A particle is executing $SHM$ along a straight line. Its velocities at distance $x_1$ and $x_2$ from the mean position are $V_1$ and $V_2$ respectively. Its time period is
AIPMT 2015,JEE MAIN 2021, Diffcult
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In $SHM,$ velocities of a particle at distances $x_{1}$ and $x_{2}$ from mean position are given by
${V_{1}^{2}=\omega^{2}\left(a^{2}-x_{1}^{2}\right)}...(i)$
${V_{2}^{2}=\omega^{2}\left(a^{2}-x_{2}^{2}\right)}...(ii)$
From equations $(i)$ and $(ii),$ we get
$V_{1}^{2}-V_{2}^{2}=\omega^{2}\left(x_{2}^{2}-x_{1}^{2}\right)$
$\omega=\sqrt{\frac{V_{1}^{2}-V_{2}^{2}}{x_{2}^{2}-x_{1}^{2}}} $
$T=2 \pi \sqrt{\frac{x_{2}^{2}-x_{1}^{2}}{V_{1}^{2}-V_{2}^{2}}}$
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