11.4 : Rolling Without Slipping

People have observed the rolling motion without slipping ever since the invention of the wheel. For example, one can look at the interaction between a car's tires and the surface of the road. If the driver presses the accelerator to the floor so that the tires spin without the car moving forward, there must be kinetic friction between the wheels and the road's surface. If the driver slowly presses the accelerator, causing the car to move forward, the tires roll without slipping. It is essential to observe that the bottom of the wheel is at rest with respect to the ground, indicating that there must be static friction between the tires and the road surface. For example, consider a bicycle in motion with the rider staying upright. The tires have contact with the road surface, and even though they are rolling, the bottoms of the tires deform slightly, do not slip, and are at rest with respect to the road surface for a measurable amount of time. There must be static friction between the tire and the road surface for this to occur.

If the tire is moving forward with spinning, then the unbalanced frictional force gives the net torque, which is responsible for the spinning of the tire. As the spinning of the tire becomes sufficiently great, the tangential velocity of the point in contact with the road becomes equal and opposite to that of the linear velocity of the center of mass of the tire, implying relative zero velocity of the point in contact with the surface. At this point, the tire moves forward without slipping. The velocity of the tire's center of mass is its radius multiplied by the angular velocity about its axis. In the rolling without slipping motion, the center of mass of the tire moves in a linear path, but the point on the rim of the tire traverses a cycloid path. In one complete rotation of the tire, the center of mass of the tire moves a distance equal to the circumference of the tire.