Question

find the time period of the spring block system. see in attachment​

Answer 1

$\huge{\underline{\sf{\red{Answer\leadsto 2\pi\sqrt\dfrac{6m}{8k}}}}}$

Given:

Two spring are in parallel of each spring constant value = k

One spring is in series with the both of the parallel ones of spring constant value= 4k

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Formula

For Spring in parallel

$K_eq = k_1+k_2$

For Spring in series

$\dfrac{1}{k_eq} = \dfrac{1}{k_1} + \dfrac{1}{k_2}$

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Solution:

$K_eq = k_1+k_2$

$\implies$ $K_eq = k+k$

$\implies$ 2k

Now We have Two Springs in series

$\dfrac{1}{k_eq} = \dfrac{1}{k_1} + \dfrac{1}{k_2}$

$\implies$$\dfrac{1}{k_eq} = \dfrac{1}{2k} + \dfrac{1}{4k}$

$\implies$ $\dfrac{1}{k_eq} = \dfrac{1}{k} \bigg( \dfrac{6}{8}\bigg)$

$\implies$ $k_eq = \dfrac{8k}{6}$

Now

Time period = $\2\pi \sqrt\dfrac{m}{k}$

$\bold\leadsto$$\large{\boxed{2\pi \sqrt\dfrac{6m}{8k}}}$