Question

1. a string with a mass per unit length of 1.5 g/m is put under a tension of 20 n. what length is required for its fundamental resonance frequency to be 220 hz?

a. 0.26 m
b. 0.96 m
c. 1.1 m
d. 1.2 m

Explanation:

Step1: Calculate the wave speed

The wave speed \(v\) on a string is given by \(v = \sqrt{\frac{T}{\mu}}\), where \(T\) is the tension and \(\mu\) is the mass per unit length.
First, convert \(\mu=1.5\ g/m = 1.5\times10^{- 3}\ kg/m\) and \(T = 20\ N\).

$$v=\sqrt{\frac{20}{1.5\times 10^{-3}}}$$

$$v=\sqrt{\frac{20000}{1.5}}\approx115.47\ m/s$$

Step2: Relate frequency, wavelength and wave speed for fundamental mode

For the fundamental resonance frequency of a string fixed at both ends, \(f=\frac{v}{2L}\), where \(f\) is the frequency and \(L\) is the length of the string.
We can re - arrange the formula for \(L\): \(L=\frac{v}{2f}\)
Given \(f = 220\ Hz\) and \(v\approx115.47\ m/s\)

$$L=\frac{115.47}{2\times220}$$

$$L=\frac{115.47}{440}\approx0.26\ m$$

Answer:

A. \(0.26\ m\)