Question

As per given figures, two springs of spring constants $K$ and $2K$ are connected to mass $m$. If the period of oscillation in figure (a) is $3s$, then the period of oscillation in figure (b) will be $\sqrt{x}s$. The value of $x$ is______

Answer 1

Answer: 2

For figure (a) :
$K_{eq}=\frac{K\times 2K}{K+2K}=\frac{2K}{3}$
$T=2\pi \sqrt{\frac{m}{K_{eq}}}=2\pi \sqrt{\frac{m}{2K/3}}=2\pi \sqrt{\frac{3m}{2K}}$
For figure (b):
$K_{eq}=3K,T^{\prime }=2\pi \sqrt{\frac{m}{3K}}$
$\frac{T^{\prime }}{T}=\sqrt{\frac{m\times 2K}{3K\times 3m}}=\frac{\sqrt{2}}{3}$
$T^{\prime }=\sqrt{2}$
$x=2$