Question

A 0.014-kg string is clamped at both ends. Its linear mass density is 0.034 kg/m. Under what tension in the string will it have a fundamental frequency

Answer 1

The fundamental frequency for the wave in the string can be found as follows:

$$f_1=\dfrac{v}{2L}.$$

From this equation we can find the velocity of the wave in the string:

$$v=2Lf_1.$$

From the other hand, the velocity of the wave in the string can be found as follows:

$$v=\sqrt{\dfrac{T}{\mu}}.$$

Equating these two equations, we get:

$$2Lf_1=\sqrt{\dfrac{T}{\mu}},$$ $$T=4L^2f_1^2\mu.$$

Since, $\mu=\dfrac{m}{L}$, we get:

$$L=\dfrac{m}{\mu},$$ $$T=4(\dfrac{m}{\mu})^2f_1^2\mu=\dfrac{4m^2f_1^2}{\mu},$$ $$T=\dfrac{4\times(0.014\ kg)^2\times(100\ Hz)^2}{0.034\ \dfrac{kg}{m}}=230.6\ N.$$