Question

A sonometer 0.60m long subjected to a tension of 50Nvibrates in resonance with a tuning fork of frequency 256hz. The wire's length is reduced by 0.01m while the tension is kept constant. If the tuning fork and the wire are sounded together determine the number of beats heard per second.

Answer 1

The frequency of a standing wave can be expressed in terms of string length L, tension T, and linear mass density μ. For the first length (60 cm) it was

$$f_1=\frac{1}{2L_1}\sqrt{\frac{T}{\mu}},\\ \space\\ \sqrt{\frac{T}{\mu}}=2f_1L_1.$$

The left part of the last equation us a constant value in our problem.

The beat frequency (the number of beats heard per second) is

$$f_b=|f_1-f_2|=\bigg|f_1-\frac{1}{2L_2}\sqrt{\frac{T}{\mu}}\bigg|=f_1\bigg|1-\frac{L_1}{L_2}\bigg|=\\ \space\\ =256\bigg|1-\frac{0.6}{0.59}\bigg|=4\text{ Hz}.$$