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To solve this problem, let’s break down the information provided:

### Given Data:
1. **Radius of the disk**: \( R = 0.400 \, \text{m} \)
2. **Force applied**: \( F = 6.00 \, \text{N} \)
3. **Time duration**: \( t = 4.00 \, \text{s} \)
4. **Final angular velocity**: \( \omega = 12.0 \, \text{rad/s} \)
5. **Moment of inertia of a uniform disk**: \( I = \frac{1}{2} m R^2 \)

### Steps to Solve:

1. **Calculate the Torque** (\( \tau \)):
   - Torque is given by \( \tau = F \cdot R \).
   - Substitute \( F = 6.00 \, \text{N} \) and \( R = 0.400 \, \text{m} \):
     \[
     \tau = 6.00 \times 0.400 = 2.4 \, \text{Nm}
     \]

2. **Calculate Angular Acceleration** (\( \alpha \)):
   - Torque is also related to angular acceleration by \( \tau = I \alpha \).
   - Substitute \( I = \frac{1}{2} m R^2 \):
     \[
     \tau = \frac{1}{2} m R^2 \cdot \alpha
     \]
   - Solving for \( \alpha \):
     \[
     \alpha = \frac{2 \tau}{m R^2}
     \]

3. **Relate Angular Acceleration and Final Angular Velocity**:
   - Using \( \omega = \alpha t \) to find \( \alpha \):
     \[
     \alpha = \frac{\omega}{t} = \frac{12.0}{4.0} = 3.0 \, \text{rad/s}^2
     \]

4. **Solve for the Mass \( m \)**:
   - Substitute \( \alpha = 3.0 \, \text{rad/s}^2 \) and \( \tau = 2.4 \, \text{Nm} \):
     \[
     2.4 = \frac{1}{2} m (0.4)^2 \cdot 3.0
     \]
     \[
     m = \frac{2.4}{0.24} = 10.0 \, \text{kg}
     \]

### Answer:
The mass of the disk is **10.0 kg** (Option C).

Would you like further explanation on any of the steps or have additional questions?

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### Additional Questions:
1. What is the relationship between torque and angular acceleration?
2. How would the mass change if the applied force were different?
3. What would be the effect of a larger radius on the angular acceleration?
4. How does the time duration affect the final angular velocity?
5. Why is the moment of inertia different for various shapes?

### Tip:
Remember, torque and force are related but act differently in rotational and linear dynamics.

A uniform disk with radius R = 0.400 m is mounted on a horizontal frictionless axle at its center. A light string is wrapped around the disk. The disk is initially at rest and then a constant force of 6.00 N is applied to the free end of the rope. After the force has been applied for 4.00 s, the disk is rotating with angular velocity 12.0 rad/s. What is the mass of the disk?

Learn how to find the mass of a rotating disk when a force of 6.00 N is applied to its edge. Given the radius, applied force, time duration, and final angular velocity, this solution explores rotational dynamics principles like torque, angular acceleration, and moment of inertia. Suitable for undergraduate physics students.

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