Two strings on a violin are stretched between two points 20 cm apart. The tensions of the strings are the same. If the ratio of the frequencies of the fundamental modes is 1.5 to 1.0, then what is the ratio of the mass per lengths in the strings?
A) 0.44
B) 0.80
C) 0.85
D) 0.94
E) 0.78

Question

Quizplus · Accepted Answer

The frequency of the fundamental mode of 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 mass per unit length of the string. Given that the tensions and lengths are the same for both strings, the ratio of the frequencies $f_1/f_2 = 1.5/1$ can be expressed in terms of the mass per lengths as $\sqrt{\mu_2/\mu_1}$. Solving for $\mu_2/\mu_1$ gives $(f_1/f_2)^2 = (1.5/1)^2 = 2.25$. Therefore, the ratio of the mass per lengths is the inverse, $1/2.25 = 0.44$.

Two Strings on a Violin Are Stretched Between Two Points