Problem 42 - University Physics with Modern Physics

Question

Adjusting Airplane Motors. 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 \(\mathrm{rpm}\) and you hear a \(2.0-\mathrm{Hz}\) beat 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 \(\mathrm{Hz}\) . 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?

Step-by-Step Solution

Answer

The original frequency of the second propeller was 455 rpm.

Step 1: Understanding rotational speed in Hz

First, we need to convert the propeller's speed from revolutions per minute (rpm) to Hertz (Hz), which is in cycles per second. Given speed is 575 rpm. Using the conversion formula: \[\text{Frequency in Hz} = \frac{\text{Frequency in rpm}}{60}\]Thus, for the first propeller:\[575 \text{ rpm} = \frac{575}{60} \approx 9.58 \text{ Hz}\]

Step 2: Calculating possible frequencies using beat frequency

The beat frequency is given as 2.0 Hz. The beat frequency is the absolute difference between the two sound frequencies:\[|f_1 - f_2| = \text{beat frequency} = 2.0 \text{ Hz}\]Since the first propeller is 9.58 Hz, the second propeller's frequency can be either:\[f_2 = 9.58 \text{ Hz} + 2.0 \text{ Hz} = 11.58 \text{ Hz}\]or\[f_2 = 9.58 \text{ Hz} - 2.0 \text{ Hz} = 7.58 \text{ Hz}\]

Step 3: Convert Hz back to rpm for possible frequencies

To find the possible frequencies of the second propeller in rpm, convert 11.58 Hz and 7.58 Hz back to rpm:\[11.58 \text{ Hz} = 11.58 \times 60 \approx 695 \text{ rpm}\]\[7.58 \text{ Hz} = 7.58 \times 60 \approx 455 \text{ rpm}\]

Step 4: Determine correct frequency with changed beat frequency

When the second propeller's speed increases slightly and the beat frequency changes to 2.1 Hz, it indicates that the difference in frequency increased. This would mean the beat formula:\[|f_1 - f_3| = 2.1 \text{ Hz}\]The increased frequency (when you get a larger beat frequency) indicates that the second frequency you assumed was less than 9.58 Hz initially. Hence, the starting second propeller frequency was 455 rpm (7.58 Hz).

Key Concepts

Rotational Speed Conversion

Understanding rotational speed is essential when it comes to exercises involving airplane propellers. Propellers are measured in revolutions per minute (rpm), a unit that provides the number of complete turns the propeller makes in one minute. However, in many physics problems, it's vital to convert rpm to Hertz (Hz) because Hz corresponds to cycles per second, which aligns with sound frequencies.
To convert rpm to Hz, use the formula:

• Frequency in Hz = Frequency in rpm / 60

By applying this to a propeller turning at 575 rpm, you get:

• 575 rpm ÷ 60 = 9.58 Hz

This conversion reveals that the motor's rotational motion is producing a frequency of 9.58 Hz.

Frequency in RPM

Knowing how to work with frequency both in cycles per second (Hz) and in revolutions per minute (rpm) opens up practical understanding of beat frequencies. In problems involving two components like propellers, beats are the result of the interference of two frequencies. The beat frequency is simply the absolute difference between the two:

• |f₁ - f₂| = Beat Frequency

For example, if the first propeller's frequency is measured at 9.58 Hz, and a beat frequency of 2.0 Hz is heard, the second propeller could have a frequency of:

• 9.58 Hz + 2.0 Hz = 11.58 Hz or
• 9.58 Hz - 2.0 Hz = 7.58 Hz

Converting these frequencies back to rpm gives:

• 11.58 Hz × 60 = 695 rpm
• 7.58 Hz × 60 = 455 rpm

This dual standard of measuring frequency allows engineers and technicians to diagnose issues and perform tweaks efficiently.

Sound Wave Frequency

Sound waves are cyclic, meaning they travel through mediums like air as repetitions of compressions and rarefactions. Frequency, in Hertz (Hz), is crucial for determining the pitch of the sound produced by machines like propellers. Higher frequencies correspond to higher pitches. When two sound waves of different frequencies reach our ear, they interfere, creating a third frequency known as the beat frequency.
The beat frequency, given as 2.0 Hz in this context, originates from the interference between the motor's sound wave of a propeller and another frequency. If you hear a beat frequency, it means:

• The frequencies of two sound sources (propellers) are close, but not the same.
• The beat frequency is the rate at which the volume seems to increase and decrease at the listener’s position.

This phenomenon is invaluable in audio and mechanical adjustments.

Propeller Tuning

Propeller tuning involves subtle adjustments aimed at balancing and fine-tuning the rotational speeds of airplane motors. The motor sound frequency should ideally match the designed frequency for optimal performance. A pilot or engineer can use the concept of beat frequencies to fine-tune propeller speeds. Adjusting the second propeller until the beat frequency is minimized indicates that both propellers are nearly in sync.
For example, an initial beat frequency of 2.0 Hz indicated some mismatch. Upon increasing the second propeller speed and noting the beat frequency shift to 2.1 Hz, you conclude:

• The original frequency of the second propeller was lower than expected, confirming the frequency was initially at 455 rpm (7.58 Hz).

Knowing these principles helps mechanics ensure that a propeller operates efficiently and harmoniously within a fleet of aircraft, minimizing vibrations and enhancing performance.