Four massless springs whose force constants are $2 \mathrm{~K}, 2 \mathrm{~K}, \mathrm{~K}$ and $2…
Four massless springs whose force constants are $2 \mathrm{~K}, 2 \mathrm{~K}, \mathrm{~K}$ and $2 \mathrm{~K}$ respectively are attached to a mass $\mathrm{M}$ kept on a frictionless plane as shown in figure, If mass $M$ is displaced in horizontal direction then frequency of oscillating system is
On the right hand side of the block, springs are connected in parallel
$\therefore \quad$ Their effective spring constant is given by
$\begin{aligned}
& \mathrm{K}_1=\mathrm{K}+2 \mathrm{~K} \\
& \mathrm{~K}_1=3 \mathrm{~K}
\end{aligned}$
On the left hand side of the block, springs are connected in series.
$\therefore \quad$ Their effective spring constant is given by,
$\begin{aligned}
& \frac{1}{\mathrm{~K}_2}=\frac{1}{2 \mathrm{~K}}+\frac{1}{2 \mathrm{~K}} \\
\therefore \quad \mathrm{K}_2 & =\mathrm{K}
\end{aligned}$
$\therefore \quad$ Effective spring constant of the system is given by,
$\begin{aligned}
& \mathrm{K}_{\mathrm{E}}=3 \mathrm{~K}+\mathrm{K}=4 \mathrm{~K} \\
\therefore \quad \omega & =\sqrt{\frac{\mathrm{K}_{\mathrm{E}}}{\mathrm{M}}}=\sqrt{\frac{4 \mathrm{~K}}{\mathrm{M}}} \\
\therefore \quad \mathrm{f} & =\frac{\omega}{2 \pi}=\frac{1}{2 \pi} \sqrt{\frac{4 \mathrm{~K}}{\mathrm{M}}}
\end{aligned}$
