A particle is vibrating in S.H.M. Its velocities are v1 and v2 when the displacements from the mean position are y1 and y2, respectively, then its time period is

v2=ω2a2−x2 ⇒v12=ω2a2−y12 .....(1) ⇒v22=ω2a2−y22 ......(2)From (1) and (2), we getT=2πy12−y22v22−v12

A particle is vibrating in S.H.M. Its velocities are v1 and v2 when the displacements from the mean position are y1 and y2, respectively, then its time period is

v2=ω2a2−x2 ⇒v12=ω2a2−y12 .....(1) ⇒v22=ω2a2−y22 ......(2)From (1) and (2), we getT=2πy12−y22v22−v12

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A particle is vibrating in S.H.M. Its velocities are v₁ and v₂ when the displacements from the mean position are y₁ and y₂, respectively, then its time period is

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$2 π \sqrt{\frac{y_{1}^{2} + y_{2}^{2}}{v_{1}^{2} + v_{2}^{2}}}$

$2 π \sqrt{\frac{y_{1}^{2} - y_{2}^{2}}{v_{2}^{2} - v_{1}^{2}}}$

$2 π \sqrt{\frac{v_{1}^{2} + v_{2}^{2}}{y_{1}^{2} + y_{2}^{2}}}$

$2 π \sqrt{\frac{v_{2}^{2} - v_{1}^{2}}{y_{1}^{2} - y_{2}^{2}}}$

answer is D.

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Detailed Solution

$v^{2} = ω^{2} (a^{2} - x^{2}) \Rightarrow v_{1}^{2} = ω^{2} (a^{2} - y_{1}^{2}) . . . . . (1) \Rightarrow v_{2}^{2} = ω^{2} (a^{2} - y_{2}^{2}) . . . . . . (2)$
From (1) and (2), we get
$T = 2 π \sqrt{\frac{y_{1}^{2} - y_{2}^{2}}{v_{2}^{2} - v_{1}^{2}}}$