The ratio of the amplitudes of the simple harmonic oscillations given by `y_1=Asinomegat` and

Options

A. `1`
B. `2`
C. `1//sqrt2`
D. `sqrt2`

Answer

D

Explanation

`y_1=Asinomegat` (i)
`y_2=A/2sinomegat+A/2cosomegat`
`=A/2sinomegat+A/2sin(pi/2+omegat)`
`=A^'sin(omegat+theta)`

`A^'=sqrt2.A/2=A/sqrt2`
`theta=45^@=pi//4`
`y_2=A/sqrt2sin(omegat+pi//4)`
`A/A^'=(A)/(A//sqrt2)=sqrt2`
OR
`y_2=A/2sinomegat+A/2cosomegat=A/2(sinomegat+cosomegat)`
`=A/2sqrt2(1/sqrt2sinomegat+1/sqrt2cosomegat)`
`=A/2sqrt2(sinomegatcospi//4+cosomegatsinpi//4)`
`=A/sqrt2sin(omegat+pi//4)`
Note: `a sin theta+b cos theta` (can be converted into a single term as)
`=sqrt(a^2+b^2)((a)/(sqrt(a^2+b^2))sintheta+(b)/(sqrt(a^2+b^2))costheta)`

`=sqrt(a^2+b^2)(sinthetacostheta+costhetasinalpha)`
`=sqrt(a^2+b^2)sin(theta+alpha)`