2D Rigid Body

A force applied off-center accelerates the center of mass and spins the body at once. What decides how much of a hit becomes rotation?

Interactive · live simulation

Drop the rod, or click anywhere to throw a fresh one. An off-center wall hit converts translation into spin through the (r×n) term — watch the spin readout jump on each glancing bounce.

2D Rigid Body — simulation render — fig 1. A spinning rod thrown across a large box. In free flight the center of mass traces an exact parabola while the angular velocity stays perfectly constant — linear and angular motion are decoupled until a wall is touched.

Extent changes everything

A rod has a moment of inertia I = \tfrac{1}{12} mL^2. Gravity acts at the center of mass and so produces no torque — meaning the COM follows a parabola and the spin rate is exactly constant in free flight. Linear and angular motion are independent until something touches a wall.

fig 2. A slightly tilted rod dropped from rest: the off-center floor contact converts a purely vertical fall into spin — the (r×n) term made visible.

Where spin comes from

A wall impulse at offset r from the COM changes velocity by j\,n/m and angular velocity by j\,(r\times n)/I. That r\times n term is the whole story: a hit straight through the COM imparts no spin; the further off-center, the more of the impulse becomes rotation. It is the rotational analogue of reduced mass.

fig 3. Energy trading between translation, rotation, and height while the total stays flat across bounces (e=1).

Honest energy

Two deliberate choices keep energy clean: no positional push-out (which would inject mg\cdot\text{depth} of potential energy per contact) and one endpoint resolved at a time (a flat rod hitting on both ends at once picks up a spurious one-step spin). With e=1, total energy stays flat across many bounces — proof the resolver isn't cheating.

This chapter is drawn from the physics-lab study notes and renders. The longer write-ups and project essays live on the blog.