Why is Torque Generated from Friction?

Question

Why does a rolling ball get torque from friction. I understand torque in relation to lever arms and the force being perpendicular to said arm. But what about wheels? There is no lever arm so how is torque generated?

Answer 1

We can't talk about lever arms before we have chosen a rotation point to look at in this system.

(Since there is no example in the question, let's say we are looking at a system where a ball rolls down a rough incline, speeding up. If the question is about a specific example, then please add a sketch.)

• If we choose the centre of rotation of the ball (the ball's centre) as our rotation point, then friction does indeed have a lever arm. It is a distance equal to the radius away from this point. And it is exactly perpendicular to the lever arm (the radius).

• We could choose other points instead. For example the contact point between ball and surface could be our chosen rotation point. In that case friction, which acts in this very same point, does not have any lever arm and thus doesn't make any difference in this rotation. But then gravity, which works from the centre-of-mass of the ball, which might be in the centre, will have a lever arm, since this force is now located at a distance from the chosen rotation point.

In any case, nomatter the choice of rotation point, the rotation and angular acceleration etc. will end up being the same.

Answer 2

Torque is just a force at a distance. The line of action of friction does not go through the center of mass, thus it creates torque about the center of mass.

If a force is applied at a location $\vec{r}$ and the center of mass is at $\vec{r}_C$ then the moment of the force (torque) is

$$ \vec{\tau} = (\vec{r} - \vec{r}_C) \times \vec{F} $$

Only when the force $\vec{F}$ is parallel to the relative position vector $(\vec{r} - \vec{r}_C)$ there is no torque generated.

_{NOTE: $\times$ is the vector cross product.}