Q60PE - College Physics (Urone)

Question

Twin jet engines on an airplane are producing an average sound frequency of 4100 Hz with a beat frequency of 0.500 Hz. What are their individual frequencies?

Step-by-Step Solution

Answer

The individual frequencies of two jet engines are 4100.25 Hz and 4099.75 Hz.

Step 1: Introduction

We know that beat frequency is modulus of difference of two frequencies. We also know average frequency of two forks is average of two frequencies. We use these two formulas to find individual frequencies.

Step 2: Given parameters and formula for beat frequency and average frequency

Beat frequency          \( = 0.5 Hz\)

Average frequency   \( = 4100 Hz\)

Beat frequency        \(f = \left| {{f_1} - {f_2}} \right|\)

Average frequency  \(f_{avg} = {f_1} + {f_2}/2\)

Here, 

\({f_1},{f_2}\) - frequency of 2 sources.

\(f\)- beat frequency.

\(f_{avg}\) -average frequency.

Step 3: Form two equations from given data

Beat frequency of engines

\(\begin{align}f &= \left| {{f_1} - {f_2}} \right|\\f &= {f_1} - {f_2}\\0.5 &= {f_1} - {f_2}\end{align}\)

This is equation1.

Average frequency of engines

\(\begin{align}f_{avg} &= {{f_1} + {f_2}}/2\\2f_{avg}  &= {f_1} + {f_2}\\8200 &= {f_1} + {f_2}\end{align}\)

This is equation2

Step 4: Solve the above two equations

\(\begin{align}0.5 &= {f_1} - {f_2}\\8200 &= {f_1} + {f_2}\\8200.5 &= 2{f_1}\\{f_1} &= 4100.25Hz\\{f_2} &= 4099.75Hz\end{align}\)

Therefore, the individual frequencies of the two jet engines are 4100.25 Hz and 4099.75 Hz.