{\alpha M}{5K}}$, $\alpha =12$