The problem appears to be from a physics textbook and involves rotational motion, specifically a uniform disk of radius $R$ and mass $M$ that is hanging by a string wound around its edge. The goal is to solve for:

1. The tension in the string ($T$),
2. The acceleration of the center of mass ($a_{cm}$),
3. The angular acceleration ($\alpha$) of the disk.

Explanation:

Concepts:

• Torque and Rotational Dynamics: Torque ($\tau$) is related to angular acceleration ($\alpha$) by: $\tau = I \alpha$ where $I$ is the moment of inertia of the disk about its center. For a uniform disk: $I = \frac{1}{2} M R^2$

• Force Analysis: The forces acting on the disk are:

  • Gravitational force: $F_g = Mg$,
  • Tension in the string: $T$.

• The linear acceleration $a_{cm}$ of the center of mass is related to the angular acceleration by: $a_{cm} = R \alpha$

Equations:

1. Force Equation (translation of the center of mass): $Mg - T = Ma_{cm}$

2. Torque Equation (rotation about the center): $\tau = T R = I \alpha$ Substituting $I = \frac{1}{2} M R^2$: $T R = \frac{1}{2} M R^2 \alpha$

   Simplify to: $T = \frac{1}{2} M R \alpha$

3. Relation between $a_{cm}$ and $\alpha$: $a_{cm} = R \alpha \implies \alpha = \frac{a_{cm}}{R}$

Solving the System:

1. From the torque equation: $T = \frac{1}{2} M a_{cm}$

2. Substitute $T = \frac{1}{2} M a_{cm}$ into the force equation: $Mg - \frac{1}{2} M a_{cm} = M a_{cm}$

   Simplify: $Mg = \frac{3}{2} M a_{cm}$

   Solve for $a_{cm}$: $a_{cm} = \frac{2}{3} g$

3. Angular acceleration $\alpha$: $\alpha = \frac{a_{cm}}{R} = \frac{\frac{2}{3} g}{R} = \frac{2g}{3R}$

4. Tension $T$: $T = \frac{1}{2} M a_{cm} = \frac{1}{2} M \cdot \frac{2}{3} g = \frac{1}{3} Mg$

Final Answers:

1. Tension in the string: $T = \frac{1}{3} Mg$,
2. Acceleration of the center of mass: $a_{cm} = \frac{2}{3} g$,
3. Angular acceleration: $\alpha = \frac{2g}{3R}$.

Would you like a deeper explanation or derivation of any specific step? Let me know!

Related Questions:

1. What would change if the object were a ring instead of a disk?
2. How does the moment of inertia influence the acceleration?
3. Can you derive these equations for a non-uniform object?
4. What happens if the string has mass? How does that affect the tension?
5. How is energy conserved in this system?

Tip:

Always check if the rotational and linear dynamics are properly coupled through $a_{cm} = R \alpha$. It ensures consistency in solving such problems.