Question

Calculate the bulk modulus of air from the following data for a sound wave of wavelength $35\,cm$ travelling in air. The pressure at a point varies between $\left( {1.0 \times {{10}^5} \pm 14} \right)\,Pa$ and the particles of the air vibrate in $SHM$ of amplitude $5.5 \times {10^{ - 6}}\,m$.

Answer 1

Hint: the bulk modulus of gases is defined as the measure of resistance relative to the compression of a substance. It is also defined as the ratio of infinitesimal increase in pressure to the relative decrease in volume. Here, we will use the formula of pressure amplitude of air to calculate the bulk modulus of air which is given below.

Formula used:
The formula used for calculating the bulk modulus is shown below
${p_0} = Bk{s_0}$
Here, ${p_0}$ is the pressure amplitude, $B$ is the bulk modulus, $k$ is the wave number and ${s_0}$ is the displacement amplitude.

Complete step by step answer:
The following terms are given in the question, as shown below;
The wavelength of the sound wave, $\lambda = 35cm = 0.35m$, Pressure amplitude, ${p_0} = 14\,Pa$. And the displacement amplitude, ${s_0} = 5.5 \times {10^{ - 6}}\,m$
Bulk modulus of air is defined as the ratio of infinitesimal increase in pressure to the relative decrease in volume. Here, we will use the formula of pressure amplitude of air to calculate the bulk modulus of air.Now, the formula of the pressure amplitude to calculate the bulk modulus of elasticity is given below
${p_0} = Bk{s_0}$
Here, ${p_0}$ is the pressure amplitude, $B$ is the bulk modulus, $k$ is the wave number and ${s_0}$ is the displacement amplitude.
$ \Rightarrow \,B = \dfrac{{{p_0}}}{{k{s_0}}}$
Here, wave number, $k = \dfrac{{2\pi }}{\lambda }$
Now, putting the values in the above formula, we get
$B = \dfrac{{{p_0}\lambda }}{{2\pi {s_0}}}$
$ \Rightarrow B = \dfrac{{14\, \times 0.35}}{{2\pi \times 5.5 \times {{10}^{ - 6}}}}$
$ \Rightarrow \,B = \dfrac{{14 \times 0.35}}{{2 \times 3.14 \times 5.5 \times {{10}^{ - 6}}}}$
$ \therefore \,B = 1.4 \times {10^5}\,N{m^{ - 2}}$

Therefore, the bulk modulus of air for a sound wave is $1.4 \times {10^5}\,N{m^{ - 2}}$.

Note: It is important to change the units of wavelength into meters because the amplitude is given in meters. Here, we got the value of the bulk modulus in newton per meter square units. Newton per meter square is the unit that is used to describe how a Pascal unit is derived from other SI units.