Out of the following function reporesenting motion of a particle which represent `SHM`
(i) ` y = sin omega t - cos omega t `
(ii) `y = sin^(3) omega t`
(iii)`y =5cos ((3pi)/(4) - 3omega t)`
(iv) ` y = 1+ omega t + omega^(2) t^(2) `

Options

A. Only `(1)`
B. Only `(4)` does not represent SHM
C. Only `(1)` and `(3)`
D. Only `(1)` and `(2)`

Answer

C

Explanation

`(1) y=sin omegat-cos omegat`
`=sqrt(2)[(1)/(sqrt(2))sin omegat-(1)/(sqrt(2)) cos omegat ]`
`=sqrt(2)[cos ((pi)/(4))sin omegat-sin ((pi)/(4))cos omegat ]`
`=sqrt(2)sin (omegat-(pi)/(4))`
It represents a SHM with the period `T=(2pi)/(omega)`
(2) `y=sin^(3)omegat =(1)/(4)[3 sin omegat-sin 3omegat ]`
It represents periodic motion with time period `T=(2pi)/(omega)` but not SHM as it does not involve single harmonic function.
`(3)` `y=5cos ((3pi)/(4)-3omegat)=5cos(3omegat-(3pi)/(4))`
`[:' cos (-theta)=costheta]`
It represents SHM with time period, `T=(2pi)/(3omega)`
`(4)` `y=1+omegat+omega^(2)t^(2)`
velocity, `upsilon=(dy)/(dt)=omega=2omega^(2)t`
Acceleration, `a=(dupsilon)/(dt)=2omega^(2)`
It represents a non-periodic motion, where y increases with increase in t .
Thus (1) and (3) represent SHMs.