Physics - SHM Question with Solution | TestHub

In the initial configuration, the mass $m$ is connected to two springs $S_1$ and $S_2$ in parallel. When the mass is displaced by a distance $x$, one spring is compressed and the other is stretched, but both exert a restoring force in the same direction.

The effective spring constant is:
$K_{eq}=K+K=2K$

The initial frequency of oscillation is:
$f=\frac{1}{2\pi }\sqrt{\frac{K_{eq}}{m}}=\frac{1}{2\pi }\sqrt{\frac{2K}{m}}$

When spring $S_2$ is removed, the mass is connected to only one spring $S_1$. The new effective spring constant is:
$K_{eq}^{\prime }=K$

The new frequency of oscillation is:
$f^{\prime }=\frac{1}{2\pi }\sqrt{\frac{K_{eq}^{\prime }}{m}}=\frac{1}{2\pi }\sqrt{\frac{K}{m}}$

Comparing the two frequencies:
$f^{\prime }=\frac{1}{\sqrt{2}}\left(\frac{1}{2\pi }\sqrt{\frac{2K}{m}}\right)=\frac{f}{\sqrt{2}}$

Answer: $\frac{\mathrm{f}}{\sqrt{2}}$