Static friction needed for rolling without slipping

[adjurovich]

If we had a wheel rolling without slipping down the inclined plane, kinematically its velocity would be 0 at the contact point to the ground since the rotational and translational components of velocity would cancel out.

Speaking of forces, forces acting on body would be static friction and the component of weight parallel to the inclined plane. The only force that will exert torque will be static friction since it’s tangential. Now, speaking theoretically, should the adequate static friction for such motion be the static friction that is able to “reduce” component of weight force and cause rotation, but so that acceleration of rotation = acceleration of linear motion?

I tend to conclude that if force of friction wasn’t strong enough, it would decrease net linear force, but also exert torque that can’t provide enough tangential acceleration so tangential acceleration would be smaller than linear and slipping would occur.

Are my statements wrong?

 

[A.T.]

acceleration of rotation = acceleration of linear motion

acceleration of rotation * radius = acceleration of linear motion

if force of friction wasn’t strong enough ... slipping would occur.

Yes. For example, in the limit without friction it would just slide down without rotating.

 

[adjurovich]

Yes. For example, in the limit without friction it would just slide down without rotating.

Just to make things clear, did you mean when there is no static friction? Obviously there would be no net torque and body would just slide down without rotating.

 

[A.T.]

Just to make things clear, did you mean when there is no static friction? Obviously there would be no net torque and body would just slide down without rotating.

Yes. And if static friction is insufficient it will slide and rotate due to kinetic friction, but to slow to match the linear motion.

 

[adjurovich]

Yes. And if static friction is insufficient it will slide and rotate due to kinetic friction, but to slow to match the linear motion.

Thanks for help!