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

Explanation

As the rope is heavy, its tension will be different at different points. The tension at thefree end will be 2 g and thatat the upper end it will be 8 g.
we have `v = nlambda`
`sqrt((T)/(mu)) = nlambda`
`sqrt((T)/(lambda)) = nsqrt(mu)`
The frequency of thewave pulse will be the same every where on the rope as it depends onlyonthe frequency ofthesource. The mass perunitlength isalso the same throughout the rope as it is uniform. Thus, we have
`As sqrt((T)/(lambda))` is constant, we have
`sqrt(2g)/(0.06m) = sqrt(8g)/(lambda_(1))`
Solving we get `lambda_(1)` = 0.12 m