Q4E - University Physics with Modern Physics

Question

A loud factory machine produces sound having a displacement amplitude of 1.00 μ-m, but the frequency of this sound can be adjusted. In order to prevent ear damage to the workers, the maxi mum pressure amplitude of the sound waves is limited to 10.0 Pa. Under the conditions of this factory, the bulk modulus of air is 1.42 x 105 Pa. What is the highest-frequency sound to which this machine can be adjusted without exceeding the prescribed limit? Is this frequency audible to the workers?

Step-by-Step Solution

Answer

The highest-frequency sound to which this machine can be adjusted without exceeding the prescribed limit is, f = 3.86 x 10³ Hz. Since f is in the range of [20 Hz - 20,000 Hz] which is the range of audible frequencies, the frequency is audible to the workers.

Step 1: Determination of the formula for Sound and Hearing

The relation that describes the pressure amplitude for a sound wave is 
 
$P_{max}=BkA$ --(1)
 
 Where the bulk modulus of the air is $B=1.42\times {10}^5 Pa$ and the displacement amplitude of the waves produced by the machine is 1 μ-m. 
 
 Use equation (1) to calculate and then use k to determine the wavelength of the wave,  

$\lambda =\frac{2\pi }{\lambda }$
 
Substitute into equation (1) with 10 Pa for P_max, 1.42 x 105 Pafor B and 1 x 10-6 m for A:
$$\begin{gathered} k=\frac{10Pa}{\left(1.42\times {10}^5 Pa\right)\times \left(1\times {10}^{-6} m\right)} \\ k=70.4m^{-1} \end{gathered}$$ 
 
 Use the following relation to calculate the wavelength:

$$\begin{gathered} \lambda =\frac{2\pi }{k} \\ =\frac{2\pi }{70.4m^{-1}} \\ \lambda =0.089m \end{gathered}$$

 

 Finally, the relation between the wavelength and the frequency of a sound wave is given by the following equation:

 

$$\begin{gathered} f=\frac{v}{\lambda } \\ f=\frac{344m/s}{0.089m} \\ f=3.86\times {10}^3 Hz \end{gathered}$$ 

Step 2: Determination of whether the frequency is audible

Since is in the range of [20 Hz - 20,000 Hz] which is the range of audible frequencies, the frequency is audible. 
 
 
 Therefore, the highest-frequency sound to which this machine can be adjusted without exceeding the prescribed limit is, f = 3.86 x 10³ Hz. Since f is in the range of [20 Hz - 20,000 Hz] which is the range of audible frequencies, the frequency is audible to the workers.