First slide

Simple hormonic motion

Question

A particle is executing SHM along a straight line. Its velocities at distances $x_{1} and x_{2}$ from the mean position are $V_{1} and V_{2}$ , respectively. Its time period is

Moderate

A

$2 π \sqrt{\frac{V_{1}^{2} + V_{2}^{2}}{x_{1}^{2} + x_{2}^{2}}}$
B

$2 π \sqrt{\frac{V_{1}^{2} - V_{2}^{2}}{x_{1}^{2} - x_{2}^{2}}}$
C

$2 π \sqrt{\frac{x_{1}^{2} + x_{2}^{2}}{V_{1}^{2} + V_{2}^{2}}}$
D

$2 π \sqrt{\frac{x_{2}^{2} - x_{1}^{2}}{V_{1}^{2} - V_{2}^{2}}}$

Solution

In SHM, velocities of a particle at distances $x_{1} and x_{2}$ from mean position are given by
$V_{1}^{2} = ω^{2} (a^{2} - x_{1}^{2})$ . . . . .(I)
$V_{2}^{2} = ω^{2} (a^{2} - x_{2}^{2})$ . . . . .(II)
From equations (i) and (ii), we get
$V_{1}^{2} - V_{2}^{2} = ω^{2} (x_{2}^{2} - x_{1}^{2})$
$ω = \sqrt{\frac{V_{1}^{2} - V_{2}^{2}}{x_{2}^{2} - x_{1}^{2}}} ∴ T = 2 π \sqrt{\frac{x_{2}^{2} - x_{1}^{2}}{V_{1}^{2} - V_{2}^{2}}}$

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