Standing waves on a string: A 1.0-g string that is 0.64 m long is fixed at both ends and is under tension. This string produces a 100-Hz tone when it vibrates in the third harmonic. The speed of sound in air is 344 m/s. The tension in the string, in is closest to
A) 2.8 N.
B) 2.3 N.
C) 1.8 N.
D) 3.4 N.
E) 3.9 N.

Question

Quizplus · Accepted Answer

The frequency of the third harmonic (n=3) for a string fixed at both ends is given by $f_n = \frac{n}{2L} \sqrt{\frac{T}{\mu}}$, where $f_n$ is the frequency of the nth harmonic, $L$ 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 $f_3 = 100 \, \text{Hz}$, $L = 0.64 \, \text{m}$, and the mass of the string $m = 1.0 \, \text{g} = 0.001 \, \text{kg}$, we can find $\mu = \frac{m}{L} = \frac{0.001}{0.64} \, \text{kg/m}$. Plugging these values into the formula and solving for $T$ gives a tension close to 2.8 N.

Standing Waves on a String: a 1