Here goes the frictional force "paradox." Note that a paradox usually means that some assumption is invalid, and we will see that in the two solutions to this paradox as presented below.
Assume that there is a wheel and it is a perfect circle, and it is rolling on a horizontal plane. As explained in class, the friction between the wheel and the plane, if it exists, is a static one, since the circle and the plane touches only at one point, and the velocity of that point is zero. Other than this possible frictional force, we do not consider any other force. From this, the following three observations can be made, which may seem like a crazy situation -- i.e., it is a paradox.
(1) The work-energy theorem says that the kinetic energy change of this wheel is only due to work done by external forces. In this case, that work is zero, since at the point of contact the mass elements of the wheel is not moving. Since power, P=dW/dt, is F dot v, it means the power delivered from the frictional force to the wheel is zero. This means that there is no work done by the frictional force, and thus, the wheel cannot change its momentum or velocity.
(2) From Newton's law, the frictional force gotta be included in the equation of motion, F_net = ma. As we will see later, this equation describes the center of mass motion for a compound object. In fact, since the frictional force is the only force along the horizontal direction, it must be that the wheel slows down (assuming the frictional force is pointed opposite to the motion).
(3) As I mentioned in class, Newton's law in the rotational form tau = I alpha is applicable witin the center of mass frame. If the frictional force is applied opposite to the motion, it is easy to see that the torque due to the frictional force with respect to the center of mass is in the direction to accelerate the wheel!
It is quite a bothersome situation -- (1),(2),(3) are claiming completely different things! What is the solution? There are two solutions. (i) On a level surface there is no friction between a non-deformable wheel and the surface, or (ii) If there is a frictional force (whether it is a small rolling friction, the frictional force that will stop the car very slowly if gas runs out and we don't bother to use the brake, or a large (near-)static friction, the frictional force that kicks in when we brake) between the surface and the wheel while the wheel is rolling on a level surface, it is due to the deformation of the wheel (i.e. the contact is not just a point).
The paradox also disappears if one considers the motion of a wheel on an incline. See the following note, which by the way also explains today's quiz (not from the velocity point of view, but from the acceleration point of view). Note that, as theta goes to 0, f_s goes to zero as well, consistent with the possible solution (i) above. Of course, on an incline, there can be additional frictions due to deformation as well.
Side remark: In this presidential election campaign, there was an episode regarding one candidate saying "keep your tires properly inflated to save energy." This is a valid argument for the reasons discussed here.
(2) From Newton's law, the frictional force gotta be included in the equation of motion, F_net = ma. As we will see later, this equation describes the center of mass motion for a compound object. In fact, since the frictional force is the only force along the horizontal direction, it must be that the wheel slows down (assuming the frictional force is pointed opposite to the motion).
(3) As I mentioned in class, Newton's law in the rotational form tau = I alpha is applicable witin the center of mass frame. If the frictional force is applied opposite to the motion, it is easy to see that the torque due to the frictional force with respect to the center of mass is in the direction to accelerate the wheel!
It is quite a bothersome situation -- (1),(2),(3) are claiming completely different things! What is the solution? There are two solutions. (i) On a level surface there is no friction between a non-deformable wheel and the surface, or (ii) If there is a frictional force (whether it is a small rolling friction, the frictional force that will stop the car very slowly if gas runs out and we don't bother to use the brake, or a large (near-)static friction, the frictional force that kicks in when we brake) between the surface and the wheel while the wheel is rolling on a level surface, it is due to the deformation of the wheel (i.e. the contact is not just a point).
The paradox also disappears if one considers the motion of a wheel on an incline. See the following note, which by the way also explains today's quiz (not from the velocity point of view, but from the acceleration point of view). Note that, as theta goes to 0, f_s goes to zero as well, consistent with the possible solution (i) above. Of course, on an incline, there can be additional frictions due to deformation as well.
Side remark: In this presidential election campaign, there was an episode regarding one candidate saying "keep your tires properly inflated to save energy." This is a valid argument for the reasons discussed here.
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