Which of the following functions represent SHM :
`sin omega t+ 2 cos omega t`

Explanation

(a) ` x = A sin omega t + B cos 2 omega t`
`(dx)/(dt) = A omega cos omega t - B (2 omega) sin 2 omega t `
` (d^2 x)/(dt^2) = - omega^2 (A sin omega t + 4B cos 2 omega t )`
`(d^2 x)/(dt^2) ne - omega^2 x `
. This differential equation is not like the differential equation of a SH M. There for `x = A sin omega t + B cos 2 omega t ` does not represent SHM
(b) ` x = Ae^(i omega t) `
`(dx)/(dt) = A omega e^(i omega t)`
`(d^2 x)/(dt^2) = - A omega^2 e^(i omega t) = - omega^2 x `
This differential equation is like the differential equation of SHM. Therefore, `x = A e^(i omega t) `represents SHM.
(iv) `x = A ln omega t`
` (dx)/(dt) = ( (A)/(omega t) ) omega = A/t`
`(d^2x)/(dt^2) = ( (A)/(omega t) ) omega = A/t`
`(d^2 x)/(dt^2) = - (A)/(t^2) rArr (d^2x)/(dt^2) ne - omega^2 x `
This differential equation is not like the differential equation of a SHM. Therefore, x = `A ln omega t ` does not represent SHM