First slide

Simple hormonic motion

Question

A system is shown in the figure. The time period for small oscillations of the two blocks will be

Moderate

A

$2 π \sqrt{\frac{3 m}{k}}$
B

$2 π \sqrt{\frac{3 m}{2 k}}$
C

$2 π \sqrt{\frac{3 m}{4 k}}$
D

$2 π \sqrt{\frac{3 m}{8 k}}$

Solution

Both the spring are in series
$∴ K_{eq} = \frac{K (2 K)}{K + 2 K} = \frac{2 K}{3}$
Time period
$T = 2 π \sqrt{\frac{μ}{K_{eq}}} where μ = \frac{m_{1} m_{2}}{m_{1} + m_{2}}$
Here $μ = \frac{m}{2}$
$∴ T = 2 π \sqrt{\frac{m}{2} . \frac{3}{2 K}} = 2 π \sqrt{\frac{3 m}{4 K}}$

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