## Question

A uniform rope of mass 0.1 kg and length 2.45 m hangs from a ceiling. (*a*) Find the speed of transverse wave in the rope at a point 0.5 *m* distant from the lower end, (*b*) Calculate the time taken by a transverse wave to travel the full length of the rope (*g* = 9.8 m/s^{2})

### Solution

2.21 m/s & 1 sec

(a) As the string has mass and it is suspended vertically, tension in it will be different at different points. For a point at a distance *x* from the free end, tension will be due to the weight of the string below it. So if *M* is the mass of string of length *L*, the mass of length *x* of the string will be (*M*/*L*)*x*.

Here *x* = 0.5 m

(b) From part (a) it is clear that the tension and so the velocity of the wave is different at different points. So if at point *x* the wave travels a distance*dx* in time *dt*,

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