Double-radius pulley

A richer case than the ordinary massive pulley is the pulley with two different radii $r_1$ and $r_2$, on whose inner and outer rim two separate ropes wind, each with its own hanging mass. This is the principle behind the windlass and the differential pulley block.

The key idea is that the pulley is a single rigid body: it rotates with one single angular acceleration $\alpha$. But the two winding points sit at different distances from the axis, so the two masses have different linear accelerations, related to $\alpha$ by the geometric constraint:

$a_i = \alpha\,r_i$

Warning — different accelerations, same $\alpha$

With different radii the linear accelerations of the two masses are different: $a_1 \neq a_2$. What is unique is the pulley’s angular acceleration. The link between the two worlds is always $a_i = \alpha\,r_i$: getting this constraint wrong is the most common mistake in these problems.

Combining Newton’s second law for the two masses with the pulley’s rotational equation $t_2\,r_2 - t_1\,r_1 = I\,\alpha$ and substituting the constraint, the angular acceleration can be isolated:

$\ev{\alpha = \frac{m_2\,g\,r_2 - m_1\,g\,r_1}{I + m_1\,r_1^2 + m_2\,r_2^2}}$

The sign of the numerator decides the direction of rotation: the mass that acts with the greater moment wins, i.e. the greater product $m\,g\,r$ — not necessarily the heavier mass, but the one hanging at the more favourable radius. This is the secret of the pulley block: a small mass on a large radius can lift a large mass on a small radius. The denominator has the structure of an effective moment of inertia for the system: that of the pulley plus the contribution $m_i r_i^2$ of each mass, as if it were a point mass at distance $r_i$ from the axis.

Collegamenti

Argomenti: Dinamica rotazionale Concetti: Momento di una forza · Accelerazione angolare Competenze: Impostazione simbolica Oggetti: Puleggia

Esercizi collegati: Problema — Puleggia a doppio raggio · Problema — Dato α, trova il momento · Problema — Tre regimi di ω(t)