Rigid bodies and rotational equilibrium

So far we have treated point-like objects, in which all forces are applied at the same point. But a rod, a beam or a ladder have an extent, and forces act on different points: this opens up the possibility that the body rotates even when its centre stays still. A richer model is therefore needed.

Principle — Rigid body

A rigid body is an extended (non point-like) object in which the distance between every pair of points is fixed and unchanging in time. Examples: a steel bar, a closed book, a brick. Counter-examples: a rope, an elastic band, a gas.

For a rigid body in static equilibrium it is not enough that the sum of the forces be zero. Imagine pushing the right edge of a book upward and the left edge downward with two equal and opposite forces: the resultant is zero, the centre of the book does not translate, and yet the book starts to rotate. To prevent this, the tendency to make the body spin must also be cancelled out, i.e. the sum of the moments of the forces.

Principle — Equilibrium of a rigid body

A rigid body is in static equilibrium if and only if both conditions are satisfied: $\begin{aligned} \sum_i \vv{F}_i = \vv{0} \quad \text{(translational equilibrium)} \\ \sum_i M_i = 0 \quad \text{(rotational equilibrium, about any pole)} \end{aligned}$

The moment (or torque) of a force $\vv{F}$ about a pole $O$ measures its ability to rotate the body about that pole, and is

$M = \pm\,\abs{\vv{F}}\cdot b$

where $b$ is the distance from the pole to the line of action of the force, called the lever arm. The lever arm is not the distance from the point of application, but from the line of action: a force pointing straight at the pole has zero lever arm and causes no rotation at all. The sign is positive if the force tends to rotate anticlockwise about the pole, negative if clockwise.

Choose the pole wisely

The second condition holds about any pole: we are free to choose the one that suits us. A pole placed at the point of application of an unknown force removes it from the moment equation, because that force will have zero lever arm. With a shrewd choice one can go from three unknowns to just one: the choice of pole is free but decisive for speeding up the working.

The concept of moment is central to much of mechanics: we shall meet it again, and go further into it, when studying the centre of mass of extended bodies and, later, the dynamics of rotation, where moments govern not only equilibrium but genuine rotational motion.

Collegamenti

Argomenti: Statica ed equilibrio Concetti: Momento di una forza Competenze: Bilancio dei momenti · Scelta del polo Oggetti: Corpo rigido

Esercizi collegati: Esercizio svolto — Sistema con due masse e puleggia · Esercizio svolto — Giostra quadrata · Esercizio svolto — Scala appoggiata a un muro liscio