A particle is describing simple harmonic motion. If its velocities are v₁ and v₂ when the displacements from the mean position are y₁ and y₂ respectively, then its time period is

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

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

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

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

Solution

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

In simple harmonic motion,

velocity

v = $ω \sqrt{A^{2} - y^{2}}$

∴ v₁ = $ω \sqrt{A^{2} - y_{1}^{2}}$

⇒ $v_{1}^{2} = ω^{2} A^{2} - ω^{2} y_{1}^{2}$ .....(i)

and $v_{2} = ω \sqrt{A^{2} - y_{2}^{2}}$

⇒ $v_{2}^{2} = ω^{2} A^{2} - ω^{2} y_{2}^{2}$ ....(ii)

Solving equations (i) and (ii), we get

$v_{2}^{2} - v_{1}^{2} = ω^{2} (y_{1}^{2} - y_{2}^{2})$

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

Time period is given by,

⇒ T = $\frac{2 π}{ω}$

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

Some More Questions From Oscillations Chapter

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