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When the gear rotates 20 revolutions, it achieves an angular velocity of omega = (30 rad) / s starting from rest. Determine its constant angular acceleration and the time required.

To solve this problem, we need to determine the constant angular acceleration \(\alpha\) and the time \(t\) required for the gear to reach an angular velocity \(\omega = 30 \, \text{rad/s}\) after rotating 20 revolutions starting from rest.

### Given Data
1. Initial angular velocity, \(\omega_0 = 0 \, \text{rad/s}\) (starting from rest).
2. Final angular velocity, \(\omega = 30 \, \text{rad/s}\).
3. Angular displacement, \(\theta = 20 \, \text{revolutions}\).

Since 1 revolution is \(2\pi\) radians,
\[ \theta = 20 \times 2\pi = 40\pi \, \text{radians} \]

### Equations of Motion for Rotational Kinematics
We use the following rotational kinematic equations:

1. \(\omega^2 = \omega_0^2 + 2\alpha\theta\)
2. \(\omega = \omega_0 + \alpha t\)

### Step-by-Step Solution

#### 1. Determine the Constant Angular Acceleration \(\alpha\)

Using the first equation:
\[ \omega^2 = \omega_0^2 + 2\alpha\theta \]
Substitute the known values:
\[ (30 \, \text{rad/s})^2 = (0 \, \text{rad/s})^2 + 2\alpha(40\pi \, \text{rad}) \]
\[ 900 = 80\pi\alpha \]
\[ \alpha = \frac{900}{80\pi} \]
\[ \alpha = \frac{900}{80\pi} \approx 3.577 \, \text{rad/s}^2 \]

#### 2. Determine the Time \(t\)

Using the second equation:
\[ \omega = \omega_0 + \alpha t \]
Substitute the known values:
\[ 30 \, \text{rad/s} = 0 \, \text{rad/s} + (3.577 \, \text{rad/s}^2) t \]
\[ t = \frac{30 \, \text{rad/s}}{3.577 \, \text{rad/s}^2} \]
\[ t \approx 8.39 \, \text{s} \]

### Final Results
- Constant angular acceleration, \(\alpha \approx 3.577 \, \text{rad/s}^2\)
- Time required, \(t \approx 8.39 \, \text{s}\)

Would you like any more details or have any other questions?

Here are some related questions that might interest you:

1. How is angular acceleration related to linear acceleration?
2. What are the units of angular displacement, velocity, and acceleration?
3. Can you explain the difference between angular velocity and angular acceleration?
4. How would the results change if the gear had an initial angular velocity other than zero?
5. What is the significance of the \(\pi\) factor in angular measurements?
6. How would you find the torque required to achieve this angular acceleration?
7. How do different materials affect the angular acceleration of a rotating object?
8. Can you derive the equations of motion for rotational kinematics from the linear kinematics equations?

**Tip:** When dealing with rotational motion, always ensure to convert revolutions to radians for consistency in units.

Learn how to determine constant angular acceleration and time required for a gear to achieve an angular velocity of 30 rad/s after rotating 20 revolutions starting from rest. Detailed step-by-step solution provided.

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