Question 1

Which body reaches the bottom of an inclined plane first when rolling without slipping?

Accepted Answer

The body with the smallest shape factor k = I/MR². The order is fixed: solid sphere (k = 2/5) first, then solid disc (k = 1/2), then hollow sphere (k = 2/3), and the ring (k = 1) last — regardless of their masses or radii.

Question 2

Does mass affect the acceleration of a body rolling down an incline?

Accepted Answer

No. For rolling without slipping, a = g·sinθ/(1 + k), where k = I/MR² is dimensionless. Substituting I = kMR² makes both M and R cancel, so a heavy body and a light body of the same shape accelerate identically.

Question 3

What is the radius of gyration?

Accepted Answer

The radius of gyration K is defined by I = MK². It is the distance from the rotation axis at which the entire mass could be concentrated to give the same moment of inertia. Note that k = I/MR² = (K/R)², so the race outcome depends only on the ratio K/R.

Question 4

Why does a ring roll slowest down an incline?

Accepted Answer

A ring has all its mass at the rim, farthest from the rotation axis, giving k = 1 — the maximum shape factor. Its acceleration a = g·sinθ/2 is the smallest, so more of its potential energy goes into rotation and less into forward speed.

Question 5

What minimum friction is needed for rolling without slipping down an incline?

Accepted Answer

The required friction force is f = kMg·sinθ/(1 + k), so the surface must provide at least μ_min = k·tanθ/(1 + k). For example, a solid sphere on a 30° incline needs μ_min ≥ (2/5)(tan30°)/(7/5) ≈ 0.165.

Rolling Race: Moment of Inertia — The Race You Cannot Rig

Practice

Learned it? Now prove it under the clock.