The problem involves a rolling disk with mass $m$ and radius $R$, subject to a constant force $F_0$ applied tangentially at its edge. The disk begins at rest, and we are asked to determine the angular velocity of the disk over time.

Steps to Solve:

1. Newton's Second Law for Rotation: The torque $\tau$ about the center of mass of the disk due to the applied force is: $\tau = F_0 \cdot R$ where $R$ is the radius of the disk.

   According to Newton's second law for rotation: $\tau = I \cdot \alpha$ where $I$ is the moment of inertia of the disk, and $\alpha$ is the angular acceleration.

   For a solid disk, the moment of inertia about its center is: $I = \frac{1}{2} m R^2$

   Substituting $\tau$ and $I$: $F_0 \cdot R = \frac{1}{2} m R^2 \cdot \alpha$

   Solving for $\alpha$: $\alpha = \frac{2 F_0}{m R}$

2. Relating Angular Velocity to Angular Acceleration: The angular velocity $\omega$ of the disk is related to angular acceleration $\alpha$ through time $t$: $\omega = \alpha \cdot t$

   Substituting $\alpha$: $\omega = \frac{2 F_0}{m R} \cdot t$

   Therefore, the angular velocity of the disk as a function of time is: $\omega(t) = \frac{2 F_0}{m R} \cdot t$

Final Answer:

The angular velocity of the disk at any time $t$ is: $\omega(t) = \frac{2 F_0}{m R} \cdot t$

Let me know if you would like further clarifications or have additional questions! 😊

5 Related Questions:

1. What is the linear acceleration of the center of mass of the disk?
2. How would the result change if the disk were a hollow cylinder instead of a solid disk?
3. What is the relationship between the linear velocity of the disk's center and its angular velocity?
4. How does the friction force at the contact point of the disk affect its motion?
5. If the force $F_0$ were applied at the center of the disk, how would the motion differ?

Tip:

Always ensure you know the shape of the object (solid or hollow) to use the correct moment of inertia in rotational motion problems!