Pulleys with mass

In elementary problems the pulley is ideal: massless, it merely redirects the rope. But a real pulley has mass, and then it no longer passively transmits the tension: to be set spinning it needs a net moment, and that moment arises from the difference in tension on the two sides of the rope.

Principle — Massive pulley

For a pulley of moment of inertia $I_p$ and radius $R$: $\ev{(t_1 - t_2)\,R = I_p\,\alpha}$ where $t_1$ and $t_2$ are the tensions on the two sides and $\alpha = a/R$ (rolling constraint of the rope on the groove).

The term $(t_1 - t_2)R$ is precisely the resultant moment of the two tensions about the pulley’s axis: they act with the same arm $R$ but in opposite directions, so their rotational effect is the difference.

Why the tensions are equal with an ideal pulley

With a massless pulley $I_p = 0$, and so $(t_1 - t_2)R = 0 \Rightarrow t_1 = t_2$: the tension is the same on both sides. Only when the pulley has mass do the two tensions differ, and that difference is what makes it rotate. As a result a massive pulley “slows down” the system: the acceleration $a$ is smaller than with an ideal pulley, because part of the masses’ action goes into spinning up the pulley itself.

Atwood machine with a massive pulley: the tensions on the two sides of the rope are different. The difference $t_1 - t_2$ generates the moment that spins the pulley.

To solve a system of this kind you write three equations: Newton’s second law for each hanging mass and the pulley’s rotational equation, linked by the constraint $a = \alpha R$. The full numerical solution is in the dedicated exercise.

Collegamenti

Argomenti: Dinamica rotazionale Concetti: Momento di una forza · Momento d’inerzia Oggetti: Puleggia · Macchina di Atwood

Esercizi collegati: Problema — Vero o falso su momento angolare e inerzia · Problema — Atwood con puleggia massiva · Problema — Carrucola a due masse