A uniform rope of mass m and length L hangs from a ceiling. The speed of transverse waves in the rope at a point at a distance x from the lower end of the rope is

Concept:

• Simple Harmonic Motion (SHM): Simple harmonic motion is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
  • Example: Motion of an undamped pendulum, undamped spring-mass system.

The speed of a transverse wave on a stretched string is given by:

Where v is the velocity of the wave, T is the tension in the string; μ is mass per unit length.

Calculation:

Let μ is the mass per unit length of the roap.

The speed of transverse waves on a stretched string is given by v = √(T/μ).

At a distance x from the lower end, we will find tension which will be

T = μ g x

So, using

  (Speed of transverse wave on a string)

v =