Question

A string of mass m and length L is under tension T. The speed of a wave in the string is v. What will be the speed of a wave in the string if the mass of the string is increased to 2m, with no change in length?

Answer 1

Answer:

$\frac{1}{\sqrt{2}}$

Explanation:

The speed of a wave in a string is given by:

$v=\sqrt{\frac{T}{m/L}}$

where

T is the tension in the string

m is the mass of the string

L is the length

In this problem, the mass of the string is increased to 2m: m' = 2 m, while the length is not changed, L'=L. If the tension in the string is not changed, then the new speed of the wave in the string will be:

$v'=\sqrt{\frac{T}{m'/L'}}=\sqrt{\frac{T}{2m/L}}=\frac{1}{\sqrt{2}}\sqrt{\frac{T}{m/L}}=\frac{v}{\sqrt{2}}$

so, the speed of the wave decreases by a factor $\frac{1}{\sqrt{2}}$