Case 2: elastic collision, hinged rod

Same hinged rod, but now the collision is elastic: the projectile bounces elastically off the end of the rod instead of embedding itself. Only one condition changes, and the whole balance changes with it.

• Angular momentum about the hinge is conserved (as in case 1, same reason: the hinge’s reaction has zero arm).
• Momentum is not conserved (as in case 1: there is the hinge’s impulsive reaction).
• Kinetic energy is conserved: the collision is elastic, $E_\text{termica} = 0$.

Now there are two unknowns: $\omega$ (angular velocity of the rod) and $v_B$ (velocity of the projectile after the collision, with sign). So two equations are needed: conservation of $L$ and of $E_\text{cin}$.

Angular momentum equation (about the hinge):

$m_p v_A L = m_p v_B L + I_\text{asta}^\text{cardine}\,\omega$

with $I_\text{asta}^\text{cardine} = \tfrac{1}{3}ML^2 = 5{,}33\;\mathrm{kg\,m^2}$ (only the rod: the projectile no longer embeds itself).

Kinetic energy equation:

$\tfrac{1}{2}m_p v_A^2 = \tfrac{1}{2}m_p v_B^2 + \tfrac{1}{2}I_\text{asta}^\text{cardine}\,\omega^2$

Summary of case 2

Two unknowns ($v_B$, $\omega$), two equations ($L$, $E_\text{cin}$): a quadratic system. It is solved like a generalised 1D elastic collision. As always in elastic collisions, the system has two solutions: the trivial one (no collision, $v_B = v_A$, $\omega = 0$) and the physical one.

Why there's no conservation of $\vec{p}$

Even in an elastic collision, if the rod is hinged the hinge exerts an impulsive constraint reaction that transmits an external impulse: total momentum is not conserved. You cannot write $m_p v_A = m_p v_B + \text{something}$. This is the most treacherous mistake in this problem.

Collegamenti

Argomenti: Dinamica rotazionale Concetti: Conservazione del momento angolare · Urto elastico Competenze: Scelta del polo Oggetti: Asta

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