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

If you pluck a string under tension, a transverse wave moves in the positive x-direction, as shown in Figure below. The small mass element oscillates perpendicular to the wave motion as a result of the restoring force provided by the string and does not move in the x-direction. The tension F_T (this will be denoted by F in the following explanation) in the string, which acts in the positive and negative x-direction, is approximately constant and is independent of position and time.

The speed of a pulse or wave on a string under tension can be found with the equation

v =

where F is the tension in the string and μ is the mass per length of the string.

μ = m/L

Here, m is the mass of the spring and L is the length of the original length of the unstretched spring.

Write the equation of the force, ie, Tension in spring.

F1 = k(x1 - L)

Write the expression for speed

v1 =

v1 =

v1²= k L (x1 - L) / m    ..... (a)

When propagate speed (v2)

F2 = k(x2 - L)

Write the expression for speed

v2 =

v2 =

v2² = k L (x2 - L) / m    ..... (b)

Divide equation (b) by equation (a)

v2² / v1²  = (kL(x2 - L)/m) / (kL(x1 - L)/m)

v2² / v1²  = (x2 - L) / (x1 - L)

Substituting the values given in question,

L = 30 cm = 0.3 m;

x1 = 60 cm = 0.6 m ; v1 = 4.5 m/s

x2 = 90 cm = 0.9 m ; v2 = ?

v2² / 4.5²  = (0.9 - 0.3) / (0.6 - 0.3)

v2² = 4.5² x (0.6 / 0.3) = 4.5² x 2 = 40.5

v2 = 6.36 m/s

The waves will travel in this spring at a speed of 6.36 m/s if it is stretched to 90 cm.