# Is friction a conservative force when a ball rolls down an inclined plane(pure rolling)?

My book says friction is non dissipative for a cylinder rolling down an inclined plane,so is it conserved?

• friction here plays the role of force of constraint. – wcc Jun 25 at 3:43

We know that for a conservative force $$F$$ we can define a potential energy function $$U$$ such that $$F=-\frac{\text d U}{\text d x}$$
and a consequence of this is that the work done by this force is given by $$W=-\Delta U$$
However, static friction does no net work on the cylinder. Therefore for all instances of rolling here $$W=0$$. This means we must have a constant potential energy, which means that $$F=0$$. This is a contradiction, since we assumed we had a static friction force. Therefore static friction cannot be conservative.