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A uniform rope of mass m and length L ha...

A uniform rope of mass `m` and length `L` hangs from a celling. (a) Show that the speed of a transverse wave on the rope is a function of `y`, the distance from the lower end, and is given by `t = 2sqrt(L//g)`.

Text Solution

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(a) As mas per unit length
`mu = (m)/(l) :. V = sqrt((gamma)/(mu)) :.` Tension at `P = muyg`
`:. V = sqrt((muyg)/(mu)) = sqrt(yg)`
(b) Now `(dy)/(dt) = sqrt(yg)`
`overset(l)underset(O)(int)(dy)/(sqrt(y)) = sqrt(g)overset(t)underset(O)(int)dt , t = 2sqrt(l//g)`
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