question_answer

A thick uniform rope of length L is hanging from a rigid support. A transverse wave of wavelength \[{{\lambda }_{0}}\] is set up at the middle of rope as shown in figure. The wavelength of the wave as it reaches to the topmost point is

A) \[2{{\lambda }_{0}}\]               

B)        \[\sqrt{2}{{\lambda }_{0}}\]

C) \[\frac{{{\lambda }_{0}}}{\sqrt{2}}\]                

D)         \[{{\lambda }_{0}}\]

Correct Answer: B

Solution :

[b] Speed of pulse at a distance x from bottom, \[v=\sqrt{gx}.\] While traveling from mid-point to the top, frequency remains unchanged. \[\frac{{{v}_{1}}}{{{\lambda }_{1}}}=\frac{{{v}_{2}}}{{{\lambda }_{2}}}\] \[\Rightarrow \]\[\frac{\sqrt{g\,\,(L\,\,/\,\,2)}}{{{\lambda }_{0}}}=\frac{\sqrt{gL}}{{{\lambda }_{2}}}\] \[\Rightarrow \]\[{{\lambda }_{2}}=\sqrt{2}{{\lambda }_{0}}\]