Question

A uniform string of length L is suspended from a point on the ceiling. A small wave pulse is now introduced at its lowest end and it starts moving up towards the ceiling.
(i) What will be the speed of the wavw pulse whenit has moved up a distance x?
(ii) How much time does this wave pulse take to reach the ceiling?

Answer 1

(i) Let the mass per unit length of the string be μ. Consider a section of the string of length x (x measured from the lower end)

The mass of this section is μx and hence its weight is μx. This weight is balanced by the tension T at the piont x distant x from the lower end.

Hece T=μxg

the wave velocity v is now given by

or v=xg−−√

(ii) as the wave pulse climbs up, x changes from zero to L.

now, V=dxdt or dxx−−√=g√dt

Integrating, we get

∫L0dxx−−√=g√∫t0dt⇒2x−−√∫L0=g√t

or 2L−−√=g√t or t=2Lg−−√.