A rope hangs from a rigid support A pulse is set by jiggling the bottom end we want to design a rope in which velocity v of pulse is independent of z the distance of the pulse from fixed end of the rope if the rope is very long the desired function for mass per unit length `mu(z)` in terms of `mu_(0)` (mass per unit length of the top (z=0) g v and z is:

Options

A. `mu(z)=mu_(0)e^(-[g//v^(2)]z`
B. `mu(z)=mu_(0)e^(+[g//v^(2)]z`
C. `mu(z)=mu_(0)log_(e)(g/v^(2))z`
D. `mu(z)=mu_(0)e+(v^(2)/g)z`

Answer

A

Explanation

`SigmaF_(z)=0`
`(T+dT)+mugdz-T=0`
`dT=-mugdz`
`also T=muv^(2)`
`dT=dmuv^(2)+2vdv dmu`
as v is independent of z
`dv=0`
`dT=v^(2)dmu`
from equation (1) and (2) we get
`mu int(dmu)/(mu)=-g/v^(2) underset(0)overset(z) intdz`
or `mu=mu_(0)e^-(g//v^(2))z`