Torque due to static friction on a rolling object

[member 392791]

I'm looking over my notes here, we have a rolling disk down an incline plane and my goal is to find its acceleration in terms of its moment of inertia

My dilemma is, when finding the torque, I look at all 3 of the forces influencing it (normal, gravity, and the f_s). The n goes through the instantaneous axis of revolution so its torque on the disk is 0, I can see the gravity's torque is mgsinθ, but I don't see why the f_s has 0 torque, even though it is perpendicular to the instantaneous axis of revolution.

Is the reason that the F_s force goes through the axis, therefore its torque on the object is 0?

 

[Bob S]

If the disc starts turning as it rolls down the inclined plane, there has to be torque. In order to get torque, the disc has to accelerate down the ramp.

The force pressing the disc against the incline is mg cos(θ), which gets smaller as the incline gets steeper. The force pushing the centroid of disc down the incline plane is mg sin(θ) which gets larger as the incline gets steeper. There is an angle at which the disc starts to slip rather than roll without slipping.

 

[member 392791]

I'm not sure if that is what I was looking for??

 

[haruspex]

Is the reason that the F_s force goes through the axis, therefore its torque on the object is 0?

Yes, any force through a point has no moment about that point. The only one of the three forces with a moment about the point of contact is gravity.

 

[PJC01]

Yes, that is correct. The reason the static friction force has zero torque on the object is because it passes through the axis of rotation. In other words, the force does not create a moment arm, or distance, from the axis of rotation, which is necessary for torque to be generated. This is why the normal force and gravity can create torque, but the static friction force cannot. Keep in mind that torque is a measure of the force's ability to rotate an object, and if the force does not create a lever arm, it cannot create rotation. So in this case, the torque on the object is only due to the normal force and gravity, and not the static friction force.