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  4. A system shown in the figure consists of a massless pulley, a spring of force constant k and a block of mass m . If the block is slightly displaced vertically downwards from its equilibrium position and released, then the period of vertical oscillations is <img class=img-fluid question-image alt=Question src=https://cdn.tardigrade.in/q/nta/p-hxejan6qecvh.png />

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Q. A system shown in the figure consists of a massless pulley, a spring of force constant $k$ and a block of mass $m$ . If the block is slightly displaced vertically downwards from its equilibrium position and released, then the period of vertical oscillations is

Question

NTA AbhyasNTA Abhyas 2020Oscillations

Solution:

In this situation, if mass $m$ moves downwards a distance $x$ from the equilibrium position, the pulley will also move by $x$ and so, the spring will stretch by $2 x$

Therefore, the spring force will be $2 k x$. The restoring force on the block
will be $4 k x$
Hence , F=-4 kx
image
$\Rightarrow m g-4 k x=m a $
$ \Rightarrow m a=-4 k x+m g $
$\Rightarrow \quad a=-\frac{4 k x}{m}+g $
$\Rightarrow \omega^2=\left|\frac{a}{x}\right| $
$ T=2 \pi \sqrt{\left|\frac{x}{a}\right|} $
$ T=2 \pi \sqrt{\left(\frac{m}{4 k}\right)}$