To determine how the fundamental frequency of a vibrating string changes when the tension is doubled, let us consider the relevant physical principles.
The fundamental frequency [tex]\( f \)[/tex] of a string is given by the formula:
[tex]\[
f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}
\][/tex]
where:
- [tex]\( L \)[/tex] is the length of the string,
- [tex]\( T \)[/tex] is the tension in the string,
- [tex]\( \mu \)[/tex] is the linear mass density of the string.
Now, suppose the tension is doubled, i.e., the new tension is [tex]\( 2T \)[/tex].
Let [tex]\( f_{\text{new}} \)[/tex] be the new fundamental frequency when the tension is [tex]\( 2T \)[/tex]. Substituting [tex]\( 2T \)[/tex] into the formula for frequency, we get:
[tex]\[
f_{\text{new}} = \frac{1}{2L} \sqrt{\frac{2T}{\mu}}
\][/tex]
We can factor [tex]\( \sqrt{2} \)[/tex] out of the square root expression:
[tex]\[
f_{\text{new}} = \frac{1}{2L} \sqrt{\frac{2T}{\mu}} = \frac{1}{2L} \sqrt{2} \sqrt{\frac{T}{\mu}} = \sqrt{2} \left( \frac{1}{2L} \sqrt{\frac{T}{\mu}} \right)
\][/tex]
Notice that [tex]\( \frac{1}{2L} \sqrt{\frac{T}{\mu}} \)[/tex] is the original fundamental frequency [tex]\( f \)[/tex]:
[tex]\[
f_{\text{new}} = \sqrt{2} f
\][/tex]
Thus, when the tension is doubled, the fundamental frequency of the vibrating string increases by a factor of [tex]\( \sqrt{2} \)[/tex].
Therefore, the correct answer is:
B. [tex]\( \sqrt{2} f \)[/tex]
