91Ó°ÊÓ

Circular motion occurs when an object moves in a path where its distance from a certain point remains constant. This point is known as the center of the circle. Imagine a tetherball tied to a pole and swinging around the stick. It keeps moving around the pole in a circle. For the movement to stay circular, the ball experiences a force directed towards the center of the circle, often called the centripetal force.
Understanding circular motion requires familiarizing oneself with certain key terms:

• **Radius ( )**: The distance from the center of the circle to any point on its circumference. In our exercise, the whistle is moving in a circle with a radius of 0.6 meters.
• **Angular Speed ( ext{ω})**: This is how fast the object travels around the circle. It's measured in radians per second, which describes the angle covered over time as the object travels along the circle. In the exercise, the given angular speed is 15 rad/s.
• **Linear Speed ( ext{v}_s)**: This is the speed that depicts how fast the object moves along the actual path of the circle. It can be found by multiplying the radius by the angular speed: ext{v}_s = r imes ext{ω}.

By calculating with these terms, we find that the whistle has a linear speed of 9 m/s while undergoing circular motion.

Frequency modulation (FM) occurs when the frequency of a wave changes in correspondence to certain stimulus or changes. In our context, the stimulus is the changing position of the moving whistle relative to the listener. The Doppler Effect is responsible for this kind of modulation.
The Doppler Effect describes how the frequency of a wave (like sound) changes for an observer moving relative to the source of the wave:

• **When the source approaches the observer:** The sound waves get "squished" together, making the observer perceive a higher frequency.
• **When the source moves away from the observer:** The waves get "stretched" out, and the observer perceives a lower frequency.

The key formula capturing this change is given by: \[ f' = \frac{f}{1 \pm \frac{v_s}{v}} \]where \( f \) represents the source's original frequency, \( v_s \) is the speed of the source (whistle, in our case), and \( v \) is the speed of sound.
By plugging in these values, we determine the frequencies that can be heard by a distant observer depending on the whistle's movement. The change in frequency we've calculated occurs because of the varying distances between the moving whistle and the listener due to circular motion.

The speed of sound is how fast sound waves travel through a medium, such as air, water, or solids. It varies based on the environment and its conditions, such as temperature and pressure. For our calculations, the sound moves through typical air conditions, at approximately 20°C, where it travels roughly at 343 m/s.
This constant speed is crucial for figuring out the effect of any change in the frequency of sound waves due to movement. When dealing with problems involving the Doppler Effect, it is essential to know the speed of sound since it acts as the basis for calculating perceived frequency changes.
The formula for perceived frequency further depends on the ratio of the source's speed to the speed of sound: \[ f'_{min} = \frac{f}{1 + \frac{v_s}{v}} \] \[ f'_{max} = \frac{f}{1 - \frac{v_s}{v}} \] where \( f \) is the original frequency, and \( v_s \) is the speed of the source. These equations allow for calculating the highest and lowest observed frequencies, considering the movement of the sound source.