Question

A uniform string of length l and mass m hangs by one end from the ceiling.

(a) Prove the speed of sound in the string a distance y above the bottom is vs = √gy, where g is the acceleration due to gravity. Hint: The tension in the string is due to string’s mass under the influence of gravity, and that tension increases as you go higher up the string.

(b) You quickly and gently hit the bottom of the string to make a short wave pulse in the string. The pulse propagates up the string. Prove that it takes time t = 2p l/g for the pulse to reach the top of the string. Hint: Yes, you need calculus to do this.

Answer 1

a )
the tension is T = ( m/L ) y g

T = m g y / L

the speed of the transverse wave is v_s = ( T/ (m/L) )^1/2

= ( L .T/ m )^1/2

= ( L m g y / L m )^1/2

v_s = (g y)^1/2

b )

dy/dt = v_s = (g y)^1/2

dt = dy / (g y)^1/2

dt = dy / (g y)^1/2

then by intgrating we get t = 2 ( L/g )^1/2