In the figure shown `A` and `B` are two ends of a string of length `100m`. `S_(1)` and `S_(2)` are two sources due to which point `'A'` and `'B'` oscillate in `'y'` and `'z'` directions respectively according to the equation `y = 2 sin (100 pi t + 30^(@))` and `z = 3 sin (100 pi t + 60^(@))` where `t` is in `sec` and `y` is in `mm`. The speed of propagation of disturbance along the string is `50 m//s`. Find the instantaneous positions vector (in `mm`) and velocity vector in `(m//s)` of a particle `'P'` of string which is at `25m` from `A`. You have to find these parameters after both the disturbance from `S_(1)` and `S_(2)` have reached `'P'`. Also find the phase difference between the waves at the point `'P'` when they meet at `'P'` first time.

Answer

Phase of difference at time `'t' = 30^(0)` constant al-ways after thy meet at `'P'`

Explanation

`y_(p) = 2 sin (100pit + 30^(@))`
`y = 2 sin [100pit (t - (25)/(50)) + 30^(@)] = 2 sub(100pt - 50p + 30^(0))`
`z = 3 sin [100pi(t - (75)/(50)) + 60^(0)]`
`= 3 sin(100 pit - 150pi + 60^(0)) = 3 sin (100pit + 60^(0))`
`overset(vec)(r_("in mm")) = xhat(i) + yhat(j) + zhat(k)`
`= 25000hat(i) + 2sin(100pi + 30^(0))hat(j) + 3 sin (100pit + 60^(0))hat(k)`
`overset(vec)(v) = (doverset(vec)r)/(dt) = 2 xx 100picos(100pit + 30^(0))hat(i) + 3 xx 10 pi cos(100pithat(j) + 60^(0))hat(k)`
Phase difference at time `'t' = 30^(0)` costant always after they meet at `'P'`.