First slide

Simple harmonic motion

Question

The displacement of a particle is represented by the equation $y=3cos [\frac{π}{4} -2ωt]$ . The motion of the particle is

Moderate

A

Simple harmonic with period $2 π/w$
B

Simple harmonic with period $π/w$
C

Periodic but not simple harmonic
D

Non periodic

Solution

Given displacement of simple harmonic oscillator is $y = 3 \cos [π/4-2ωt]$
to find Velocity differentiate above equation with respect to time t,
then $V= \frac{dy}{dt} = \frac{d}{dt} [3cos [π/4-2ωt]]$

$V=6ωsin [π/4-2ωt]$
Acceleration $a= \frac{dV}{dt} = \frac{d}{dt} [6ωsin (π/4-2ωt)]$
$=6ω× (-2ω) cos (π/4-2ωt)$
$= - 12 ω^{2} cos (π/4-2ωt)$
$= - 4 ω^{2} [3cos [π/4-2ωt]]$
${a=-4ω}^{2} y$
$a α -y$ .
Hence, due to negative sign motion is SHM.
Clearly, from the equation
$ω^{1} =2ω$ [standard equation y=acos( $ω$ t+ $φ$ )]
$\frac{2π}{T^{1}} =2ω$
$T^{1} = \frac{π}{ω}$ . So motion is SHM with period $π/ω$ .

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