Friction during rolling: troubling student questions

[Dr Ringuette]

Greeting fellow physicists,

I have been having some questions from my AP Physics C: Mechanics students about the direction of friction in rolling. Here are a few situations.

1. For a wheel that is accelerating to the right while rolling smoothly, it must also be rotating clockwise. In order to prevent sliding between the bottom of the wheel and the surface, frictional force must also be pointed to the right. A typical example is a car accelerating from rest. The problem I run into here is since the friction is pointing forward, is causes a positive (ccw) torque, yet the wheel begins rolling in the cw direction?! Do only the magnitudes matter in Newton's second of rotation when considering rolling (net torque = rot. inertia * ang. acc)?! It seems so simple, but I've obviously forgotten something here.

2. For a wheel that is rolling smoothly at a constant velocity to the right (or the left), the frictional force on the bottom of the wheel must be zero (assuming no opposing torques). In this case, friction is not needed to keep the wheel rotating. That is taken care of by the wheel's rotational inertia. No problems here; its what happens when you are driving and hit a patch of ice. No (or very little) friction acts on the tires, but your tires keep spinning at the same rate and the car still moves forward at the same speed (ignoring air resistance).

3. For a wheel that is rolling smoothly to the right with a decreasing velocity, it must still be rotating clockwise. This is where my students loose it. They say since the frictional force must be pointed to prevent sliding (correctly), and the bottom of the wheel would slide left (I think the problem is here), then friction must still be pointed to the right. I use the situation where the car is rolling to the right (no longer on the ice) at a constant speed and a huge gust of wind blows from the front, causing the car to slow down. From observations, we see the CoM of the car slow down and the wheels angular speed also decrease. Without friction, the wheel's angular velocity would not change, even when the car comes to a stop (sure wish I had perfect ice to show this). Thus, friction must act to slow down the angular speed of the wheel, pointing backward to slow it down. This is counter-intuitive again, since the direction of the torque caused by the friction seems to be in the wrong direction.

I appreciate you reading all this. Maybe it's just the end of the day, but I'd appreciate some thoughtful feedback on this. Thank you.

 

[brainpushups]

I'll get us started!

1. For a wheel that is accelerating to the right while rolling smoothly, it must also be rotating clockwise. In order to prevent sliding between the bottom of the wheel and the surface, frictional force must also be pointed to the right. A typical example is a car accelerating from rest. The problem I run into here is since the friction is pointing forward, is causes a positive (ccw) torque, yet the wheel begins rolling in the cw direction?! Do only the magnitudes matter in Newton's second of rotation when considering rolling (net torque = rot. inertia * ang. acc)?! It seems so simple, but I've obviously forgotten something here.

It may help to specify the point about which the torque is being evaluated (a good habit to instill in students). Taking the axle of the wheel as the axis of rotation the force of friction is not the only torque about the axis. Consider this: what if you take the (idealized) point of contact with the road as the (instantaneous) axis of rotation? Friction produces no torque about this axis. Does that offer any insight?

2. For a wheel that is rolling smoothly at a constant velocity to the right (or the left), the frictional force on the bottom of the wheel must be zero (assuming no opposing torques). In this case, friction is not needed to keep the wheel rotating. That is taken care of by the wheel's rotational inertia. No problems here; its what happens when you are driving and hit a patch of ice. No (or very little) friction acts on the tires, but your tires keep spinning at the same rate and the car still moves forward at the same speed (ignoring air resistance).

Right. Consider rolling with 'slipping' on a frictionless surface. Spin the wheel and set is down and it stays in place spinning. Give it a little push and it 'rolls' forward with linear speed less than the linear speed of a point on its edge.

3. For a wheel that is rolling smoothly to the right with a decreasing velocity, it must still be rotating clockwise. This is where my students loose it. They say since the frictional force must be pointed to prevent sliding (correctly), and the bottom of the wheel would slide left (I think the problem is here), then friction must still be pointed to the right. I use the situation where the car is rolling to the right (no longer on the ice) at a constant speed and a huge gust of wind blows from the front, causing the car to slow down. From observations, we see the CoM of the car slow down and the wheels angular speed also decrease. Without friction, the wheel's angular velocity would not change, even when the car comes to a stop (sure wish I had perfect ice to show this). Thus, friction must act to slow down the angular speed of the wheel, pointing backward to slow it down. This is counter-intuitive again, since the direction of the torque caused by the friction seems to be in the wrong direction.

To be clear: are the brakes being applied or not? I bet these images will help. I remember discovering them several years ago.

 

[vela]

1. For a wheel that is accelerating to the right while rolling smoothly, it must also be rotating clockwise. In order to prevent sliding between the bottom of the wheel and the surface, frictional force must also be pointed to the right. A typical example is a car accelerating from rest. The problem I run into here is since the friction is pointing forward, is causes a positive (ccw) torque, yet the wheel begins rolling in the cw direction?! Do only the magnitudes matter in Newton's second of rotation when considering rolling (net torque = rot. inertia * ang. acc)?! It seems so simple, but I've obviously forgotten something here.

In addition to what @brainpushups said, consider if the car is lifted off the ground so the tire isn't in contact with the road. You put the car in gear and give it some gas. The drive wheel starts spinning. Clearly, it's not friction with the road that causes the wheel to rotate.

2. For a wheel that is rolling smoothly at a constant velocity to the right (or the left), the frictional force on the bottom of the wheel must be zero (assuming no opposing torques). In this case, friction is not needed to keep the wheel rotating. That is taken care of by the wheel's rotational inertia. No problems here; its what happens when you are driving and hit a patch of ice. No (or very little) friction acts on the tires, but your tires keep spinning at the same rate and the car still moves forward at the same speed (ignoring air resistance).

I don't understand what the problem is here. You noted at constant velocity, no friction is needed, so what's the big deal if you end up on a frictionless surface?

3. For a wheel that is rolling smoothly to the right with a decreasing velocity, it must still be rotating clockwise. This is where my students loose lose it. They say since the frictional force must be pointed to prevent sliding (correctly), and the bottom of the wheel would slide left (I think the problem is here), then friction must still be pointed to the right. I use the situation where the car is rolling to the right (no longer on the ice) at a constant speed and a huge gust of wind blows from the front, causing the car to slow down. From observations, we see the CoM of the car slow down and the wheels angular speed also decrease. Without friction, the wheel's angular velocity would not change, even when the car comes to a stop (sure wish I had perfect ice to show this). Thus, friction must act to slow down the angular speed of the wheel, pointing backward to slow it down. This is counter-intuitive again, since the direction of the torque caused by the friction seems to be in the wrong direction.

What do you mean by "backward"? The students are correct: the force of friction will point to the right, causing a torque in the correct direction to slow the wheel. If you think friction will point to the left, could you explain how you concluded that?

 

[zwierz]

why not to follow the formal argument? for slipping surfaces the friction force is directed opposite to the corresponding relative velocity ; for non-slipping case the friction force is found from the equations of motion