Springs of spring constants $K, 2K, 4K, 8K,$ ..... are connected in series. A mass $40\, gm$ is attached to the lower end of last spring and the system is allowed to vibrate. What is the time period of oscillation ..... $\sec$. (Given $K = 2\, N/cm$)
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$\frac{1}{K_{eq}}=\frac{1}{K}+\frac{1}{2 K}+\frac{1}{4 K}+\frac{1}{8 K}+\ldots \ldots=\frac{1}{K}\left[1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots\right]$
$\frac{1}{K_{e q}}=\frac{1}{K}\left[\frac{1}{1-\frac{1}{2}}\right]=\frac{2}{K} \Rightarrow K_{e q}=\frac{K}{2}$
$\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{m}}{\mathrm{K}_{\mathrm{eq}}}}$
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