Question

An unstretched spring is 30 cm long. When the spring is stretched to a total length of 60cm, it supports transverse waves moving at 4.5 m/s. How fast will waves travel in this spring if it is stretched to 90cm?

Answer 1

For a stretched string the velocity of transverse wave across the string is given by

T is the tension on the string and is the linear mass density

Tension acting on a wire of force constant k due to a stretching/compressing by a distance is given by

The un-stretched length of the spring is

When the spring is stretched to a length of x=60 cm, we have

Hence, tension on the wire is

So velocity of the wave at this length is

Now, the spring is stretched to a length of x=90 cm=0.9 m

So, tension on the wire as calculated previously is

So, new velocity is

Taking ratio of equation (i) and (ii) we have

This is the velocity of the waves.