Collisions with Orbital and Spin Angular Momentum

When a translating object collides with an object that can rotate about a fixed axis, both orbital and spin angular momentum may be exchanged. The key to solving such problems lies in identifying the correct conservation law and reference point.

Choosing the Reference Point

A hinge or pivot exerts a constraint force during the collision. This force can be large and impulsive, preventing conservation of linear momentum. However, because the force acts at the pivot itself, it produces zero torque about that point. Angular momentum about the pivot is therefore conserved:

$$L_i = L_f$$

Solving for Missing Variables

Consider a translating object of mass $m$ moving with velocity $v_i$ that collides with a hinged rigid object of moment of inertia $I$ rotating at angular velocity $\omega_i$. After the collision, the translating object bounces off the hinged object with some new velocity $v_f$, which then rotates with angular velocity $\omega_f$.

Taking angular momentum about the pivot, conservation gives:

$$mv_ir +I\omega_i= I\omega_f + mv_f r$$

where $r$ is the perpendicular distance from the pivot to the line of motion of the translating object at the point of impact, and $v_f$ is the final velocity of the object. This equation lets you solve for one unknown when all other quantities are known.

Classifying the Collision

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