If a solid sphere, disc, and hollow cylinder of the same outer radius R and mass M are allowed to roll down on an inclined plane from the same height, then which one of these will reach the bottom first?

Concept:

• Moment of Inertia (I): Moment of inertia is the inertia of a rotating body concerning its rotation.
  • Moment of inertia is also known as rotational inertia.

I = Σ MR2

Where I is the moment of inertia of the body, M is the mass of the body, and R is the radius.

• Moment of Inertia of a circular hollow cylinder 

​I_C = ½ M (R02 + Ri2)

​​⇒ IC = ½ M (R2 + Ri2)

• Moment of Inertia of a circular disc ID =  ½ M R2
• Moment of Inertia of a solid sphere Is = 2/5 M R2

Explanation:

• The relation of rotational inertia among a circular disc, a circular ring, and a solid sphere is

IC > ID > Is, that is

Hollow cylinder > circular disc > solid sphere;

So, the sphere will reach the bottom first.