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

Answer

A

Explanation

`x = A sin omegat`
`v Aomega cos omegat`
`a = omega^(2)A sin omegat`
`a^(2)T^(2) + 4pi^(2)v^(2) = A^(2)omega^(4) sin^(2) omegat((2pi)/(omega))^(2) + 4pi^(2)A^(2)omega^(2)cos^(2)omegat` ,
` = A^(2)4pi^(2) omega^(2) sin^(2)omegat + A^(2)omega^(2)4pi^(2) cos^(2)omegat`
`= A^(2)omega^(2)4pi^(2)` which can change with time.
`(aT)/(x) = (-omega^(2)x + T)/(x)`
` = -(4pi^(2))/(T)` (A is constant)