The Verification Venue · settling an argument the internet keeps getting wrong
The Wheel That Gets the Same
Ask the internet what an open differential does with one wheel on ice and you get two camps, both wrong: "100% of the torque goes to the wheel that spins" and "0% goes to it." The truth is stranger and provable — it sends equal torque to both wheels, always. That is exactly why the car goes nowhere.
Not equal speed — equal torque. The two output shafts of an open differential are forced to the same twist at every instant, no matter how fast either wheel turns. So when one wheel sits on ice, the wheel with grip is choked down to the pathetic torque its frozen twin can hold. Drag a grip slider below and watch the two torque bars stay locked together while one wheel spins uselessly and the gripped wheel sits dead still.
▲ the two bars are ALWAYS equal — that's the whole law ▲
left wheel
200 rpm
carrier (avg)
100 rpm
right wheel
0 rpm
ωL + ωR = 2·ωcarrier → 200 + 0 = 200 (= 2·100)
Drag the left wheel onto ice and the whole car is hostage to it.
Perfect grip here buys nothing if its twin is on ice — watch it get starved.
More throttle can't beat the law: past the weaker wheel's limit, the surplus just spins the loose wheel faster. It never reaches the road.
The law lives in one small, freely-spinning gear. Inside the differential, the input (the ring gear, or carrier) carries a spider gear that meshes with both output (side) gears at the same radius. Because that spider gear is free to turn on its pin, it can't hold a net moment — so it must push both side gears with equal and opposite force. Equal force at equal radius is equal torque. The proof is the little gear that can't take sides:
That equal split is fine on dry pavement — both wheels are far from their grip limit, both push, the car drives. It turns cruel the instant the two surfaces differ. A wheel can only deliver torque up to the friction it can find, T_cap = μ·N·r. Since both wheels are forced to the same torque, the deliverable torque is capped by the worse wheel — so on ice the good wheel is starved to a tenth of what it could do, and the total pull at the road is twice that tiny number. Throttle doesn't help: anything beyond the weak wheel's limit has nowhere to go but into spinning it.
The check — every number recomputed in front of you
For the grips you've set right now, here is the whole computation, live:
Per-wheel friction-torque capacity, T_cap = μ·N·r, with the named inputs r = 0.32 m, N = 4000 N per wheel:
| surface | μ | μ·N·r (N·m) | force μ·N (N) |
|---|
The two output torques are equal by the spider-gear statics above (independent of speed); the speeds obey ω_L + ω_R = 2·ω_carrier; the tractive limit is 2·μ_min·N·r. Run it yourself: node research/the-wheel-that-gets-the-same/verify-the-wheel-that-gets-the-same.mjs — 20 checks, all green.
What's idealised, what's exactly true, and what's out of scope
Exactly true. In an ideal symmetric (open) bevel differential the two output torques are equal at all times and equal to T_carrier/2 — a pure statics consequence of the free spider gear meshing both side gears at equal radius, independent of the wheels' speeds. The kinematic identity ω_L + ω_R = 2·ω_carrier is exact. So is the conclusion that tractive torque is capped at twice the lower-grip wheel's friction torque.
The free choices. Wheel radius r = 0.32 m and per-wheel load N = 4000 N (≈408 kg) are representative of a mid-size car's driven axle; the "stuck / moves" verdict compares the available pull to the force needed to pull away up an 8% grade at 1500 kg (≈1177 N) — all named, all changeable in the equations. The friction coefficients (ice ≈0.10, snow ≈0.30, wet ≈0.50, dry asphalt ≈0.90) are representative ranges, not exact for any one tire or surface; the slider interpolates linearly between them.
Idealised. We model a frictionless open differential. A real open diff has a little internal friction and preload, so it actually sends slightly more than 50% to the higher-grip side — a few percent, not the dramatic bias folklore imagines; the equal-torque law is the limit it sits just beside. We treat the static/kinetic friction transition as a single threshold (real tires lose a bit of grip once they break loose, which only makes the spinning wheel worse). The car-motion check is a quasi-static traction limit, not a full vehicle dynamics model.
Out of scope (the contrast case). A limited-slip or locking differential exists precisely to defeat this law: it adds internal resistance (clutches, gears, or a clutch pack) so the two shafts can carry different torques, and the gripping wheel is no longer hostage to the spinning one. That is a different mechanism — the whole point of this page is the open diff, which has none of it.