First slide

Simple hormonic motion

Question

Four springs of constant as shown are attached to a pair of masses m each as shown. The time period will be 2π times

Moderate

A

$\sqrt{\frac{m}{k}}$
B

$\sqrt{\frac{2 m}{k}}$
C

$\sqrt{\frac{4 m}{k}}$
D

$\sqrt{\frac{3 m}{4 k}}$

Solution

$\begin{array}{l} \frac{1}{k_{eff}} = \frac{1}{2 k} + \frac{1}{2 k} + \frac{1}{2 k} = \frac{3}{2 k} \Rightarrow k_{eff} = \frac{2 k}{3} & μ = \frac{m \times m}{m + m} = \frac{m}{2} . \\ T = 2 π \sqrt{\frac{μ}{k_{eff}}} = 2 π \sqrt{\frac{m}{2 \times 2 \frac{k}{3}}} \Rightarrow T = 2 π \sqrt{\frac{3 m}{4 k}} \end{array}$

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