Answer:
(a) increase by [tex]\sqrt{2}[/tex] times
Explanation:
Natural frequency of a wave in a string is given by:
[tex]f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}[/tex]
where, L is the length of the string, T is the tension in the string and [tex]\mu[/tex] 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:
[tex]f\propto \sqrt{T}[/tex]
[tex]\frac{f_n}{f}=\sqrt{\frac{T_n}{T}}\\ \Rightarrow f_n =\sqrt{\frac{2T}{T}}f = \sqrt{2} f[/tex]
Thus, the natural frequency of the string would increase by [tex]\sqrt{2}[/tex] times.
