Directions: In the following question more than one of the answers given may be correct. Select the correct answer and mark it according to the code:
A wave disturbance in a medium is described by\[y(x,t)=0.02\,\cos \left( 50\pi t+\frac{\pi }{2} \right)\cos (10\pi x),\]where\[x\]and y are in metre and t is in second? (1) A node occurs at\[x=0.15\text{ }m\] (2) An antinode occurs at\[x=0.3\text{ }m\] (3) The speed of wave is\[15\text{ }m{{s}^{-1}}\] (4) The wavelength is 0.2 mA) 1 and 2 are correct
B) 2 and 3 are correct
C) 1 and 4 and correct
D) 1, 2 and 4 are correct
Correct Answer: D
Solution :
In the given case, \[y(x,t)=0.02\,\cos \left( 50\pi t+\frac{\pi }{2} \right)\cos (10\pi x)\] \[\cong A\cos \left( \omega t+\frac{\pi }{2} \right)\cos kx\] Node occurs when \[kx=\frac{\pi }{2},\frac{3\pi }{2},\]etc., \[10\pi x=\frac{\pi }{2},\frac{3\pi }{2}\] \[\Rightarrow \] \[x=0.05\,m,0.15\,m\] Antinode occurs when \[kx=\pi ,3\pi \]etc., \[\Rightarrow \] \[10\pi x=\pi ,3\pi \] \[x=0.1m,\,0.3\,m\] Speed of wave is given by \[v=\frac{\omega }{k}=\frac{50\pi }{10\pi }=5\,m{{s}^{-1}}\] Wavelength is given by \[\lambda =\frac{2\pi }{k}=\frac{2\pi }{10\pi }=\frac{1}{5}\,m=0.2\,m\]
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