91Ó°ÊÓ

Let's start by understanding tangential velocity. In circular motion, tangential velocity is the linear speed of a point on the edge of a rotating object. It is given by the formula:
\[ v = r \times \text{rotational speed} \]
Here, *r* represents the radius of the circle, and the rotational speed is how fast the object is spinning measured in radians per second (rad/s).
For the given problem, the whistle has a radius (r) of 0.60 meters and a rotational speed of 15.0 rad/s. Therefore, the tangential velocity (v) can be calculated as:
\[ v = 0.60 \text{ m} \times 15.0 \text{ rad/s} = 9.0 \text{ m/s} \]
This means that any point on the whistle's edge moves at 9.0 meters per second along the circular path.

Next, let's delve into the concept of frequency shift due to the Doppler effect. The Doppler effect describes how the frequency of a wave like sound changes for an observer moving relative to the wave's source.
When a sound source moves towards an observer, the frequency appears higher (this is called a frequency increase). Conversely, when it moves away, the frequency appears lower (this is called a frequency decrease).
For a source moving towards the observer, the observed frequency (\[ f_{\text{high}} \]) can be calculated using:
\[ f_{\text{high}} = f \times \frac{v_{\text{sound}}}{v_{\text{sound}} - v_{\text{source}}} \]
where *f* is the emitted frequency, *v_{\text{sound}}* is the speed of sound, and *v_{\text{source}}* is the tangential velocity of the source.
Using the values from our problem:
* Emitted frequency (f): 540 Hz * Speed of sound (\[ v_{\text{sound}} \]): 343 m/s * Tangential velocity (\[ v_{\text{source}} \]): 9.0 m/s
We find: \[ f_{\text{high}} = 540 \text{ Hz} \times \frac{343 \text{ m/s}}{343 \text{ m/s} - 9.0 \text{ m/s}} = 555.87 \text{ Hz} \] Similarly, for a source moving away from the observer, the frequency (\[ f_{\text{low}} \]) is calculated using:
\[ f_{\text{low}} = f \times \frac{v_{\text{sound}}}{v_{\text{sound}} + v_{\text{source}}} \]
Plugging in the values, we get: \[ f_{\text{low}} = 540 \text{ Hz} \times \frac{343 \text{ m/s}}{343 \text{ m/s} + 9.0 \text{ m/s}} = 526.25 \text{ Hz} \]

Finally, let's discuss rotational speed in the context of circular motion.
Rotational speed describes how fast an object spins around a central point or axis. It is usually measured in radians per second (rad/s).
In the problem given, the whistle has a rotational speed of 15.0 rad/s. This quantity affects the tangential velocity, which we previously determined using the formula \[ v = r \times \text{rotational speed} \].
Knowing the rotational speed and radius allows us to calculate how fast the edges of the rotating object are moving linearly. This linear speed (or tangential velocity) is crucial when applying the Doppler effect to understand the highest and lowest frequencies an observer would hear. To summarize the key points:

• Rotational speed (rad/s) quantifies how fast something spins.
• Tangential velocity (m/s) tells us the linear speed due to rotation.
• This linear speed affects the observed frequency due to the Doppler effect.

Understanding these concepts allows us to solve problems involving circular motion and sound frequency changes effectively.