Question

Two simple harmonic motions are represented by the following equations
`y_1=10sin(3pit+pi/4)`
`y_2=5(sin 3pit+sqrt3cos3pit)`
Find out the ratio of their amplitudes. What are the time periods of two motions?

Answer 1

`y_(1)=10sin((pi)/(4))(12t+1)=10sin(3pit+(pi)/(4))` …(i)
`y^(2)=5[sin3pit+sqrt(3)cos 3pit]`
`=5xx2[sin3pitxx(1)/(2)+(sqrt(3))/(2)cos3pit]`
`=10[sin3pit cos ((pi)/(3))+sin((pi)/(3))cos3pit]`
`=10[sin(3pit+pi//3)]` …(2)
The general equation of SHM is
`y=Asin[omega t + phi_(0)]=A sin[(2pi)/(T)t+phi_(0)]` ...(iii)
Comparing equation (i) and (ii) with equation (iii), we get
`A_(1)=10, A_(2)=10, (2pi)/(T_(1))=3pi=(2pi)/(T_(2))`
`:. (A_(1))/(A_(2))=(10)/(10)=1:1` and ` T_(1)=(2pi)/(3pi)=(2)/(3)s=T_(2)`