### Rotational Motion Problem**Problem Statement:**Initially at rest, a gear starts rotating with a constant angular acceleration \(\alpha = 5 \, \text{rad/s}^2\). What is the number of revolutions of the gear within the first 6 seconds?**Options:**- ⭕ \(90\pi \, \text{rev}\)- ⭕ \(\frac{45}{\pi} \, \text{rev}\)- ⭕ \(\frac{180}{\pi} \, \text{rev}\)- ⭕ \(10 \, \text{rev}\)For this problem, you need to apply the principles of rotational kinematics to find the answer. Here is a brief guide on how to tackle this type of problem:### Explanation:1. **Given Data:** - Initial angular velocity (\(\omega_0\)) = 0 (since the gear starts from rest) - Angular acceleration (\(\alpha\)) = 5 \(\text{rad/s}^2\) - Time (\(t\)) = 6 s2. **Rotational Kinematics Equation:** To find the angular displacement (\(\theta\)) of the rotating gear, we use the equation: \[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \] Since \(\omega_0 = 0\): \[ \theta = \frac{1}{2} \alpha t^2 \]3. **Calculate Angular Displacement:** \[ \theta = \frac{1}{2} \times 5 \, \text{rad/s}^2 \times (6 \, \text{s})^2 \] \[ \theta = \frac{1}{2} \times 5 \times 36 \] \[ \theta = 90 \, \text{rad} \]4. **Convert Angular Displacement to Revolutions:** Since one revolution equals \(2\pi\) radians: \[ \text{Number of revolutions} = \frac{\theta}{2\pi} = \frac{90 \, \text{rad}}{2\pi} = \frac{90}{2\pi}

Introduction: A pseudoscalar, angular acceleration, exists. If the angular speed rises…

Initially at rest, a gear starts rotating with a constant angular acceleration a = 5 rad/s². What is the number of revolution of the gear within the first 6 seconds? O 90π rev O 45 π 180 π rev rev O 10 rev

Initially at rest, a gear starts rotating with a constant angular acceleration a = 5 rad/s². What is the number of revolution of the gear within the first 6 seconds? O 90π rev O 45 π 180 π rev rev O 10 rev

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Author:Sadiku, Matthew N. O.

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Question

### Rotational Motion Problem

**Problem Statement:**
Initially at rest, a gear starts rotating with a constant angular acceleration \(\alpha = 5 \, \text{rad/s}^2\). What is the number of revolutions of the gear within the first 6 seconds?

**Options:**
- ⭕ \(90\pi \, \text{rev}\)
- ⭕ \(\frac{45}{\pi} \, \text{rev}\)
- ⭕ \(\frac{180}{\pi} \, \text{rev}\)
- ⭕ \(10 \, \text{rev}\)

For this problem, you need to apply the principles of rotational kinematics to find the answer. Here is a brief guide on how to tackle this type of problem:

### Explanation:
1. **Given Data:**
- Initial angular velocity (\(\omega_0\)) = 0 (since the gear starts from rest)
- Angular acceleration (\(\alpha\)) = 5 \(\text{rad/s}^2\)
- Time (\(t\)) = 6 s

2. **Rotational Kinematics Equation:**
To find the angular displacement (\(\theta\)) of the rotating gear, we use the equation:
\[
\theta = \omega_0 t + \frac{1}{2} \alpha t^2
\]
Since \(\omega_0 = 0\):
\[
\theta = \frac{1}{2} \alpha t^2
\]

3. **Calculate Angular Displacement:**
\[
\theta = \frac{1}{2} \times 5 \, \text{rad/s}^2 \times (6 \, \text{s})^2
\]
\[
\theta = \frac{1}{2} \times 5 \times 36
\]
\[
\theta = 90 \, \text{rad}
\]

4. **Convert Angular Displacement to Revolutions:**
Since one revolution equals \(2\pi\) radians:
\[
\text{Number of revolutions} = \frac{\theta}{2\pi} = \frac{90 \, \text{rad}}{2\pi} = \frac{90}{2\pi}

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