A) the velocity of the wave is 4 m/s
B) the two consecutive points on the wave having equal and opposite displacement are separated by a distance 2 cm.
C) the phase difference between two points separated by a distance of 0.42 m is\[\pi /2\]
D) the displacement of the point distant 200 cm from the source will be zero at \[t=\frac{201}{400}\]sec.
Correct Answer: C
Solution :
: \[y=5\cos \pi (0.5x-200t)\] \[=5\cos (0.5\pi x-200\pi t)\] \[=5\cos (200\pi t-0.5\pi x)\] Standard equation is\[y=a\cos \left( \frac{2\pi }{\lambda }vt-\frac{2\pi }{\lambda }x \right)\] \[\therefore \] \[a=5cm\] ... (i) \[\frac{2\pi x}{\lambda }=0.5\pi x\] or \[\lambda =4\text{ }cm\] ... (ii) Again\[\frac{2\pi vt}{\lambda }=200\pi t\] Or \[v=\frac{200\times \lambda }{2}=\frac{200\times 4}{2}=400\,cm/s\] or \[v=4\text{ }m/s\] ... (iii) Now consider the options given in the question [a] is correct from (iii) [b] is correct from (ii) where from\[\frac{\lambda }{2}=2\text{ }cm\] [c] is incorrect \[\frac{\phi }{2\pi }=\frac{x}{\lambda }\] or \[\phi =\frac{2\pi x}{\lambda }=\frac{2\pi \times 0.42\times 100}{4}=21\pi \] [d] is correct \[y=5\cos \pi t(0.5\times 200-200t)\] \[0=5\cos \pi (100-200t)\] \[\cos \frac{\pi }{2}=\cos \pi (100-200t)=\cos \pi (200t-100)\] Or \[\frac{\pi }{2}=\pi (-100+200t)\]or\[\frac{1}{2}=-100+200t\] Or \[200t=\frac{201}{2}\] or \[t=\frac{201}{400}\]sec This is correct.
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