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Q.
A particle of mass $m$ is attached to three identical massless springs of spring constant $k$ as shown in the figure. The time period of vertical oscillation of the particle is
When the particle of mass $m$ at $O$ is pushed by $y$ in the direction of $A$, the spring $A$ will be compressed by $y$ while spring $B$ and $C$ will be stretched by $y^{'}=y \cos 45^{\circ} .$ So, that the total restoring force on the mass $m$ is along $O A$
$F_{\text {net }}=F_{A}+F_{B} \cos 45^{\circ}+F_{C} \cos 45^{\circ}$
$=k y+2 k y^{'} \cos 45^{\circ}$
$=k y+2 k\left(y \cos 45^{\circ}\right) \cos 45^{\circ}$
$=2 k y$
Also, $F_{\text {net }}=k^{'} y $
$\Rightarrow k^{'} y=2 k y$
$ \Rightarrow k^{'}=2 k$
$T=2 \pi \sqrt{\frac{m}{k^{\prime}}}=2 \pi \sqrt{\frac{m}{2 k}}$
