Solutions to Week 11 Assignment

 

  1. From a proportion with the intensities treating the jet airplane as an isotropic source.
      Ij/IE = (P/4prj2)/(P/4prE2)  = rE2/rj2
         Ij  = (rE/rj)2IE = (1/5.2)2(1400W/m2) = 52W/m2.
  2. (a) v = Dx/Dt = (1.80m - 1.50m)/0.20s = 1.5m/s
         x = x0 + vDt = 1.80m + (1.5m/s)(3.00s - 0.20s) = 6.0m.
    (b) t = (x - x0)/v + t0 = (4.00m - 1.80m)/(1.5m/s) + 0.02s = 1.7s.
  3. f = v/l = (120m/s)/(30.0 x 10-2m) = 400Hz.
  4. A = 0.120m, l = 0.300m, v = 6.40m/s, and y(x, t) = Asin(w t + kx)
    w =2pv/l = 2p(6.40m/s)/(0.300m) = 134s-1
    k = 2p/l = 2p/(0.300m) = 20.9m-1
    So,  y(x, t) = (0.120m)sin[(134s-1)t + (20.9m-1)x]
  5. (a) ymax = 2.6cm, so A = 2.6cm.
    (b) l = Dx = 16m - 2m = 14m.
    (c) v = Dx/Dt = (7.5m - 5.5m)/(0.10s) = 20m/s.
    (d) f = v/l = (2.0 x10-1m/s)/(14m) = 1.4Hz.
    (e) T = 1/f = 1/(1.4Hz) = 0.70s.
  6. Let y1 = Asin(w t + kx) and y2 = Asin(w t + kx - f) and use the trigonometric identy
    sina + sinb = 2sin[(a + b)/2]cos[(a - b)//2]. Use the principle of superposition.
    y = y1+ y2= Asin(w t + kx) + Asin(w t + kx - f) = 2Asin(w t + kx - f/2)cos(f//2) = A'sin(w t + kx - f/2)
    where A' = 2Acos(f//2) = A.
    Find f
    2Acos(f//2) = A
       2cos(f//2) = 1
         cos(f//2) = 1/2
                 f//2 = cos-1(1/2)
                    f/ = 120°
  7. f = v/l and the frequency is the same in both mediums.
    va/la  = vw/lw
         lw = (vw/va)la
              =0.750(0.500 x 10-6m) = 375nm.
  8. Intensity is proportional to the amplitude squared. Find A1/A2.
     A1/A2 = Ö(I1/I2) = Ö(25/15) = Ö(5.0/3.0)
    For constructive interference, the resultant amplitude is the sum of the original amplitudes.
     A = A1 + A2 = A2Ö(5.0/3.0) + A2 = A2[1 + Ö(5.0/3.0)]
    Ö(I/I2) = A/A2 = 1 + Ö(5.0/3.0)
    I = [1 + Ö(5.0/3.0)]2I2 = [1 + Ö(5.0/3.0)]2(15mw/m2) = 79mW/m2.
  9. (a)  fn = nv/2L and v = ÖT/m
           f1 = v/2L = 1/2LÖT/m = 1/2(1.5m)Ö12N/1.2 x10-3kg/m = 33Hz.
    (b)  f3 = 3v/2L = 3/2LÖT/m  
              4L2f32/9 = T/m
                         T = 4mL2f32/9
                            = 4(1.2 x 10-3kg/m)(1.5m)2(0.50 x10Hz)2/9
                            = 300N.