First slide

Simple hormonic motion

Question

A particle executing SHM while moving from one extremity is found at distance $x_{1}, x_{2} and x_{3}$ from the centre at the end of three successive seconds. The time period of oscillation is (where $θ = \cos^{- 1} (\frac{x_{1} + x_{3}}{2 x_{2}})$ )

Moderate

A

$\frac{2 π}{θ}$
B

$\frac{π}{θ}$
C

$θ$
D

$\frac{π}{2 θ}$

Solution

Displacement-time equation of the particle will be
$X = A \cos ωt$
Given that $x_{1} = A \cos ω$ and $x_{2} = A \cos 2 ω$
Now, $\frac{x_{1} + x_{3}}{2 x_{2}} = \frac{A (\cos θ + \cos 3 ω)}{2 Acos 2 ω}$
= $\frac{2 A \cos 2 ω \cos ω}{2 A \cos 2 ω}$
= $\cos ω$
$ω = \cos^{- 1} (\frac{x_{1} + x_{3}}{2 x_{2}}) = \frac{2 π}{T}$
or $T = \frac{2 π}{θ} where θ = \cos^{- 1} (\frac{x_{1} + x_{3}}{2 x_{2}})$

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