If x, and a denote the displacement, the velocity and the acceler 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+2piupsilon`
C. `aT//upsilon`
D. `a^(2)T^(2)+4pi^(2)upsilon^(2)`

Answer

A

Explanation

`x=rsiniomegat, upsilon=(dx)/(dt)=romegacosomegat,`
`a=(dupsilon)/(dt)=-romega^(2)sinomegat =-omega^(2)x.` So,
`(aT)/(x)=(-omega^(2)xxxT)/(x)=-omega^(2)T=-(4pi^(2))/(T^(2))xxT`
`=-(4pi^(2))/(T)=a ` constant
`(aT)/(upsilon)=(-omega^(2)rsinomegat xxT)/(omegar cos omegat )=-omegaTtanomegat`
`=(2pi)/(T)xxTtanomegat =(`not constant )
`aT+2piupsilon` is not constant.
`a^(2)T^(2)+4pi^(2)upsilon^(2)` is also not constant.