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Q.
The time period of vertical oscillations of the system shown in figure is $\left(\right.$ given $k_{1}=k_{2}=k$ and $k_{3}=2 k$ )
Solution:
The spring constant of the parallel combination of springs of spring constants $k_{1}$ and $k_{2}$ is
$k_{p}=k_{1}+k_{2}=k+k=2 k$
This combination is in series with the spring of spring constant $k_{3} .$ So the effective spring constant of the system is.
$\frac{1}{k_{\text{eff}}}=\frac{1}{k_{p}}+\frac{1}{k_{3}}=\frac{1}{2 k}+\frac{1}{2 k}=\frac{1}{k} $
$\therefore k_{\text{eff}}=k$
and $T=2 \pi \sqrt{\frac{m}{k_{e f f}}}=2 \pi \sqrt{\frac{m}{k}}$
