The time period of simple harmonic motion of mass $M$ in the given figure is $π \sqrt{\frac{α M}{5 K}}$ , where the value of $α$ is _______.

Question

The time period of simple harmonic motion of mass M in the given figure is π α M 5 K , where the value of α is _______.

TestHub · Accepted Answer

For series combination of the spring, equivalent spring constant is given by, $\frac{1}{k_{e q}} = \frac{1}{k_{1}} + \frac{1}{k_{2}}$ .

Therefore, for combined system net spring constant $k_{e q} = \frac{2 k \times k}{2 k + k} + k = \frac{5 k}{3}$

Angular frequency of oscillation $(ω) = \sqrt{\frac{k_{e q}}{m}}$

$\Rightarrow (ω) = \sqrt{\frac{5 k}{3 m}}$

Period of oscillation $(τ) = \frac{2 π}{ω} = 2 π \sqrt{\frac{3 m}{5 k}}$

$= π \sqrt{\frac{12 m}{5 k}}$

Therefore, $α = 12$ .

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