If x, v and a denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period T, then, which of the following does not change with time?

Options

A. `(aT)/(x)`
B. `aT + 2piv`
C. `(aT)/(v)`
D. `a^(2)T^(2) + 4pi^(2)v^(2)`

Answer

A, D

Explanation

`x = A sin omegat`
`v = Aomega cos omegat`
`a = -Aomega^(2) sin omegat`
(1) `(aT)/(x) = ((-Aomega^(2) sinomegat)/(A sinomegat))T = -omega^(2)T = -2piomega`
(2) `aT + 2piv = -Aomega^(2)T sin omegat + 2pi. Aomega cosomegat`
`(aT)/(v) = (-Aomega^(2)sinomegat xx T)/(Aomega cos omegat)`
`a^(2)T^(2) + 4pi^(2)v^(2) = + A^(2)omega^(4)T^(2) sin^(2)omegat + 4pi^(2)A^(2)omega^(2) cos^(2)omegat = + A^(2)omega^(4) ((2pi)/(omega))^(2) sin^(2)omegat + 4pi^(2)A^(2)omega^(2) cos^(2)omegat = 4pi^(2)A^(2)omega^(2)`