This problem involves comparing the angular speeds ($\omega_1, \omega_2, \omega_3$) of three rolling objects: a solid cylinder, a solid sphere, and a thin-walled hollow cylinder, all having the same mass, radius, and angular momentum about their center of mass.

Key Concepts:

1. Angular Momentum: $L = I \omega$, where $I$ is the moment of inertia and $\omega$ is the angular velocity.
2. Moments of inertia for the objects:
   • Solid Cylinder: $I = \frac{1}{2} m R^2$
   • Solid Sphere: $I = \frac{2}{5} m R^2$
   • Thin-walled Hollow Cylinder: $I = m R^2$
3. Since the angular momentum ($L$) is the same for all three objects, the relationship between $\omega$ and $I$ is $\omega = \frac{L}{I}$.

Step-by-Step Analysis:

• The object with the largest moment of inertia will have the smallest angular velocity ($\omega$), and vice versa, since $\omega$ is inversely proportional to $I$.

Compare Moments of Inertia:

• $I_{\text{solid sphere}} = \frac{2}{5} m R^2$ (Smallest)
• $I_{\text{solid cylinder}} = \frac{1}{2} m R^2$ (Middle)
• $I_{\text{hollow cylinder}} = m R^2$ (Largest)

Thus: $\omega_{\text{hollow cylinder}} < \omega_{\text{solid cylinder}} < \omega_{\text{solid sphere}}$ or equivalently: $\omega_3 < \omega_1 < \omega_2$

Correct Answer:

The correct option is: $\boxed{\omega_3 < \omega_1 < \omega_2}$

Let me know if you'd like further clarification or details! Here are five related questions to expand your understanding:

1. How does the moment of inertia depend on the shape of an object?
2. What happens to angular velocity if the mass or radius of the objects changes?
3. How does angular momentum conservation apply in this scenario?
4. Can you explain why the thin-walled hollow cylinder has the largest moment of inertia?
5. How would rolling friction affect the angular velocities of these objects?

Tip: When solving rotational dynamics problems, always consider the relationship between moment of inertia and angular velocity—they are often inversely related!