Question

Two simple harmonic motions are represented by the following equations
`y_1=10sinpi/4(12t+1)`
`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=10sinpi/4(12t+1)=10sin(3pit+pi//4)`
`=a_1sin(omega_1t+pi//4)` (i)
`y_2=5(sin 3pit+sqrt3cos3pit)`
`=5xx2(1/2sin3pit+sqrt3/2cos3pit)`
`=10(cospi/3sin3pit+sinpi/3cos3pit)`
`=10sin(3pit+pi//3)=a_2sin(omega_2t+pi//3)` (ii)
Ratio of amplitudes: `a_1/a_2=10/10=1`
Ratio of time periods: `T_1/T_2=(2pi//omega_1)/(2pi//omega_2)=(omega_2)/(omega_1)=(3pi)/(3pi)=1`