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The displacement of a particle represent...

The displacement of a particle represented by the equation `y= 3 cos((pi)/(4)-2 omega t)`. The motion of the particle is

A

simple harmonic with period `(2pi)/(omega)`

B

simple harmonic with period `(pi)/(omega)`

C

periodic but not simple harmonic

D

non-periodic

Text Solution

Verified by Experts

The correct Answer is:
B
The displacement of motion of particle
`y= 3 cos ((pi)/(4)- 2omega t)`
`therefore v= (d)/(dt)[3 cos ((pi)/(4)- 2omega t)]`
`= 3(-2omega) [-sin((pi)/(4)- 2omega t)]`
`therefore v = 6 omega sin ((pi)/(4)-2omega t)`
Acceleration, `a= (dv)/(dt)= (d)/(dt) [6omega sin ((pi)/(4)-2omega t)]`
`therefore a = 6omega xx (-2omega)cos ((pi)/(4)- 2omega t)`
`= -12omega^(2) cos ((pi)/(4)- 2omega t)`
`= -4omega^(2) [3cos ((pi)/(4)- 2omega t)]`
`therefore a= -4omega^(2)y" ""........"(1)`
`therefore a propto -y`
This is condition of simple harmonic motion, the motion is SHM and
`omega.= 2 omega" "[therefore " Comparing equation (1) with "a= -(omega.)^(2)y]`
`therefore (2pi)/(T.)= 2omega`
`therefore T.= (2pi)/(2omega)= (pi)/(omega)`
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