Question

A rope of length l and mass m hangs freely from the ceiling the velocity of transfer to transverse wave as a function of position x along the rope is proportional to

Answer 1

A rope of length l and mass m hangs freely from the ceiling the

proportional to velocity of transfer to transverse wave as a function of

position x along the rope is proportional  to ..

a rope of length l  and mass  m hangs freely from the ceiling.

l=length ,m=mass

the velocity of transverse waves on rope is

v=√T/μ   ...1

now rope is hanging ,tension of string depend upon the position  x along rope

T=μgx  

put values in of T in  equation  (1) so

v=√T/μ=√μgx/μ=√gx

hence answer of transverse of velocity is v=√gx

Answer 2

we know, if T is tension experienced by sting of length L , $\mu$ linear mass density {linear mass density means mass per unit length } then, velocity of wave propagated in string is given by, $v=\sqrt{\frac{T}{\mu}}$

here, we have to find velocity of wave as a function of position x along the rope.
length of rope from lowest point = x
mass of rope of length x , $\mu x$

so, tension experienced by string of length x = $\mu gx$

now, velocity of transverse wave , v = $\sqrt{\frac{\mu gx}{\mu}}$ = $\sqrt{gx}$

hence, velocity of transfer to transverse wave as a function of position x along the rope is proportional to square root of x. e.g., v = √gx