# How does acceleration work for rolling?

I have difficulties understanding acceleration for rolling. Consider a magic wheel you have control over. Suppose the wheel is initially rolling on the ground without slipping at a constant velocity. You then make it roll faster by increase the rate at which it spins. During this time, the wheel does not slip. This causes a force of static friction to point in the direction the wheel is going. So far, this makes sense; the force of static friction causes a linear acceleration in the forward direction which is what you would expect if you were on a bike and decided to pedal faster. However, that force of static friction also causes a torque opposing the spin of your wheel. This is the part that doesn't make sense to me. Friction is simultaneously linearly accelerating your wheel but also slowing down your wheel. How does this work? On a somewhat related note, is there a way to quantify the static friction force based on the angular acceleration?

Suppose the wheel is initially rolling on the ground without slipping at a constant velocity.

So the wheel is moving horizontally and the frictional force is zero.

You then make it roll faster by increase the rate at which it spins. During this time, the wheel does not slip.

You are applying a force on the wheel and for a force not applied at the axle think of the string of a Yo-Yo applying the force.

This causes a force of static friction to point in the direction the wheel is going.

When this force is applied a static frictional force (as there is no slipping) is applied to the point of contact in the direction which is backwards to the motion of the wheel and you have reasoned correctly why the frictional force is not in the forward direction.

The backward frictional force applies a torque about the centre of mass of the wheel which increases its angular speed (makes it rotate faster) and, in a sense, reduces the forward acceleration which would have been produced by the applied force alone to preserve the no slipping condition.

• The case of a yoyo and a string is different than the case I had in mind. I referring to applying a couple on the wheel. If I am not mistaken, this case is different than the yoyo case because in the yoyo case, both linear acceleration and a torque are applied simultaneously. – Serendipitous Epiphany Apr 4 '19 at 13:14
• @TypicalHighschooler With a couple trying to increase the speed of rotation the frictional force is in the forward direction providing a translational acceleration of the wheel whilst at the same time keeping a check on the net torque applied to the wheel so that the no slipping condition is satisfied. – Farcher Apr 4 '19 at 13:19