In this problem, you're analyzing the order in which three objects (a hollow sphere, a solid sphere, and a hoop) reach the bottom of a ramp when released from rest. All objects have the same mass $M$ and radius $R$ and roll down without slipping. The problem requires you to rank their descent times based on rotational inertia and explain your reasoning without detailed calculations.

Here's a breakdown of the key points:

1. Moment of Inertia: The moment of inertia ($I$) depends on the shape and mass distribution. The more mass is concentrated away from the axis of rotation, the larger the moment of inertia.

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

2. Kinetic Energy and Acceleration: Since all objects roll without slipping, they each have both translational and rotational kinetic energy. The total energy of an object rolling down a slope is partitioned between translational kinetic energy ($\frac{1}{2}mv^2$) and rotational kinetic energy ($\frac{1}{2}I\omega^2$).

3. Reasoning for Ranking:

   • The solid sphere has the lowest moment of inertia relative to its mass, meaning more of its potential energy will go into translational kinetic energy, allowing it to accelerate faster down the ramp.
   • The hollow sphere has a higher moment of inertia than the solid sphere but lower than the hoop, so it comes next.
   • The hoop, with the highest moment of inertia, will have the least acceleration and will reach the bottom last.

Ranking:

1. Solid Sphere
2. Hollow Sphere
3. Hoop

The reasoning aligns with your answer, which correctly ranks the solid sphere first, then the hollow sphere, and finally the hoop.

Do you have questions or would you like further details on any part of this explanation?

Here are five related questions that may deepen your understanding:

1. How does the angle $\theta$ of the ramp affect the acceleration of each object?
2. What would happen if the objects had different masses but the same shapes and sizes?
3. How would the outcome change if the objects were allowed to slide without rolling?
4. Can we rank these objects if they had different radii but the same mass?
5. What role does friction play in ensuring that these objects roll without slipping?

Tip: When analyzing rolling motion, always remember to consider both translational and rotational kinetic energy to determine the object’s overall energy distribution and speed.