A stretched string, clamped at its ends, vibrates in its fundamental frequency. To double the fundamental frequency, one can change the string tension by a factor of:
A) 2
B) 4
D) 1/2

Question

A stretched string, clamped at its ends, vibrates in its fundamental frequency. To double the fundamental frequency, one can change the string tension by a factor of:
A) 2 
B) 4 
D) 1/2

Quizplus · Accepted Answer

The fundamental frequency of a stretched 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 mass density of the string. Doubling the frequency means increasing $f$ by a factor of 2, so
\begin{align*}
2f&=\frac{1}{L}\sqrt{\frac{2T}{\mu}}\
4f^2&=\frac{4T}{\mu L^2}\
4f^2&= \frac{T}{\mu}\left(\frac{4}{L^2}ight).
\end{align*}
Since $4/L^2$ is a constant (since the length $L$ and linear mass density $\mu$ are fixed), then to double the frequency, $T$ must be multiplied by a factor of 4. Therefore, the answer is (B).

A Stretched String, Clamped at Its Ends, Vibrates in Its