Presets
Playback
Cone
Size and sharpness of the double cone.
Rails and incline
V opening, slope and rail length.
Damping
Small losses in the rolling.
Speed
Slow motion shows the axis height change.
Camera
Layers
Quality
Model checks
The science
What happens
A double cone is placed at the narrow end of two rails that form a V and rise toward the wide end. Released, it rolls up the incline. Nothing pushes it uphill: as the rails spread apart the cone settles lower between them, because the rails touch it nearer its pointed tips where its radius is smaller. The height of its centre of mass falls even though it travels up the slope.
Equations
Let the rails have half-opening angle phi (tan phi = t), the cone have equator radius R and half-length l, so its surface radius at lateral distance z from the equator is rho(z) = R - (R/l) z. For the equator centred at distance s along the V, the axis height above the rail plane is the largest clearance H(s)^2 = max_z [ rho(z)^2 - (z/t - s)^2 ].
The centre of mass height is y = s sin(beta) + H(s) cos(beta) for an incline beta. Rolling without slipping with omega = v / rho_c and I = 0.3 m R^2 gives a one-degree-of-freedom system: (m + I/rho_c^2) s'' + (1/2) M' s'^2 + dU/ds + c s' = 0 with U = m g y.
Near the contact point dH/ds = -tan(phi) tan(alpha), where alpha is the half-apex angle of the cone (tan alpha = R/l). The cone climbs while tan(beta) < tan(phi) tan(alpha), which is exactly the condition that its centre of mass descends.
Honest simplifications
The rails are thin lines, the cone is a rigid solid of revolution with a uniform density (I = 3/10 m R^2), the contact is a single point on each rail without slipping, and the spin rate is taken as the speed divided by the contact radius. A real cone rolling between diverging rails must slip a little because the contact radius changes along the rails; that slip, air drag and rolling resistance are replaced here by a small viscous damping term. The rail ends and the wide end are idealised stops.
Checks in the page
Check 1 finds the tipping incline by bisection on the simulated start acceleration and compares it with tan(phi) tan(alpha). Check 2 samples 6000 points along a rail and verifies the cone surface touches it at one point and never overlaps it. Check 3 verifies energy conservation (kinetic + potential + damping heat) over a rolling run with a fourth-order Runge-Kutta integrator.