Determine the bulk modulus of air using the given data for a sound...

The pressure amplitude $\Delta P_m$ of a sound wave is related to the bulk modulus $B$, the wave number $k$, and the displacement amplitude $A$ by the relation:

$\Delta P_m = B k A$

The wave number $k$ is given by $k = \dfrac{2\pi}{\lambda}$. Substituting this into the equation gives:

$\Delta P_m = B \left(\dfrac{2\pi}{\lambda}\right) A$

Rearranging for the bulk modulus $B$:

$B = \dfrac{\Delta P_m \lambda}{2\pi A}$

Given the values:
Pressure amplitude, $\Delta P_m = 14 \text{ Pa}$
Wavelength, $\lambda = 35 \text{ cm} = 0.35 \text{ m}$
Displacement amplitude, $A = 5.5 \times 10^{-6} \text{ m}$

Substituting these values into the formula:

$B = \dfrac{14 \times 0.35}{2 \times \pi \times 5.5 \times 10^{-6}}$

Using $\pi \approx \dfrac{22}{7}$:

$B = \dfrac{4.9}{11 \times \dfrac{22}{7} \times 10^{-6}}$

$B = \dfrac{4.9 \times 7}{242 \times 10^{-6}}$

$B = \dfrac{34.3}{242} \times 10^6$

$B \approx 1.41 \times 10^5 \text{ Pa}$

Rounding to two significant figures, we get $1.4 \times 10^5 \text{ Pa}$.