First slide

Simple hormonic motion

Question

A particle in S.H.M. is described by the displacement function x(t) = a cos ( $ωt + θ$ ).If the initial (t = 0) position of the particle is I cm and its initial velocity is $π cm / s$ . The angular frequency of the particle is $π rad / s$ , then its amplitude is

Moderate

A

1 cm
B

$\sqrt{2} cm$
C

2 cm
D

2.5 cm

Solution

$x = a \cos (ωt + θ)$ ------------(i)
And $v = \frac{dx}{dt} = - aω \sin (ωt + θ)$ --------(ii)
Given at t = 0, x = 1 cm and v = $π$ and $ω = π$
Putting these values in equation (i) and (ii), we will get
$\sin θ = \frac{- 1}{a} and \cos θ = \frac{1}{A}$
$\Rightarrow \sin^{2} θ + \cos^{2} θ = {(- \frac{1}{a})}^{2} + {(\frac{1}{a})}^{2} \Rightarrow a = \sqrt{2} cm$

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