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Attwo points S_(1) and S_(2) on a liquid surface two coherent wave sources are set in motion at t = 0 with the same phase. The speed of the waves in the liquid v = 0.5 m/s, the frequency of vibration eta = 5 Hz and the amplitude A = 0.04 m. At a point P of the liquid surface which is at a distance x_(1) = 0.30m from S_(1) and x_(2) = 0.34 m from S_(2) a piece of cork floats: (a) Find the displacement ofthe cork at t = 3 s. (b) Find the time t_(0)that elapse from the moment the wave sources were set in motion until the moment that the cork passes through the equilibrium position for the first time. |
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Answer» `Delta = 0.04m` wavelength `lambda = (v)/(n) = (0.5)/(5)` = 0.1m = `2pi xx 0.04 = (4pi)/(5)` The waves from `S_(1)` and `S_(2)` arrive at point P at DIFFERENT time `t_(1)` and `t_(2)` given as = `t_(1) = (0.3)/(0.5)` = 0.6s and `t_(2) = (0.34)/(0.5)` = 0.68s equation of motion of cork it `t_(0) = 3s` is `y = y_(1) + y_(2)` = `Asin(omega + t_(0) - t) Asin(omega+(t_(0) - t_(2)))` `A[sin(10pi(2.4)) + Asin(omega(t_(0) - t_(2)))]` = `0.04 xx sin (23.2pi)` = - 0.02344m. If we consider t = 0 the time when cork starts it motion then the resulting oscillation at cork is given as `y = A sin omegat + A sin(omega -(4pi)/(5))` = `Rsin(omegat -THETA)` `theta = tan^(-1) ((Asin(4pi//5))/(A + Acos(PI//5)))` = `70^(@) = (2pi)/(5)` rad Here initial phaseof `70^(@)` `(or (2pi)/(5) rad` will be there in cork motion when SECOND wave arrives at it. Thus time after which cork will pass through mean position is given as ` t = 0.68 + pi - (2pi)/((5)/(omega))` = `0.68 + (3pi)/((5)/(10pi))` = 0.68 + 0.06 = 0.74s. |
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