Conservation of energy seems to contradict conservation of angular momentum.

If a disc is spinning then every point on that disc is undergoing linear acceleration perpendicular to its trajectory. For anything with mass to be accelerated linearly, force must be applied. If force is applied over a distance, work is done, and energy expended.

If that energy comes from within the system, it will eventually run out, and the disc will stop spinning. If it comes from outside the system, then that's not a closed system. In neither case does conservation of angular momentum in a closed system seem to apply.

What's the problem in my thinking here?