A block of mass ' $\mathrm{m}$ ' attached to one end of the vertical spring produces extension…

A block of mass ' $\mathrm{m}$ ' attached to one end of the vertical spring produces extension $\mathrm{x}^{\prime}$. If the block is pulled and released, the periodic time of oscillation is
  1. $2\pi\sqrt{\frac{2 x}{g}}$
  2. $2 \pi\sqrt{\frac{\mathrm{x}}{\mathrm{g}}}$
  3. $2 \pi \sqrt{\frac{\mathrm{x}}{2 \mathrm{~g}}}$
  4. $2 \pi \sqrt{\frac{\mathrm{x}}{4 \mathrm{~g}}}$

Solution

$\mathrm{mg}=-\mathrm{kx} \quad \therefore \mathrm{k}=\frac{\mathrm{mg}}{\mathrm{x}}$ $\omega^{2}=\frac{k}{m}$ $\left(\frac{2 \pi}{T}\right)^{2}=\frac{k}{m}$ $\frac{4 \pi^{2}}{T^{2}}=\frac{k}{m}$ $\therefore T^{2}=\frac{4 \pi^{2} m}{k}=\frac{4 \pi^{2} m x}{m g} \Rightarrow T=2 \pi \sqrt{\frac{x}{g}}$

Asked in: MHT CET 2020 (16 Oct Shift 2)

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