If two `S.H.M.'s` are represented by equation `y_(1) = 10 "sin" [3pit+(pi)/(4)]` and `y_(2) = 5[sin(3pit)+sqrt(3)cos(3pit)]` then find the ratio of their amplitudes and phase difference in between them.
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As `y_(2) = 5[sin(3pit)+sqrt(3)cos(3pit)]`……(1) So if `5 = A cosphi` and `5sqrt(3) = A sinphi` Then `A = sqrt(5^(2) + (5sqrt(3))^(2)) = 10` and `tanphi = (5sqrt(3))/(5) = sqrt(3)` so `phi = (pi)/(3)` The above equation `(i)` becomes `y_(2) = Acosphi sin(3pit) + Asinphi cos(3pit)` `rArr y_(2) = Asin(3pit + phi) = 10 sin[3pit + ((pi)/(3))]` so, `(A_(1))/(A_(2)) = (10)/(10)` `rArr A_(1) : A_(2) = 1 : 1`, Phase difference `= (pi)/(4) - (pi)/(3) = -(pi)/(12)`
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