Question

The string is fixed at two ends with distance 1.5 m. Its mass is 5 g and the tension in the string is 50N and it vibrates on its third harmonic.
a) What is the wavelength of waves of the string.
b) What is the frequency of the waves.
c) The vibrations produce the sound with the same frequency. What is the wavelength of the sound emitted by the string?

Answer 1

Answer:

a) $\lambda=1\ m$

b) $f=122.47\ Hz$

c) $\lambda_s=2.8\ m$

Explanation:

Given:

distance between the fixed end of strings, $l=1.5\ m$

mass of string, $m=5\ g=0.005\ kg$

tension in the string, $F_T=50\ N$

a)

Since the wave vibrating in the string is in third harmonic:

Therefore wavelength λ of the string:

$l=1.5\lambda$

$\lambda=\frac{1.5}{1.5}$

$\lambda=1\ m$

b)

We know that the velocity of the wave in this case is given by:

$v=\sqrt{\frac{F_T}{\mu} }$

where:

$\mu=$ linear mass density

$v=\sqrt{\frac{50}{(\frac{m}{l}) } }$

$v=\sqrt{\frac{50}{(\frac{0.005}{1.5}) } }$

$v=122.47\ m.s^{-1}$

Now, frequency:

$f=\frac{v}{\lambda}$

$f=\frac{122.47}{1}$

$f=122.47\ Hz$

c)

When the vibrations produce the sound of the same frequency:

$f_s=122.47\ Hz$

Velocity of sound in air:

$v_s=343\ m.s^{-1}$

Wavelength of the sound waves in air:

$\lambda_s=\frac{v_s}{f_s}$

$\lambda_s=2.8\ m$