The motion of a particle is given by `x=A sin omegat+Bcos omegat`. The motion of the particle is

Options

A. not simple harmonic
B. simple harmonic with amplitude A + B
C. simple harmonic with amplitude `((A + B))/(2)`
D. simple harmonic with amplitude `sqrt(A^(2)+B^(2))`

Answer

D

Explanation

`x = A sin omega t + B cos omega t`

`= sqrt(A^(2)+B^(2)) [(A)/(sqrt(A^(2)+B^(2)))sin omega t + (B)/(sqrt(A^(2)+B^(2)))cos omega t]`
`= sqrt(A^(2)+B^(2))sin (omega t + phi)" "["where, tan" phi = (B)/(A)]`
Thus, the motion is SHM with amplitude `sqrt(A^(2) + B^(2))`.