Case 3: elastic collision, rod free on a rail

Now the rod is free to slide on a frictionless rail: no fixed hinge. This radically changes the balance, because the impulsive constraint reaction that ruined conservation of momentum disappears.

• Momentum is conserved: there are no impulsive external horizontal forces.
• Kinetic energy is conserved: elastic collision.
• Angular momentum about any fixed pole is conserved: no impulsive external force, hence no external moment (weight is balanced by the rail’s reaction, and both are negligible during the instantaneous collision).

With all three conservation laws available, there are three unknowns:

• $v_B$ = final velocity of the projectile;
• $v_\text{CM}$ = velocity of the rod’s centre of mass (translation);
• $\omega$ = angular velocity of the rod (rotation about its CM).

And three equations: conservation of $p_x$, $L$, $E_\text{cin}$.

Momentum:

$m_p v_A = m_p v_B + M v_\text{CM}$

Angular momentum (about a fixed pole, for example the initial position of the rod’s CM):

$m_p v_A\cdot\tfrac{L}{2} = m_p v_B\cdot\tfrac{L}{2} + I_\text{CM}\,\omega$

with $I_\text{CM} = \tfrac{1}{12}ML^2$ — the rod’s moment of inertia about its own CM, because here the rod is not hinged.

Kinetic energy:

$\tfrac{1}{2}m_p v_A^2 = \tfrac{1}{2}m_p v_B^2 + \underbrace{\tfrac{1}{2}M v_\text{CM}^2}_{E_\text{cin,CM}} + \underbrace{\tfrac{1}{2}I_\text{CM}\omega^2}_{E_\text{cin,int}}$

König's theorem in action

In case 3 the rod’s kinetic energy splits into two contributions (König’s theorem): the translational one of the centre of mass plus the internal rotational one about the CM. This is the decomposition that keeps the accounting consistent and that distinguishes the free body from the hinged body. With three unknowns and three equations, this is the most complete of the three cases.

Collegamenti

Argomenti: Dinamica rotazionale Concetti: Conservazione del momento angolare · Conservazione della quantità di moto · Urto elastico Metodi: Teorema di König Oggetti: Asta

Esercizi collegati: Problema — Urto elastico 2D su un biliardo · Urto elastico tra due biglie · Vero o falso sugli urti