The motors that drive airplane propellers are, in some cases, tuned by using beats. The whirring motor produces a sound wave having the same frequency as the propeller. (a) If one single-bladed propeller is turning at 575 rpm and you hear 2.0-Hz beats when you run the second propeller, what are the two possible frequencies (in rpm) of the second propeller? (b) Suppose you increase the speed of the second propeller slightly and find that the beat frequency changes to 2.1 Hz. In part (a), which of the two answers was the correct one for the frequency of the second single-bladed propeller? How do you know?

Verified step by step guidance

1. First, understand the concept of beats. Beats occur when two sound waves of slightly different frequencies interfere with each other, resulting in a new sound wave whose amplitude varies at a frequency equal to the difference between the two original frequencies.
2. Convert the frequency of the first propeller from revolutions per minute (rpm) to hertz (Hz). Since 1 rpm is equal to 1/60 Hz, the frequency of the first propeller is 575 rpm * (1/60) Hz/rpm.
3. For part (a), use the beat frequency formula: |f1 - f2| = beat frequency, where f1 is the frequency of the first propeller and f2 is the frequency of the second propeller. Given that the beat frequency is 2.0 Hz, set up the equation |f1 - f2| = 2.0 Hz to find the two possible values for f2.
4. Convert the possible frequencies of the second propeller from hertz back to revolutions per minute by multiplying by 60, since 1 Hz is equal to 60 rpm.
5. For part (b), when the speed of the second propeller is increased and the beat frequency changes to 2.1 Hz, determine which of the two possible frequencies from part (a) results in a beat frequency of 2.1 Hz when the second propeller's speed is increased. This will indicate the correct frequency of the second propeller in part (a).

Key Concepts

Beat Frequency

Beat frequency is the phenomenon that occurs when two sound waves of slightly different frequencies interfere with each other, resulting in a new sound wave with a frequency equal to the absolute difference between the two original frequencies. This is perceived as a periodic variation in volume or 'beats'. In this problem, the beat frequency helps determine the frequency of the second propeller.

Frequency and RPM Conversion

Frequency, measured in Hertz (Hz), is the number of cycles per second, while revolutions per minute (rpm) is a measure of rotational speed. To convert between these, use the relation: 1 Hz = 60 rpm. Understanding this conversion is crucial for solving the problem, as it involves comparing frequencies in Hz with rotational speeds in rpm.

Interference and Frequency Adjustment

Interference occurs when two waves overlap, affecting the resultant wave's amplitude and frequency. In this context, adjusting the speed of the second propeller changes its frequency, altering the beat frequency. By observing how the beat frequency changes with adjustments, one can deduce the correct frequency of the second propeller, as a decrease or increase in beat frequency indicates the direction of frequency adjustment needed.