To solve this problem, we need to apply the principles of the Doppler effect, which describes how the frequency of sound changes for an observer moving relative to the source of the sound. In this case, we have two tuning forks, both with a frequency of 340 Hz, but one is moving away from the observer while the other is moving towards the observer. The observer hears beats with a frequency of 3 Hz, which indicates the difference in frequencies perceived from the two forks.
Understanding the Doppler Effect
The Doppler effect states that when a source of sound moves towards an observer, the frequency of the sound increases, and when it moves away, the frequency decreases. The formula for the observed frequency (f') can be expressed as:
- For a source moving towards the observer: f' = f (v + vo) / (v - vs)
- For a source moving away from the observer: f' = f (v - vo) / (v + vs)
Where:
- f = original frequency of the source (340 Hz)
- v = speed of sound in air (340 m/s)
- vo = speed of the observer (0 m/s, since the observer is stationary)
- vs = speed of the source (the tuning forks)
Calculating the Frequencies
Let's denote the speed of the tuning forks as vs. For the fork moving towards the observer, the observed frequency (f1) is:
f1 = 340 Hz (340 m/s + 0 m/s) / (340 m/s - vs)
For the fork moving away from the observer, the observed frequency (f2) is:
f2 = 340 Hz (340 m/s - 0 m/s) / (340 m/s + vs)
Finding the Beat Frequency
The beat frequency is the absolute difference between the two observed frequencies:
Beat frequency = |f1 - f2| = 3 Hz
Setting Up the Equation
Now, we can set up the equation using the expressions for f1 and f2:
|(340 * (340) / (340 - vs)) - (340 * (340) / (340 + vs))| = 3
To simplify this, we can factor out 340:
340 * |(340 / (340 - vs)) - (340 / (340 + vs))| = 3
This simplifies to:
|(1 / (340 - vs)) - (1 / (340 + vs))| = 3 / 340
Solving for vs
Now, let's solve the equation:
Let A = 1 / (340 - vs) and B = 1 / (340 + vs). Then:
|A - B| = 3 / 340
Calculating A and B gives:
A - B = (340 + vs - (340 - vs)) / ((340 - vs)(340 + vs)) = (2vs) / ((340 - vs)(340 + vs))
Setting this equal to 3 / 340, we have:
|(2vs) / ((340 - vs)(340 + vs))| = 3 / 340
Cross-multiplying leads to:
2vs * 340 = 3 * (340 - vs)(340 + vs)
Expanding and simplifying this equation will allow us to isolate vs. After some algebra, you will find:
vs = 15 m/s
Final Thoughts
Thus, the speed of each tuning fork is 15 m/s. This example illustrates how the Doppler effect can be applied to real-world scenarios, such as sound waves from moving sources, and how we can derive meaningful information from the observed frequencies and beat frequencies.
