Question

If the tension in a string is doubled, then a natural frequency will (a) increase, (b) decrease by ________?

Answer 1

Answer:

(a) increase by $\sqrt{2}$ times

Explanation:

Natural frequency of a wave in a string is given by:

$f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$

where, L is the length of the string, T is the tension in the string and $\mu$ is the linear density of the string.

Considering the length and linear density of the string are constant, if the tension in a string is doubled, the natural frequency of the string would:

$f\propto \sqrt{T}$

$\frac{f_n}{f}=\sqrt{\frac{T_n}{T}}\\ \Rightarrow f_n =\sqrt{\frac{2T}{T}}f = \sqrt{2} f$

Thus, the natural frequency of the string would increase by  $\sqrt{2}$ times.