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

Answer

A

Explanation

We know, instantaneous displacement, `x= r sin omega t`
`because` Instantaneous velocity, `v=(dx)/(dt)= r omega cos omega t`
`because` Instantaneous acceleration , `a= (dv)/(dt)= - r omega^(2) sin omega t= -omega^(2)x`
So, `(aT)/(x)= (-omega^(2)x xx T)/(x)= - omega^(2)T= (-4pi^(2))/(T^(2)) xx T= (-4pi^(2))/(T)`= constant
`rArr (aT)/(V)= (-omega^(2) r sin omega t xx T)/(omega r cos omega t)= -omega T tan omega T= -(2pi)/(T)xx T tan omega t`= not constant
Similarly, `aT+ 2pi v and a^(2)T^(2) + 4pi^(2)v^(2)` is also not constant, i.e. both are function of t.