Question

Two simple harmonic motions are represented by the equations
`y_(1) = 10 sin(3pit + pi//4) and y_(2) = 5(sin 3pit + sqrt(3)cos 3pit)` their amplitude are in the ratio of ………… .

Answer 1

`y_(1) = 10 sin(3 pi t+ pi//4)`….(i)
`y_(2) = 5sin 3 pi + t + 5 sqrt(3) cos t` ….(ii)
In Eq.(ii), let
`5 = a cos theta`…(iii)
and `5 sqrt(3) = a sin theta`….(iv)
`:. y_(2) = a sin 3 pi t cos^(2) theta + a sin theta cos 3 pi t`
`y_(2) = a sin(3 pi t + theta)`....(v)
Squaring and additing Eqs. (iii) and (iv), we get
`5^(2) +(5 sqrt(3))^(2) = a^(2) cos^(2) theta + a^(2) sin^(2) theta`
`rArr 100 = a^(2)`
`rArr a = 10`
Therefore, Eqs.(v) cab be written as
`y_(2) = 10 sin (3 pi t + theta)`....(vi)
From Eqs (i) and (vi), the ratio of amplitudes is `10 : 10`, i.e., `1 : 1`.