The displacement of a particle is repersented by the equation `y=3cos((pi)/(4)-2omegat)`. The motion of the particle is
(b)
(c) (d)

Options

A. non-periodic
B. periodic but not simple harmonic
C. simple harmonic with period `2pi//omega`
D. simple harmonic with period `pi//omega`

Answer

B

Explanation

(b) Given , equation of motion is
`y="sin"^(3)omegat=(3 "sin"omegat-4"sin"omegat)//4" "[because "sin"3 theta =3 "sin"theta-4"sin"^(3)theta]`
`implies (dy)/(dt)=[(d)/(dt)(3"sin"omegat)-(d)/(dt)(4 "sin"omegat)]//4`
`4(dy)/(dt)=3omega"cos"omegat-4xx[3omega"cos"3 omegat]`
`implies4xx(d^(2)y)/(dt^(2))=-3omega^(2)"sin"omegat+12 omega^(2)"sin"omegat`
`implies(d^(2)y)/(dt^(2))=-(3omega^(2)"sin"omegat+12omega^(2)"sin"3omegat)/(4)`
`implies(d^(2)y)/(dt^(2))` is not proportional to y.
Hence, motion is not SHM.
As the expression is involing sine function ,hence it will be periodic.