Question

A string of length L, mass per unit length mu, and tension T is vibrating at its fundamental frequency. What effect will the following have on the fundamental frequency? The length of the string is doubled, with all other factors held constant. The mass per unit length is doubled, with all other factors held constant. The tension is doubled, with all other factors held constant.

Answer 1

Fundamental frequency of the vibrating string is $f=\frac{1}{2L}\sqrt{\frac{T}{\mu}}$

a) Length of the string is doubled.$f_1=\frac{1}{2L_1}\sqrt{\frac{T}{\mu}}=\frac{1}{4L}\sqrt{\frac{T}{\mu}}=\frac{f}{2}$

that is the frequency if halved.

b) Mass per unit length is doubled. $f_1=\frac{1}{2L}\sqrt{\frac{T}{\mu_1}}=\frac{1}{2L}\sqrt{\frac{T}{2\mu}}=\frac{f}{\sqrt{2}}$

c) Tension is doubled $f_1=\frac{1}{2L}\sqrt{\frac{T_1}{\mu}}=\frac{1}{2L}\sqrt{\frac{2T}{\mu}}=\sqrt{2}f$