A uniform rope of length l and mass M hangs vertically from a rigid support. A block of mass m is attached to the free end of the rope. A transverse pulse of wavelength `lambda` is produced at the lower end of the rope. The wavelength of the pulse, when it reaches the top of the rope, is:

Options

A. `lambda sqrt((M-m)/m)`
B. `lambda(M+m)/m`
C. `lambda sqrt(m/(M+m))`
D. `lambda sqrt((M+m)/m)`

Answer

D

Explanation

Using `v = sqrt(T/m)`
If `v_(1)` is velocity at top and vb is velocity at bottom then
`v_(t)/v_(b) = sqrt(T_(t)/T_(b)) = sqrt(((M +m)g)/(mg)) = sqrt((M+m)/m)`
but `v_(t) = v_(t)lambda_(t)` and `v_(b) = v_(b)lambda_(b), therefore (v_(t)lambda_(t))/(v_(b)lambda_(b)) = sqrt((M+m)/m)`, since `v_(t) = v_(b)` we get
`lambda_(t) = lambdasqrt((M+m)/m)`