Question

A uniform rope of length 10 cm and mass 4 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 wavelength 0.06 m is produced at the lower end of the rope, what is the wavelength of the pulse when it reaches the top of the rope?

Answer 1

As rope is heavy, Tension is different at different points. T at lower end is 2g and at upper end is 6g

$\bf V = v \lambda \\ \\ \bf \sqrt{ \frac{F}{ \mu} } = v \lambda \\ \\ \bf \frac{ \sqrt{F} }{ \lambda} = v \sqrt{ \mu} \\ \\ \bf \frac{ \sqrt{F} }{ \lambda} \: is \: a \: constant \\ \\ \bf\frac{ \sqrt{2g} }{0.00} = \frac{ \sqrt{8g} }{ \lambda_1} \\ \\ \large \boxed{ \sf \orange{ \lambda_1 = 0.12 \: m}}$