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Let's go through each part of the problem to see where errors might be occurring and confirm the calculations.

### Given Information
- **Mass of disk, \( m = 145.0 \, \text{kg} \)**
- **Radius of disk, \( r = 1.8 \, \text{m} \)**
- **Force applied, \( F = 40.0 \, \text{N} \)**
- **Time of application, \( t = 56.0 \, \text{s} \)**

#### Part (a): Angular Acceleration \( \alpha \)

The angular acceleration can be calculated using the formula:
\[
\alpha = \frac{\tau}{I}
\]
where:
- \( \tau \) is the torque, given by \( \tau = F \times r \).
- \( I \) is the moment of inertia of a solid disk, \( I = \frac{1}{2} m r^2 \).

1. **Calculate Torque \( \tau \):**
   \[
   \tau = F \times r = 40.0 \, \text{N} \times 1.8 \, \text{m} = 72.0 \, \text{Nm}
   \]

2. **Calculate Moment of Inertia \( I \):**
   \[
   I = \frac{1}{2} m r^2 = \frac{1}{2} \times 145.0 \, \text{kg} \times (1.8 \, \text{m})^2 = 234.9 \, \text{kg} \cdot \text{m}^2
   \]

3. **Calculate Angular Acceleration \( \alpha \):**
   \[
   \alpha = \frac{\tau}{I} = \frac{72.0}{234.9} \approx 0.3065 \, \text{rad/s}^2
   \]
   This matches the answer provided in the solution, \( 0.3065 \, \text{rad/s}^2 \, \hat{k} \).

#### Part (b): Final Angular Velocity \( \omega \)

The final angular velocity \( \omega \) can be found using the formula:
\[
\omega = \alpha \times t
\]
where \( \alpha = 0.3065 \, \text{rad/s}^2 \) and \( t = 56.0 \, \text{s} \).

1. **Calculate \( \omega \):**
   \[
   \omega = 0.3065 \times 56.0 \approx 17.564 \, \text{rad/s}
   \]

However, this answer is marked incorrect. The reason might be due to rounding errors or requiring more precision in intermediate steps.

#### Part (c): Translational Speed \( v \) at the Edge

The translational speed at the edge can be calculated using:
\[
v = \omega \times r
\]

1. **Using \( \omega \approx 17.564 \, \text{rad/s} \):**
   \[
   v = 17.564 \times 1.8 \approx 31.6152 \, \text{m/s}
   \]

This answer is also close but might require more precision. To improve accuracy, consider using all intermediate steps with more decimal places.

Would you like detailed calculations with extended precision, or help on any particular section? 

### Related Questions
1. How does changing the force affect the angular acceleration?
2. What would happen to the final angular velocity if the mass of the disk increased?
3. How would a change in the radius affect the moment of inertia and torque?
4. Why might rounding errors impact the final answer?
5. Can we calculate the kinetic energy of the rotating disk after 56 seconds?

### Tip:
Always carry extra significant figures in intermediate calculations to minimize rounding errors in the final answer.

A large stone disk is viewed from above and is initially at rest. The disk has a mass of 145.0 kg and a radius of 1.800 m. A constant force of 40.0 N is applied tangent to the edge of the disk for 56.0 s, causing the disk to spin around the z-axis. Calculate the angular acceleration, final angular velocity, and translational speed at the edge of the stone after 56.0 s.

Learn how to calculate the angular acceleration, final angular velocity, and translational speed of a rotating stone disk with a mass of 145.0 kg, a radius of 1.8 m, and a tangential force of 40.0 N applied for 56.0 s. This problem covers key rotational motion concepts including torque, moment of inertia, and angular motion.

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