What is the period of small oscillations of t | vedclass

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(A) Let the displacement of the block of mass $m$ be $x$ downwards.Since the block is supported by two segments of a string,each with tension $T$,the

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

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(A) Let the displacement of the block of mass $m$ be $x$ downwards.Since the block is supported by two segments of a string,each with tension $T$,the total upward force on the block is $2T = mg$ in equilibrium.The lower pulley is connected to the block. If the block moves down by $x$,the string on the right side of the lower pulley is fixed,so the left side of the string must move down by $2x$.This string goes over the upper pulley. If the string moves by $2x$,the spring connected to the upper pulley must stretch by $4x$ because the pulley arrangement provides a mechanical advantage.The effective spring constant $k_{eff}$ is determined by the relation $F = k_{eff} x$. Here,the restoring force $F = 4(k(4x)) = 16kx$ is incorrect; let's re-evaluate.Let the spring stretch be $y$. The tension in the spring is $ky$. This tension $ky$ acts on the upper pulley. The force transmitted to the lower pulley is $2(ky) = 2ky$. The block is supported by two such segments,so the total upward force is $2(2ky) = 4ky$.If the block moves by $x$,the spring stretches by $4x$. Thus,the restoring force is $F = 4k(4x) = 16kx$.Therefore,$k_{eff} = 16k$.The time period $T = 2\pi \sqrt{\frac{m}{k_{eff}}} = 2\pi \sqrt{\frac{m}{16k}} = 2\pi \frac{1}{4} \sqrt{\frac{m}{k}} = \frac{\pi}{2} \sqrt{\frac{m}{k}}$.

What is the period of small oscillations of the block of mass $m$ if the springs are ideal and pulleys are massless?

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