Problem — Solid or hollow, which wins

Problem

Two cylinders of the same mass and radius roll without slipping down the same inclined plane. One is solid, the other is hollow (a tube). Which arrives first, and why?

Solution

The rolling acceleration is $a = \dfrac{g\sin\theta}{1 + I/(MR^2)}$: whoever has the smaller factor $I/(MR^2)$ wins, i.e. whoever has the mass closer to the axis.

• Solid cylinder ($I = \tfrac{1}{2}MR^2$): $a = \dfrac{g\sin\theta}{1{,}5}$;
• Hollow cylinder ($I \approx MR^2$): $a = \dfrac{g\sin\theta}{2}$.

The solid one has the greater acceleration, so it arrives first. The hollow one has all its mass far from the axis: a larger moment of inertia to “load”, which slows its descent. $\ev{\text{the solid cylinder wins}}$

Links

Topics: Rotational dynamics Concepts: Rolling · Moment of inertia Objects: Rotating disc · Inclined plane