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Chapter 16: Q82. (page 454)

A rope of mass m and length L hangs from a ceiling. a. Show that the wave speed on the rope a distance y above the lower end is v = 1gy. b. Show that the time for a pulse to travel the length of the string is 鈭唗 = 22L/g.

Short Answer

Expert verified

a: v=gy

b: 鈭�t=2Lg

Step by step solution

01

Given Information

Mass of rope: m

Length of rope: L

y is an assumed length measured from bottom of the rope.

02

Part a:

We know that

v=Ts渭

Where, v is speed of sound

Tsis tension in the string

and 渭is the linear density of the string.

Ts=Mg, where M is mass of string of y length.

g is acceleration due to gravity

Similarly 渭=My

Now substituting these values in equation (1) we get,

v=MgMy鈬�v=gy

03

Part b:

Now since we have the velocity as a function of length from bottom end of the rope, we can calculate time taken by wave to travel by integrating inverse of velocity over length L i.e.

鈭�t=鈭�0Ldyv

鈭�t=鈭�0Ldyv=鈭�0Ldygy=1g鈭�0Ldyy=1g[2y]0L=2LG

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