Center of Mass — Balancing a Multi-Mass Beam

Masses on the Beam

Selected mass 1.0 kg

Selected position 0.0 m

Pivot & Physics

Pivot position 0.0 m

Adjust masses, then release.

Live Readouts

Center of mass x̄

0.00 m

Total mass

0.0 kg

Net torque τ

0.00 N·m

Moment of inertia I

0.00 kg·m²

How it works

The center of mass of a set of discrete point masses on a line is the mass-weighted average position:

x̄ = Σ(mᵢ · xᵢ) / Σ mᵢ

When you press Release on pivot, the beam becomes a rigid body free to rotate about the pivot. Gravity pulls down on every mass, so each one contributes a torque about the pivot equal to force times lever arm:

τᵢ = mᵢ · g · (xᵢ − x_pivot) · cos(θ)
τ_net = Σ τᵢ
I = Σ mᵢ (xᵢ − x_pivot)²   (moment of inertia about the pivot)
α = τ_net / I               (angular acceleration, Newton's 2nd law for rotation)

The beam's tilt angle θ integrates α forward each frame with a small damping term (representing pivot friction), exactly like a physical pendulum. If the pivot sits exactly under the center of mass, x̄ = x_pivot, every lever arm's torque contribution cancels on average and the beam stays essentially level — the definition of static equilibrium, Στ = 0. Move the pivot away from x̄ and the net torque never reaches zero at θ = 0, so the beam keeps rotating until it swings past vertical or settles at the bottom, just like a real seesaw loaded off-center.

Mass rows / sliders — select a mass, then drag its own value and beam position; the center of mass, total mass, torque and moment of inertia all update live, even while the beam is swinging.
+ Add mass / Shuffle — add a new point mass, or randomize every mass and position at once to build a fresh composite body.
Pivot slider — moves the support point that the beam rotates around.
Release on pivot — starts the real rotational-dynamics integration (α = τ/I) from the current layout.

Drag left/right on a sphere to move it

Center of mass Pivot support

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