Rotational motion sandbox

Torque and equilibrium, a rolling-shapes race, and a spinning disk that conserves angular momentum — three animated sandboxes built on the real rotational dynamics.

Lessons

Pick a topic, read the steps, then try them right in the sandbox below.

Class 11 — Torque and rotational equilibrium

Concept: a rod balances on its pivot when the net torque is zero — torque from each side is force (weight) × distance from the pivot, and the two sides must cancel.

Switch to Torque & balance mode.
Set Left mass m₁ = 2 kg at Left distance s₁ = 1 m, and Right mass m₂ = 2 kg at Right distance s₂ = 1 m. Check the State stat reads "balanced".
Now change Right distance s₂ to 2 m without touching anything else — the rod should start rotating, and State should switch to "rotating" with a nonzero Net torque.
Fix the balance again by adjusting Right mass m₂ down to 1 kg (since 2×1 = 1×2).

Takeaway: balance depends on the product of mass and distance on each side, not on the masses or distances alone.

Class 11 — Moment of inertia: the rolling-shapes race

Concept: objects rolling down the same incline don't all arrive together — the one with the smallest moment-of-inertia factor k (in I = kMR²) always wins, regardless of its mass or radius.

Switch to Rolling race mode.
Set any Incline angle and start the race with all shapes released together.
Watch the finish order: the solid sphere (k = 2/5) wins, then the solid cylinder, then the hollow shapes, with the ring (k = 1) always last.

Takeaway: more mass concentrated near the rim (a larger k) means more energy goes into spinning instead of moving forward — so it accelerates down the slope more slowly, no matter what it weighs.

Class 11 — Conservation of angular momentum

Concept: with no external torque, angular momentum L = Iω stays constant — so if I decreases, ω must increase to compensate.

Switch to Angular momentum mode.
Note the starting Moment of inertia and Angular momentum L values with the arms extended.
Drag Arm radius r down, pulling the weights inward. Watch the Moment of inertia fall while the spin visibly speeds up — but Angular momentum L stays constant.

Takeaway: this is exactly how a figure skater spins faster by pulling their arms in — nothing pushes them faster, their moment of inertia just drops.

What each mode covers

Torque & balance — two masses sit at adjustable distances on either side of a pivoted rod. Torque is force × perpendicular distance (τ = mgs·cosθ here, since the rod itself rotates); drag the masses in or out and watch the rod settle wherever the net torque is zero — or swing all the way down if it can't balance.

Rolling race — a solid sphere, a solid cylinder, a ring and a hollow sphere all roll from rest down the same incline. Mass and radius don't matter; only how the mass is distributed does. Acceleration is g sinθ/(1+k), where k = I/(mr²) — the solid sphere (k=2/5) always wins, the ring (k=1) always finishes last.

Angular momentum — pull the two arm-weights in and the disk spins faster, automatically, because L = Iω stays constant while I = Iᵢ + 2mr² drops. This is the figure-skater spin-up, and the readouts show the twist: kinetic energy goes up as the arms come in — that energy comes from the work you do pulling them against the outward pseudo-force, not from nowhere.

Idealisations: no friction/air resistance anywhere, rolling is always "without slipping", and the rod/disk are treated with simple textbook moments of inertia — the same spirit as the ideal battery in Circuit lab and the no-air- resistance free fall in Mechanics.

Part of Forces — see Learn for the class 11 rotational mechanics topics this maps to.

Test yourself

Pick Class 11 in the physics quiz to test yourself on rotational motion — 10 random questions, with an explanation for every answer.

Take the physics quiz →