Which of the following represent simple harmonic motion?
(i) `x=A sin omegat+Bcos omegat` (ii) `x=Asin omegat+Bcos2omegat` (iii) `x=Ae^(iomegat)` (iv) `x=Alnomegat`

Explanation

(i) `x=Asinomegat+Bcosomegat`
`(dx)/(dt)=Aomegacosomegat-Bomegasinomegat`
`(d^(2)x)/(dt^(2))=-omega^(2)(Asinomegat+Bcosomegat)`
`(d^(2)x)/(dt^(2))=omega^(2)x`
This differential equation is similar to the differential equation of SHM. Therefore, `x=Asinomegat+Bcosomegat` represents SHM.
(ii) `x=Asinomegat+Bcos2omegat`
`(dx)/(dt)=Aomegacosomegat-B(2omegat)sin2omegat`
`(d^(2)x)/(dt^(2))=-omega^(2)(2sinomegat+4Bcos2omegat)`
`(d^(2)x)/(dt^(2))ne-omega^(2)x`
This differential equation is not like the differential equation of a SHM. Therefore, `x=Asinomegat+Bcos2omegat` does not represent SHM.
(iii) `x=Ae^(iomegat)`
`(dx)/(dt)=Aomegat^(iomegat)`
`(d^(2)x)/(dt^(2))=-Aomegat^(2)e^(iomegat)=-omega^(2)x`
This differential equation is like the differential equation of SHM. Therefore, `x=Ae^(iomegat)` represents SHM.
(iv) `x=Alnomegat`
`(dx)/(dt)=((A)/(omegat))omega=(A)/(t)`
`(d^(2)x)/(dt^(2))=-(A)/(t^(2))implies(d^(2)x)/(dt^(2))ne-omega^(2)x`
This differential equation is not like the differential equation of a SHM. Therefore, `x=Alnomegat` does not represent SHM.