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.

The physics behind it

Formulas used in this lab

Torque and balance Class 11

Torque: τ = rF sinθ = F × (perpendicular distance)
Rotational form of Newton's second law: τ = Iα
Rotational equilibrium: Στ = 0, i.e. m₁s₁ = m₂s₂ for a horizontal rod
Tilted rod (as in the lab): net torque = (m₁s₁ − m₂s₂) g cosθ

Moment of inertia Class 11

Definition: I = Σ mr²
Solid sphere: I = (2/5)MR², k = 2/5
Solid cylinder or disc: I = ½MR², k = 1/2
Hollow sphere: I = (2/3)MR², k = 2/3
Ring or hoop: I = MR², k = 1
Shape factor: k = I/(MR²)

Rolling without slipping Class 11

Condition: v = ωR
Kinetic energy: KE = ½mv² + ½Iω² = ½mv²(1 + k)
Acceleration down an incline: a = g sinθ / (1 + k)
Speed at the bottom (height h): v = √(2gh/(1 + k))
Order of finishing: solid sphere, solid cylinder, hollow sphere, ring (smaller k wins), independent of mass and radius

Angular momentum Class 11

Angular momentum: L = Iω
Conservation (no external torque): I₁ω₁ = I₂ω₂
Disc with two arm masses (as in the lab): I = I_disc + 2mr²
Kinetic energy: KE = ½Iω² = L²/2I, which rises as I falls
Work done pulling the masses in: W = ΔKE