Question

A whistle of frequency \(538 \mathrm{~Hz}\) moves in a circle of radius \(71.2 \mathrm{~cm}\) at an angular speed of \(14.7 \mathrm{rad} / \mathrm{s}\). What are \((a)\) the lowest and (b) the highest frequencies heard by a listener a long distance away at rest with respect to the center of the circle?

Short answer

The lowest frequency heard by the listener is approximately \(515 \mathrm{~Hz}\) and the highest frequency is approximately \(565 \mathrm{~Hz}\).

Step-by-step solution

Identify given values

Here, the original frequency \(f_0\) is \(538 \mathrm{~Hz}\), the radius \(r\) of the circle is \(0.712 \mathrm{~m}\) (converting cm to m), and the angular speed \(\omega\) is \(14.7 \mathrm{~rad/s}\). The speed \(v_{source}\) of the whistle can be calculated as \(v_{source} = r \cdot \omega\). The speed of sound in air (\(v_{sound}\)) is approximately \(343 \mathrm{~m/s}\).

Calculate speed of the source (whistle)

Plug in the given values into the equation: \[ v_{source} = r \cdot \omega = 0.712 \mathrm{~m} \cdot 14.7 \mathrm{~rad/s} = 10.46 \mathrm{~m/s} \]

Find the lowest frequency

The lowest frequency occurs when the whistle is moving away from the observer. This is given by Doppler's Effect formula for frequency shift: \[ f = f_0 \left(\frac{v_{sound}}{v_{sound} + v_{source}}\right) = 538 \mathrm{~Hz} \left(\frac{343 \mathrm{~m/s} }{343 \mathrm{~m/s} + 10.46 \mathrm{~m/s}}\right) \approx 515 \mathrm{~Hz} \]

Find the highest frequency

The highest frequency occurs when the whistle is moving towards the observer. Again, using the formula for Doppler's Effect: \[ f = f_0 \left(\frac{v_{sound}}{v_{sound} - v_{source}}\right) = 538 \mathrm{Hz} \left(\frac{343 \mathrm{~m/s} }{343 \mathrm{~m/s} - 10.46 \mathrm{~m/s}}\right) \approx 565 \mathrm{~Hz} \]

Key concepts

Frequency Shift

When we talk about the frequency shift, we're diving into a core part of the Doppler Effect. This effect happens when there's a change in frequency (and in turn the pitch) of a sound as the source moves closer or farther from the observer.
In our exercise, think of the whistle that's moving in a circle. As it comes closer, the sound waves pile up, which increases the frequency, leading to the highest frequency we hear. When it moves away, the opposite happens, the waves stretch out, lowering the frequency.
Key points about frequency shifts:

• When the source moves towards the observer, the frequency perceived is higher.
• When the source moves away, the frequency perceived is lower.

This is what explains why we hear differences in sound as the position of the sound source changes in motion.

Angular Speed

Angular speed is all about how fast an object is rotating along a circular path. It measures the rate of change of the angular position of the object, in radians per second.

For the whistle in the exercise, its angular speed is given as 14.7 rad/s. This helps us understand how quickly the whistle is making a complete circle. Some important things to note about angular speed:

• It is a vector quantity, meaning it has both a magnitude and a direction.
• It's different from linear speed, which measures how fast something moves along a path, rather than rotating.

It's crucial to grasp this concept because, through angular speed, we can determine the linear speed of the whistle using its radius.

Circular Motion

When an object travels around a path and the distance from the center remains constant, it's in circular motion. In the context of this exercise, circular motion means our whistle follows a constant-radius path or circle.

Several factors come into play during circular motion:

• Radius: It's the distance from the center of the circle to the path. Here, it's 0.712 m.
• Linear speed: Derived from angular speed and radius, showing how fast the whistle moves along the circle.

In real-world terms, it's similar to how the hands of a clock move—always keeping the same distance from the center (the axis in this case). Understanding circular motion helps us see the bigger picture of how movement around a fixed path alters perception of sound, like the changing frequencies we hear.

Speed of Sound

The speed of sound is a key factor that influences how we perceive frequency changes in moving objects due to the Doppler Effect.

In the step-by-step solution, the speed of sound in air is approximated as 343 m/s. This number represents how fast sound propagates through the air under normal conditions. Why is this important?

• It's the baseline against which other speeds are compared when calculating frequency shifts.
• Variations in this speed (such as changes in temperature or pressure) can alter how we calculate and perceive frequency shifts.

Understanding the speed of sound helps us make accurate calculations regarding how sound behaves in various conditions, essential for applying principles like the Doppler Effect in real-life scenarios.