A uniform disk initially at rest undergoes constant angular acceleration, rotating about its

center. When the angular speed is 3.00 [𝑟𝑎𝑑

𝑠 ], the rotates an angle 2.50 [𝑟𝑎𝑑]. What is the

angular speed when the disk rotates another 2.50 [𝑟𝑎𝑑]?

To solve this problem, let’s use the equations of rotational kinematics. Here’s what we know:

1. **Initial angular speed, \( \omega_0 = 0 \, \text{rad/s} \)** (since the disk starts from rest).
2. **Initial angular displacement, \( \theta_0 = 0 \)**.
3. **Angular speed after first 2.5 radians, \( \omega = 3.00 \, \text{rad/s} \)**.
4. **Angular displacement after first 2.5 radians, \( \theta = 2.50 \, \text{rad} \)**.

We are tasked with finding the angular speed after the disk has rotated an additional \( 2.50 \, \text{rad} \) (total displacement \( \theta = 5.00 \, \text{rad} \)).

---

### Step 1: Determine Angular Acceleration
We can use the following rotational kinematic equation to find the angular acceleration \( \alpha \):

\[
\omega^2 = \omega_0^2 + 2 \alpha (\theta - \theta_0)
\]

Substituting the known values:

\[
(3.00 \, \text{rad/s})^2 = (0 \, \text{rad/s})^2 + 2 \alpha (2.50 \, \text{rad})
\]

\[
9.00 = 5.00 \alpha
\]

\[
\alpha = \frac{9.00}{5.00} = 1.80 \, \text{rad/s}^2
\]

### Step 2: Find the Angular Speed at \( \theta = 5.00 \, \text{rad} \)

Now, we want to find the angular speed \( \omega \) after the disk has rotated another \( 2.5 \, \text{rad} \), making the total angular displacement \( 5.0 \, \text{rad} \).

Using the same kinematic equation:

\[
\omega^2 = \omega_0^2 + 2 \alpha (\theta - \theta_0)
\]

\[
\omega^2 = (0)^2 + 2 (1.80) (5.0)
\]

\[
\omega^2 = 18.0
\]

\[
\omega = \sqrt{18.0} = 4.24 \, \text{rad/s}
\]

---

### Answer
The angular speed of the disk after it has rotated a total of \( 5.0 \, \text{rad} \) is **4.24 rad/s**.

Would you like a detailed breakdown of the equations or more examples of similar problems?

Here are some related questions to consider:
1. How would the angular speed change if the angular acceleration were doubled?
2. What would be the time taken to reach 4.24 rad/s in this scenario?
3. How much time would it take to reach an angular displacement of 2.50 rad under the same conditions?
4. How would a change in the disk’s mass affect the outcome in this problem?
5. What if the disk started with an initial angular speed instead of from rest?

**Tip:** In rotational kinematics, always identify if the acceleration is constant, as it determines which kinematic equations to use.

A uniform disk initially at rest undergoes constant angular acceleration, rotating about its center. When the angular speed is 3.00 rad/s, the disk has rotated an angle of 2.50 rad. What is the angular speed when the disk rotates another 2.50 rad?

Learn how to calculate the angular speed of a disk rotating with constant angular acceleration. This example problem explores rotational kinematics, focusing on angular displacement, initial and final angular velocities, and angular acceleration. Suitable for Grades 11-12 physics students.

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