Case 1: inelastic collision, hinged rod

The rod is hinged and the projectile embeds itself in it. Before calculating anything, the decisive question is: what is conserved and what isn’t?

• Angular momentum about the hinge is conserved: the hinge’s constraint reaction has zero arm about the hinge itself, so it produces no moment.
• Momentum is not conserved: the hinge’s impulsive constraint reaction transmits an external impulse to the system.
• Kinetic energy is not conserved: the collision is inelastic, part of the energy turns into heat.

Pole: the hinge

The hinge is the only sensible choice. About any other pole, the hinge’s reaction would have a non-zero arm and would contribute an external angular impulse, making angular momentum non-conserved.

One unknown ($\omega$, the angular velocity after the collision with the embedded projectile) and one equation (conservation of $L$ about the hinge):

$\underbrace{m_p v_A L}_{L^A} = \underbrace{I_\text{tot}\,\omega}_{L^B}$

The system’s moment of inertia (rod + point-like projectile at the end) about the hinge is

$I_\text{tot} = \tfrac{1}{3}ML^2 + m_p L^2 = \tfrac{1}{3}\cdot 4\cdot 4 + 0{,}3\cdot 4 = 5{,}33 + 1{,}2 = 6{,}53\;\mathrm{kg\,m^2}$

from which

$\omega = \frac{m_p v_A L}{I_\text{tot}} = \frac{0{,}3\cdot 900\cdot 2}{6{,}53} \approx 82{,}7\;\mathrm{rad/s}$

Energy tells us how violent the collision was:

$E_\text{cin,A} = \tfrac{1}{2}m_p v_A^2 = \tfrac{1}{2}\cdot 0{,}3\cdot 810\,000 = 121\,500\;\mathrm{J}$

$E_\text{cin,B} = \tfrac{1}{2}I_\text{tot}\omega^2 = \tfrac{1}{2}\cdot 6{,}53\cdot 82{,}7^2 \approx 22\,320\;\mathrm{J}$

$E_\text{termica} = 121\,500 - 22\,320 \approx 99\,180\;\mathrm{J}$

About 82% of the initial kinetic energy went into heat: the hallmark of a strongly inelastic collision.

A body pinned at a fixed point

For a rigid body hinged at a fixed point $P$ you can use directly $L = I_P\,\omega$ and $E_\text{cin} = \tfrac{1}{2}I_P\,\omega^2$, where $I_P$ is the moment of inertia about $P$ (not about the centre of mass).

Collegamenti

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

Esercizi collegati: Esercizio svolto — proiettile contro asta incernierata · Problema — Disco rotante colpito tangenzialmente · Problema — La pattinatrice che si stringe