Chapter 12, Problem 55P

To determine

To Find: The beat frequency that can be heard.

Answer to Problem 55P

  $f_{beat}=2.18\text{ Hz}$

Explanation of Solution

Given:

Train car frequency, $\left(f\right)$

  $=75\text{ Hz}$

The train car approached the station $\left(v\right) =10.0\text{ m/s}$ .

Formula used:

  $f^1=f\left(\frac{v-v_0}{v}\right)\text{ Hz}$

  $f_{beat}=f-f^1$

  $f=$ Train frequency

  $f=$ Observer frequency

  $v$ = Speed of sound waves

  $v_0$ = Observer velocity

  $$

Calculation:

The train is an observer and moving toward the source.

To calculate the beat frequency, use the Doppler effect,

  $f^1=f\left(\frac{v-v_0}{v}\right)\text{ Hz}$

Where,

  $v=343\text{ m/s}$

  $v_s=10\text{ m/s,}$

  $f^1=f\left(\frac{v+v_0}{v}\right)\text{ Hz}$

  $f^1=75\left(\frac{343+10}{343}\right)\text{ Hz}$

  $f^1=77.18\text{ Hz}$

Now, the beat frequency is the difference of frequencies,

  $f_{beat}=f-f^1$

  $f_{beat}=77.18-75\text{ Hz}$

  $f_{beat}=2.18\text{ Hz}$

The beat frequency is $2.18\text{ Hz}.$

Conclusion:

The beat frequency that can be heard, if the train car approached to the station is $2.18\text{ Hz}.$