In this problem, you're analyzing three objects (a hollow sphere, a solid sphere, and a hoop) rolling down an inclined plane without slipping. Given that they have the same mass $M$ and radius $R$, the objective is to rank them based on the order they reach the bottom of the ramp.

Explanation of the Physics

The key principle here is that each object will have a different moment of inertia, which affects its rotational and translational kinetic energy distribution as it rolls down the ramp. Objects with lower moments of inertia relative to their mass will generally accelerate faster because they convert less of their potential energy into rotational kinetic energy, leaving more for translational kinetic energy, which increases their linear speed.

1. Moment of Inertia:

   • Solid Sphere: $I = \frac{2}{5}MR^2$
   • Hollow Sphere: $I = \frac{2}{3}MR^2$
   • Hoop: $I = MR^2$

   The smaller the moment of inertia, the less rotational energy is required for a given amount of translational energy, allowing the object to reach the bottom faster.

2. Order of Speed (from fastest to slowest) Based on Moment of Inertia:

   • Solid Sphere (smallest $I$) will reach the bottom first.
   • Hollow Sphere comes next.
   • Hoop (largest $I$) will be the slowest.

Justification

In the student's response, they correctly noted that the solid sphere has a lower moment of inertia relative to its mass, which allows it to convert more potential energy into translational kinetic energy rather than rotational. This means it can achieve a higher velocity and therefore reaches the bottom first.

Ranking

1. Uniform Solid Sphere (Rank 1)
2. Hollow Sphere (Rank 2)
3. Hoop (Rank 3)

Your answer and justification align well with the physics principles. The moment of inertia difference among these shapes directly influences their acceleration down the ramp, with lower moments of inertia resulting in faster descent times.

Would you like further details on the derivation of the kinetic energy for each object or have any questions about the concepts?

Here are five related questions that could deepen your understanding:

1. How would the ranking change if the masses of the objects were different?
2. What if the ramp was frictionless—how would that affect the rankings?
3. Why does the hoop, with the highest moment of inertia, roll the slowest?
4. How does the incline angle $\theta$ affect the speed of each object?
5. Can you calculate the exact speeds of each object at the bottom if the length $L$ is given?

Tip: Always check the moment of inertia when comparing rotational motion; lower moments allow for faster speeds in rolling scenarios.