Question

Two waves passing through a region are represented by
$y_1=5mm\sin \left[\right(2\pi c{m}^{-1})x-(50\pi s^{-1}\left)t\right]$
and $y_2=10mm\sin \left[\right(\pi c{m}^{-1})x-(100\pi s^{-1}\left)t\right]$
Find the vertical displacement of a particle at x = 1 cm at time t = 5.0 ms

• A. 5 mm
• B. 6 mm
  
• C. 7 mm
  
• D. 8 mm
  

Answer 1

The correct option is A 5 mm

According to the principle of superposition each wave produces its disturbance independent of the other and he resultant disturbance is equal to the vector sum of the individual disturbance The displacements of the particle at x = 1 cm at time t = 5.0 ms due to the two waves are
$y_1=5mm\sin \left[\right(2\pi c{m}^{-1})x-(50\pi s^{-1}\left)t\right]$
$y_1=5mm\sin \left[\right(2\pi c{m}^{-1})\times 1cm-(50\pi s^{-1})5\times {10}^{-3}sec]$
$=5mm\sin [2\pi -\frac{\pi }{4}]=-5mm$
and $y_2=10mm\sin \left[\right(\pi c{m}^{-1})x-(100\pi s^{-1}\left)t\right]$
$y_2=10mm\sin \left[\right(\pi c{m}^{-1})\times 1cm-(100\pi s^{-1})5\times {10}^{-3}sec]$
$=10mm\sin [\pi -\frac{\pi }{2}]=10mm$
The net displacement is $y=y_1+y_2=10mm-5mm=5mm$