Question

A uniform rope of length \(12~m\) and mass \(6~kg\) hangs vertically from a rigid support. A block of mass \(2~kg\) is attached to the free end of the rope. A transverse pulse of wavelengths \(0.06~m\) is produced at the lower end of the rope. What is the wavelength (in \(m\)) of the pulse when it reaches the top of the rope?

Answer 1

Given, length of the rope \(=12~m\),
Mass of the rope \(=6~kg\)
Mass of the block \(=2~kg\)
Wavelength of the produced transverse pulse \(0.06~m\)
Tension in the string at lowest point \(A\)
\(T_{A}=2g=20~N\)
Tension in the string at highest point \(B\)
\(T_{B}=(2+6)g=80~N\)
Speed of transverse wave \(v=\sqrt{{T}/{\mu}}\)
\(v~\propto~\sqrt{T}\)
\(v=n\lambda\), \(n~\colon\) frequency is independent of medium
\(v~\propto~\sqrt{T}~\Rightarrow~\lambda~\propto~\sqrt{T}\)

\(\dfrac{\lambda_{B}}{\lambda_{A}}=\sqrt{\dfrac{T_{B}}{T_{A}}}\)
\(\dfrac{\lambda_{B}}{0.06}=\sqrt{\dfrac{80}{20}}~\Rightarrow~\lambda_{B}=0.12~m\)
Final answer: \((0.12)\)