Disc with an unwinding rope

Another classic example of conservation of angular momentum: a disc rotating about a central pivot, with a package hanging from a rope that progressively unwinds from the rim, moving away from the centre.

Choosing the centre of the disc as the pole, the pivot’s reaction and the (radial) tension produce no moment: the system’s total angular momentum is conserved. With the constraint $v = \omega r$ between the package’s velocity and the angular velocity, the conservation law reads

$\left(\tfrac{1}{2}m_d R^2 + m_p r_A^2\right)\omega_A = \left(\tfrac{1}{2}m_d R^2 + m_p r_B^2\right)\omega_B$

Numerically, with $m_d = 4\;\mathrm{kg}$, $R = 1\;\mathrm{m}$, $m_p = 7\;\mathrm{kg}$, $r_A = 1\;\mathrm{m}$, $r_B = 2\;\mathrm{m}$ and $\omega_A = 1\;\mathrm{rad/s}$, one gets $\omega_B = 0{,}3\;\mathrm{rad/s}$.

The system slows down: as the package moves away from the centre, its contribution $m_p r^2$ to the moment of inertia grows (from $7\cdot 1^2 = 7$ to $7\cdot 2^2 = 28$), and since $L = I\omega$ stays constant, the angular velocity must fall. It is the same physics as the skater, run in reverse: spreading out the mass distribution decreases $\omega$. The numerical solution is in the dedicated problem.

Collegamenti

Argomenti: Dinamica rotazionale Concetti: Conservazione del momento angolare · Momento d’inerzia Oggetti: Disco rotante

Esercizi collegati: Problema — Vero o falso su momento angolare e inerzia · Problema — Disco con fune che si svolge · Problema — Disco rotante colpito tangenzialmente