Question

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

Answer 1

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