Newton's second law for rotation

Just as in translational dynamics the net force causes an acceleration, in rotational dynamics it is the moment (or torque) that causes an angular acceleration. The equation linking the two quantities is the perfect analogue of $\sum F = ma$.

Fundamental law of rotational dynamics

$\ev{\sum M = I\,\alpha}$ where $\sum M$ is the sum of the moments of the forces about the axis of rotation and $\alpha$ is the angular acceleration.

Reading the equation component by component: the moment plays the role of the force, the moment of inertia that of the mass, the angular acceleration that of the linear acceleration. A large moment produces a lot of angular acceleration; a large moment of inertia damps it. To set a heavy flywheel spinning you need a sizeable moment, exactly as accelerating a lorry needs a sizeable force.

The moment of a force about an axis is calculated as $M = F\,r\sin\theta$, where $r$ is the distance from the point of application to the axis and $\theta$ the angle between the force and the arm: only the component of the force perpendicular to the arm makes the body rotate, while the radial component is ineffective (it pushes towards the axis or away from it, without producing rotation).

The inverse problem

The equation can also be read the other way round: if you know the angular acceleration (for example from $\alpha = \Delta\omega/\Delta t$), you can work out the net moment applied, $\sum M = I\,\alpha$. This is the method used to measure a motor’s torque experimentally, by observing how quickly it spins up a flywheel of known moment of inertia.

Collegamenti

Argomenti: Dinamica rotazionale Concetti: Momento di una forza · Momento d’inerzia · Accelerazione angolare Competenze: Impostazione simbolica

Esercizi collegati: Problema — Vero o falso su momento angolare e inerzia · Problema — Atwood con puleggia massiva · Problema — Puleggia a doppio raggio