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

Answer

C, D

Explanation

`a = -omega^(2)x, v = omegasqrt(A^(2)-x^(2))`
`a^(2)T^(2) + 4pi^(2)v^(2) = omega^(2)x^(2)T^(2) + 4pi^(2)omega^(2)(A^(2) - x^(2))`
`= (4pi^(2))/(T^(2)) omega^(2)x^(2)T^(2)+ 4pi^(2)omega^(2)(A^(2) - x^(2))`
`= 4pi^(2)omega^(2)A^(2) = "constant"`
`(aT)/(x) = -(omega^(2)xT)/(x) = -omega^(2)T = "constant"`