A ring, disc and solid sphere are left from the top of an incline which has sufficient friction for pure rolling. Which will reach the bottom first, if they all have the same mass ‘M’ & radius ‘R’ ?

Solid sphere

The equation of motions will be: Mgsinθ – f = Ma,

... where f is the friction & a is acceleration.

Also, fR = Ia/R.

From this equation ‘f’ will be substituted in the first equation: Mgsinθ – Ia/R² = Ma.

∴ a = Mgsinθ / (I/R²+ M).

Thus, more the value of I lesser will be the acceleration.

Moment of inertia of ring = MR²

Moment of inertia of disc= MR²/2

Moment of inertia of solid sphere = 2MR²/5

The solid sphere has the least moment of inertia, so it will have the maximum acceleration & hence it will take the least time. Note that if there was no friction their accelerations would be the same and they would reach together.