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Angular velocity is a measure of how quickly an object moves through an angle. In circular motion, it describes the rate of rotation around a circle. Imagine a clock's hand moving steadily around its dial; this uniformity is known as constant angular velocity. For circular motion, it is calculated by dividing the total angle traversed (2Ï€ radians for one complete revolution) by the time it takes to complete one full circle.
In our scenario, the angular velocity is given by the formula \[ \omega = \frac{2\pi}{T} \] where

• \(\omega\) is the angular velocity,
• and \(T\) is the time period for one complete revolution.

Here, with a period of 20.0 seconds per revolution, we find the angular velocity, \(\omega\), to be \[ \omega = \frac{2\pi}{20.0} = \frac{\pi}{10} \text{ rad/s}. \] This steady motion is pivotal in determining other dynamic properties like displacement and velocity when an object travels around a circular path.

The position vector in a circular motion reveals the exact location of a particle at any given time. This vector starts from the circle's center and stretches out to the particle's location along the circumference. In terms of coordinates, it tells us both the distance and direction to that point.
The position vector, \(\vec{r}(t)\), can be calculated using:\[ \vec{r}(t) = r(\cos \theta, \sin \theta) \] where

• \(r\) is the circle's radius,
• \(\theta\) is the angular position.

For example, at \(t = 5.00 s\), the angular position is \(\frac{\pi}{2} \) radians. Thus, the position vector is \[ \vec{r}(5s) = 3.00 \cdot (0, 1) = (0, 3.00) \text{ m}. \] Each position vector shows where the particle is relative to the center and provides essential data for further calculations involving displacement or velocity.

Displacement in circular motion refers to the straight-line distance and direction from the starting point to the final position over a specific time period. Unlike the path length which follows the curve of the circle, displacement is straightforward and can have different directions or angles based on the initial and final positions.
To calculate displacement, \(\Delta \vec{r}\), you subtract the initial position vector \(\vec{r}(t_1)\) from the final position vector \(\vec{r}(t_2)\):\[ \Delta \vec{r} = \vec{r}(t_2) - \vec{r}(t_1). \] For instance, between times \(t = 5.00 \text{s}\) to \(t = 10.00 \text{s}\), if \(\vec{r}(5s) = (0, 3.00) \text{ m}\) and \(\vec{r}(10s) = (-3.00, 0) \text{ m}\), the displacement is:\[ \Delta \vec{r} = (-3.00, 0) - (0, 3.00) = (-3.00, -3.00) \text{ m}. \] Hence, displacement is indicated both in magnitude and direction, which here is pointed at a 225° angle from the positive x-axis.

Average velocity helps in identifying how fast something is moving overall and in which direction during a time interval in uniform circular motion. It always aligns with the net displacement and not the path taken around a circle.
Mathematically, the average velocity, \(\vec{v}_{\text{avg}}\), is calculated using:\[ \vec{v}_{\text{avg}} = \frac{\Delta \vec{r}}{\Delta t} \] where:

• \(\Delta \vec{r}\) is the total displacement,
• \(\Delta t\) is the time duration.

For the interval between 5.00 seconds and 10.00 seconds, the average velocity is:\[ \vec{v}_{\text{avg}} = \frac{(-3.00, -3.00) \text{ m}}{5.00 \text{ s}} = (-0.600, -0.600) \text{ m/s}. \] The direction of average velocity follows the displacement, indicating a 225° angle from the positive x-axis. Thus, average velocity provides a simplified overview of motion over time.

Centripetal acceleration is fundamental in understanding the forces that keep a particle moving in a circular path. It points toward the center of the circle and is necessary to change the direction of an object's velocity, while its speed remains constant. Unlike regular acceleration, it does not speed up or slow down the particle but continuously alters its direction.
The formula for centripetal acceleration, \(a_c\), is:\[ a_c = \frac{v^2}{r} \] where:

• \(v\) is the linear speed of the particle,
• \(r\) is the radius of the circular path.

After calculating the linear speed where \(v = r\omega\), using \(\omega = \frac{\pi}{10}\), the centripetal acceleration becomes:\[ a_c = \frac{(0.300\pi)^2}{3.00} \approx 0.300\pi^2 / 3 \text{ m/s}^2. \] This value assures that the particle stays in a circular path, with acceleration vectors at different points like (0, 270°) at \(t = 5.00s\), symbolizing a direction toward the center of rotation.