First slide

Simple hormonic motion

Question

Springs of spring constants K, 2K, 4K, 8k, .... 2048 K are connected in series. A mass ‘m’ is attached to one end the system is allowed to oscillation. The time period is approximately

Easy

A

$\large 2\pi \sqrt {\frac{m}{{2K}}}$
B

$\large \,2\pi \sqrt {\frac{{2m}}{K}}$
C

$\large 2\pi \sqrt {\frac{m}{{4K}}}$
D

$\large 2\pi \sqrt {\frac{{4m}}{K}}$

Solution

$\large \frac{1}{{{k_{eff}}}}\, = \,\frac{1}{k} + \frac{1}{{2k}} + \frac{1}{{4k}} + ....$
$\large \frac{1}{{{k_{eff}}}}\, = \,\frac{1}{k}\left[ {1 + \frac{1}{2} + \frac{1}{{{2^2}}} + ...} \right]$
$\large \frac{1}{{{k_{eff}}}}\, = \,\frac{1}{k}\left[ {\frac{a}{{1 - r}}} \right]\, = \,\frac{1}{k}\left[ {\frac{1}{{1 - \frac{1}{2}}}} \right]\, = \,\frac{2}{k}$
$\large {k_{eff}} = \,\frac{k}{2};\,T\, = \,2\pi \sqrt {\frac{m}{{{k_{eff}}}}}$
$\large T\, = \,2\pi \sqrt {\frac{{2m}}{k}}$

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