Spin: Angular Velocity, Center of Mass and Inertia

A gust catches one end of the long gangplank as the crew carries it aboard, and the plank swings round. Push a crate squarely in the middle, though, and it just slides. Where a force lands decides whether a body spins, and how its mass is spread out decides how easily.

The rule

Push through the center: it slides. Push off-center: it spins too.

Angular velocity $\omega$ (omega) is how fast a body turns, in radians per second. In 2D it is a single number, positive for counter-clockwise. It updates exactly like position: angle += omega * dt.

The center of mass is the mass-weighted average position of a body, and it's the point the body naturally spins about. A 15 kg crate at $x = 0$ strapped to a 5 kg anchor at $x = 2$ has its center of mass at $(15 \times 0 + 5 \times 2) / 20 = 0.5$.

A force applied at an offset $\mathbf{r}$ from the center of mass creates a torque, which in 2D is

$\tau = r_x F_y - r_y F_x$

A force aimed straight through the center of mass has zero torque: it only slides the body. An off-center force slides it and spins it.

The moment of inertia $I$ plays the part of mass for turning: angular acceleration = $\tau / I$. It grows with the mass and with the square of how far that mass sits from the center. For a thin plank of mass $m$ and length $L$, turning about its middle, $I = mL^2 / 12$. Double the length and it becomes four times harder to spin.

The shortcut

Only the sideways part of a push, times its distance from the center, makes spin. After that, treat turning exactly like moving: torque ÷ I gives angular acceleration, just as force ÷ mass gives acceleration.

The trap is measuring $\mathbf{r}$ from the wrong point, such as the plank's end or the world origin. Torque that drives spin is always measured from the center of mass.

Drills

Answer fast — trust the rule.

Remember

• Angular velocity updates like position: angle += omega × dt.
• Torque = the sideways part of a push times its distance from the center of mass; a push through the center makes no spin.
• Angular acceleration = torque ÷ I, and I grows with the square of distance, so long things are slow to spin.