A projectile hitting a rod

This is the classic problem illustrating conservation of $\vec{L}$: a projectile that strikes a hinged rod and embeds itself in it, setting it into rotation. It is worth fully understanding the logic, because it is the model for countless situations, from a revolving door hit with a shoulder to a rotational ballistic pendulum.

The heart of the reasoning is the choice of pole. During the collision the hinge exerts an intense impulsive reaction: if we ignored it, we would be wrong. But by choosing the hinge as the pole, this reaction has zero arm and therefore contributes nothing to the angular momentum. The rod’s weight also has an arm, but it is not impulsive (it acts weakly during the very brief instant of the collision). The result: the angular momentum about the hinge is conserved.

Before the collision only the projectile has angular momentum (the rod is at rest); afterwards, rod and projectile rotate together as a rigid body:

$m\,v_0\,L = \left(\tfrac{1}{3}ML^2 + mL^2\right)\omega$

On the left, the angular momentum of the projectile (arm $L$, because it strikes the end); on the right, the product of the system’s total moment of inertia by the common angular velocity. The rod’s moment of inertia about the hinge is $\tfrac{1}{3}ML^2$ (end), while that of the point-like projectile at the end is $mL^2$.

Why momentum is not conserved

In an inelastic collision with a rigid body rotating about a hinge, momentum is not conserved: there is the hinge’s impulsive reaction, which is a force external to the system. But angular momentum about the hinge is: it is the only quantity that survives the presence of the constraint, and that is why it is so valuable.

The collision is strongly inelastic: much of the projectile’s initial kinetic energy turns into heat, sound and deformation. The full numerical solution, with a check of the energy dissipated, is in the dedicated worked exercise.

Collegamenti

Argomenti: Dinamica rotazionale Concetti: Conservazione del momento angolare · Urto anelastico · Momento d’inerzia Competenze: Scelta del polo Oggetti: Asta · Proiettile

Esercizi collegati: Problema — Disco rotante colpito tangenzialmente · Esercizio svolto — proiettile contro asta incernierata · Problema — Vero o falso su momento angolare e inerzia