Summary of the conservation laws

Comparing the three cases in a single table, a common thread emerges: every time we add a conservation law, we gain one more unknown that we can determine.

What is conserved in the three cases

	Case 1 inelastic hinged	Case 2 elastic hinged	Case 3 elastic free
Conservation of $\vec{p}$	no	no	yes
Conservation of $L$	yes	yes	yes
Conservation of $E_\text{cin}$	no	yes	yes
Unknowns	1 ($\omega$)	2 ($v_B,\omega$)	3 ($v_B,v_\text{CM},\omega$)
Equations	1 ($L$)	2 ($L$, $E_\text{cin}$)	3 ($\vec{p}, L, E_\text{cin}$)

Angular momentum about the right pole is conserved in all three cases: it is the most robust quantity, the one that survives even the presence of the hinge. Momentum, on the other hand, is the most fragile, because the fixed constraint destroys it; it reappears only when the rod is free.

The unknowns–equations balance

In all three cases the number of unknowns matches exactly the number of available conservation laws. This is not a coincidence: it is the consistency check that guarantees the problem has a solution. If, on counting, you find more unknowns than equations, you have forgotten a conservation law or set up the problem wrongly.

Collegamenti

Argomenti: Dinamica rotazionale Concetti: Conservazione del momento angolare · Conservazione della quantità di moto Competenze: Ragionamento di ranking

Esercizi collegati: Problema — La pattinatrice che si stringe · Esercizio svolto — proiettile contro asta incernierata · Problema — Perché il pattinatore ruota più veloce