Rotation
Rolling down an incline
Section titled “Rolling down an incline”A disc rolling down a slope arrives later than a block sliding down a frictionless one, because part of the energy goes into spinning. A ring arrives later still: more of its mass is far from the axis. And rolling needs friction. With a slippery slope the disc slides without turning. Lower the friction slider until the slip event appears.
For a round body with moment of inertia rolling without slipping,
which needs static friction . A disc has , a ring .
Rolling is not a special mode in the engine. The contact point’s friction is decided like any other, and when it holds, the point does not slide, so falls out. When is too small the same rule produces slipping and the textbook sliding acceleration. Validated to 1e-8 in both regimes.
Atwood machine with a massive pulley
Section titled “Atwood machine with a massive pulley”In the ideal Atwood machine the string’s tension is the same on both sides. Give the pulley mass and that can no longer be true: something has to spin the pulley up, and the only thing touching it is the string. So the heavier side pulls a little harder than the lighter side, and the difference supplies the torque. The whole system accelerates more slowly than with a massless pulley, because the pulley’s inertia is along for the ride.
With a uniform disc of mass , , and the no-slip condition ,
The torque on the pulley is .
The pulley is a rigid disc pinned at its centre, and the rope wraps it as a wheel: the rope’s tangent points are treated as material points of the rim, so the rope segment on each side is its own constraint and the no-slip coupling falls out of the geometry. Validated to 1e-8 for the acceleration, both tensions and the rim speed.
Flywheel with a hanging mass
Section titled “Flywheel with a hanging mass”Wind a string around a heavy wheel and hang a mass from it. The mass falls, but slowly: most of the energy goes into spinning the wheel. Double the wheel’s mass and watch the fall slow further. This is how a flywheel stores energy.
Same wheel node as the massive pulley, with one end of the rope fixed to the rim. Validated to 1e-8.
Seesaw
Section titled “Seesaw”A seesaw balances when the torques match, not the masses. A child far from the pivot balances a heavier adult close to it. Move a mass outward with the slider and the plank tips, slowly at first, because the whole system has to be spun up.
Balance when . Otherwise the initial angular acceleration is
counter-clockwise positive.
The plank is pinned to the ground and each mass is pinned to the plank: three exact constraints solved together. Balanced, the plank’s angular velocity stays below 1e-9. Tipping, the initial angular acceleration matches the formula to 1e-9. This scenario is also the one that caught a solver bug: the position projection has to be Gauss–Seidel, not Jacobi, or three coupled pins diverge.