Bodies that roll

When a body rolls without slipping, its motion is simultaneously translation (the centre of mass moves forward) and rotation (the body spins on itself). The two are not independent: they are linked by the rolling constraint, which arises from the fact that the point of contact with the surface does not slide.

Rolling constraint

$v = \omega\,r$ The velocity of the centre of mass is related to the angular velocity by the radius. Differentiating: $a = \alpha\,r$.

The energy consequence is profound. The kinetic energy of a rolling body splits into two contributions: the translational one of the centre of mass and the rotational one about the centre of mass (this is König’s theorem applied to rolling).

$E_\text{cin} = \underbrace{\tfrac{1}{2}m v^2}_{\text{translation}} + \underbrace{\tfrac{1}{2}I_\text{CM}\,\omega^2}_{\text{rotation}}$

Since part of the available energy ends up in rotation, a body that rolls down a slope arrives at the bottom slower than a block that slides without friction from the same height: the block puts all its potential energy into translation, whereas the rolling body has to split it.

Using the constraint $\omega = v/r$ and conservation of energy $mgh = \tfrac{1}{2}mv^2 + \tfrac{1}{2}I_\text{CM}\,\omega^2$, the velocity at the bottom can be found. The larger the moment of inertia (mass further from the axis), the slower the body.

Velocity at the bottom of the slope (starting from rest, height $h$)

Body	Velocity at the base
Block (slides, no rotation)	$v = \sqrt{2gh}$
Solid cylinder (rolls)	$v = \sqrt{\tfrac{4}{3}gh}$
Solid sphere (rolls)	$v = \sqrt{\tfrac{10}{7}gh}$

The further the mass is from the axis, the slower the body: in a downhill race the solid sphere wins, the ring loses.

The acceleration along the slope can be written compactly as

$a = \frac{g\sin\theta}{1 + I_\text{CM}/(mr^2)}$

which shows the hierarchy at a glance: the factor $I_\text{CM}/(mr^2)$ equals $\tfrac{2}{5}$ for the solid sphere, $\tfrac{1}{2}$ for the solid cylinder, $1$ for the ring — and it does not depend on the radius, so that two spheres of different sizes arrive together. The detailed numerical solution is in the worked exercise on the sphere.

Collegamenti

Argomenti: Dinamica rotazionale Concetti: Rotolamento · Energia cinetica rotazionale · Conservazione dell’energia meccanica Oggetti: Sfera che rotola · Piano inclinato

Esercizi collegati: Esercizio svolto — sfera che rotola giù da un piano · Problema — Sfera che rotola, velocità alla base · Problema — Asta incernierata che cade dall’orizzontale