First slide

Simple harmonic motion

Question

The displacement of a particle is represented by the equation $y=3cos [\frac{π}{4} -2wt]$ . 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 $y = 3 \cos [π/4-2wt]$
Velocity $V= \frac{dy}{dt} = \frac{d}{dt} [3cos [π/4-2wt]]$
$= 3(-2w) [-sin [π/4-2wt]]$
$V=6wsin [π/4-2wt]$
Acceleration $a= \frac{dV}{dt} = \frac{d}{dt} [6wsin (π/4-2wt)]$
$=6w× (-2w) cos (π/4-2wt)$
$= - 12 w^{2} cos (π/4-2wt)$
$= - 4 w^{2} [3cos [π/4-2wt]]$
${a=-4w}^{2} y$
$a α -y$ .
Hence, due to negative sign motion is SHM.
Clearly, from the equation
$w^{1} =2w$ [standard equation y=acos(wt+ $φ$ )]
$\frac{2π}{T^{1}} =2w$
$T^{1} = \frac{π}{w}$ . So motion is SHM with period $π/w$ .

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