Problem 75 - College Physics

Question

A turntable 1.50 \(\mathrm{m}\) in diameter rotates at 75 \(\mathrm{rpm}\) . Two speakers, each giving off sound of wavelength \(31.3 \mathrm{cm},\) are attached to the rim of the table at opposite ends of a diameter. A listener stands in front of the turntable. (a) What is the greatest beat frequency the listener will receive from this system? (b) Will the listener be able to distinguish individual beats?

Step-by-Step Solution

Answer

The greatest beat frequency is approximately 37.5 Hz, which is likely not distinguishable by the listener as individual beats.

Step 1: Calculate Linear Velocity of the Speakers

The turntable rotates at 75 rpm, which we convert to radians per second. First, convert rpm to radians per second: \[ 75 \text{ rpm} = \frac{75 \times 2\pi}{60} \text{ rad/s} \approx 7.85 \text{ rad/s}.\]The radius of the turntable is \( \frac{1.50}{2} = 0.75 \text{ m}.\)So, the linear velocity \( v \) of the speakers is:\[ v = \omega r = 7.85 \times 0.75 \approx 5.89 \text{ m/s}.\]

Step 2: Calculate Doppler Shifted Frequencies

Given the speed of the sound is approximately \( c = 343 \text{ m/s} \), find the Doppler shift for each speaker:- For the approaching speaker:\[ f' = \frac{v_0}{v_0 - v_s} f = \frac{343}{343 - 5.89} f.\]- For the receding speaker:\[ f'' = \frac{v_0}{v_0 + v_s} f = \frac{343}{343 + 5.89} f.\]

Step 3: Calculate Beat Frequency

Find the difference between the shifted frequencies:\[ \Delta f = f' - f'' = f \left( \frac{343}{343 - 5.89} - \frac{343}{343 + 5.89} \right).\] With \( f = \frac{c}{\lambda} = \frac{343}{0.313} \approx 1096.48 \text{ Hz} \), we substitute \( f \) to find \( \Delta f \).

Step 4: Determine if Beats Are Noticeable

Calculate \( \Delta f \) using numerical values:\[ \Delta f \approx 1096.48 \times 0.0342 \approx 37.5 \text{ Hz}.\]Beats are generally distinguishable if \( \Delta f \leq 20 \text{ Hz} \). Since 37.5 Hz > 20 Hz, the listener may struggle to distinguish individual beats.

Key Concepts

Angular Velocity

Angular velocity is a concept that describes how fast an object rotates. It is usually measured in radians per second (rad/s). To find the angular velocity of a rotating object like a turntable, we often need to convert from rotations per minute (rpm) to radians per second. Understanding this conversion is key. When a turntable spins at 75 rpm, we convert this using the formula:

\[\text{Angular velocity } (\omega) = \frac{75 \times 2\pi}{60} \approx 7.85 \text{ rad/s}.\]This formula incorporates the full circle's angle measure, which is \(2\pi\) radians. In everyday terms, angular velocity tells us how quickly an object spins or rotates, conveying how many radians it traverses in a second. In the context of circular motion, understanding angular velocity is crucial as it connects to other concepts like linear velocity. Each section of the rotating system shares this single angular speed, irrespective of its position from the center.

Linear Velocity

Linear velocity signifies the speed at which a point on a rotating object moves along its path. This velocity results from the object’s angular velocity and the radius of the circular path the object travels. If you have the angular velocity of a turntable, you can easily find its linear velocity by using the formula:

\[v = \omega r,\]where \(\omega\) is the angular velocity in rad/s, and \(r\) is the radius of the turntable. In our example, the turntable’s diameter is 1.50 m, giving a radius of 0.75 m. Substituting in the known values:

\[v = 7.85 \times 0.75 \approx 5.89 \text{ m/s}.\]This means each point on the rim of the turntable moves at nearly 5.89 meters per second along its circular path. Linear velocity is crucial in problems involving rotational systems because it determines how fast different parts of the object move through space. It also influences Doppler effects and wave interference, especially in sound systems.

Beat Frequency

Beat frequency is a fascinating phenomenon that results from wave interference, particularly when two sound waves of slightly different frequencies interact. This interaction creates a sound that seems to 'pulsate' in amplitude, forming audible "beats." In the context of our problem, this occurs because one speaker moves towards the listener while the other moves away, resulting in slightly different frequencies due to the Doppler effect.

The listener receives one frequency from the approaching speaker,\(f'\), and another from the receding speaker, \(f''\). The beat frequency is the absolute difference between these two frequencies:

\[\Delta f = |f' - f''|.\]For our example with sound waves, we find

\[\Delta f \approx 37.5 \text{ Hz},\]which is greater than the typical upper limit of human beat distinguishability, typically around 20 Hz. Therefore, these rapid beats may blend into a single sound, challenging our ability to distinguish individual beats. Appreciating the calculation and significance of beat frequency enhances our understanding of acoustic phenomena and sound interactions.

Wave Interference

Wave interference is the interaction of waves that travel in the same medium. This can create a variety of new wave patterns based on how the waves align and combine. When two or more waves overlap, they interfere with each other through either constructive interference (waves add to form a larger amplitude wave) or destructive interference (waves add to form a smaller amplitude wave), depending on their phase relationship.

In the case of our turntable problem, wave interference is particularly interesting because the speakers emit sound waves that continuously interfere. As the turntable spins, each speaker's sound wave experiences a Doppler shift due to their motion toward or away from the listener. These Doppler-shifted waves then overlap at the listener's position, causing the interference that produces beats.

The principles of wave interference are vital in various fields, including acoustics, optics, and electromagnetic theories. They help explain how interference patterns form and how certain sound or light phenomena are observed, such as the changing loudness in beats or the rainbow patterns in light diffraction. Understanding wave interference allows us to predict and harness effects in technologies ranging from musical acoustics to advanced communication systems.