A fibre's angular momentum
Clairaut's relation is a conservation law. A bead coasting on a surface of revolution with no friction and no gravity keeps constant speed and constant angular momentum about the axis, so R sin α is constant: it follows a geodesic and turns at the same radius as a geodesic fibre. Friction lets R sin α drift; that drift is steering.
Predict · at 1:05 in the video
A bead and a geodesic fibre start on the same funnel, at the same radius and the same angle. Do they turn around at the same radius?
Yes. Both keep R sine alpha constant, so both turn where R equals that constant. In fact they trace the same path.
This course is our own. These books go deeper.
- Composite Filament Winding, S. T. Peters (ed.), ASM International, 2011
- Filament Winding: a Unified Approach, S. Koussios, Delft University Press, 2004
Simulations in this lesson are our own. Where a number rests on an assumption, the video marks it with an asterisk and says where it comes from.
