question_answer

A stretched string is vibrating according to the equation \[y=5\,\sin \,\left( \frac{\pi x}{2} \right)\cos \,4\pi t\] where \[y\] and \[x\] are in cm and t in second. The distance between two consecutive nodes on the string is:

A)  2 cm         

B)  4 cm

C)  8 cm         

D)  16 cm

Correct Answer: A

Solution :

 Key Idea: Distance between consecutive nodes is half of wavelength. The standard equation of wave is \[y=a\,\sin \,kx\,\cos \omega \,t\] ... (i) where a is amplitude and co is angular velocity. Given equation is \[y=5\sin \left( \frac{\pi x}{2} \right)\cos 4\pi t\] ... (ii) Comparing Eq. (i) with Eq. (ii), we get \[k=\frac{2\,\pi }{\lambda }=\frac{\pi }{2}\] \[\Rightarrow \] \[\lambda =4\,cm\] Hence, distance between two consecutive nodes is N = Node \[=\frac{\lambda }{2}=\frac{4}{2}=2\,cm\]