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In the problem of cylinder rolling without slipping on a horizontal floor,both the cylinder and floor are generally treated as rigid bodies in normaltextbooks. When the air resistance is ignored, the equation of motion has asolution with a constant velocity. However, in the real world, permanent motiondoes not occur. The difficulty cannot be solved only by the horizontal force,because a horizontal force opposite the translational direction increases theangular velocity of rotation around the center. Therefore other mechanisms needto be examined. There are two main reasons for this result. 1) Both a cylinderand a floor are not perfect circle and perfect plane, but have uneven surfaces.The micro bumps on the surface yield small collisions in the directionperpendicular to the floor. The collisions generate a rolling friction torquearound the center. 2) A strong force acts on the contact part which isdeformed. The high-speed deformation produces a history effect on therelationship between stress and strain, because the compressed wave in thecontact part diffuses to the outside at the speed of sound. Therefore a rollingfriction torque is also generated. Both torques are caused by forcesperpendicular to the floor. The rolling friction torque eliminates thediscrepancy between the textbook results and reality by solving thesimultaneous differential equations of rotation and translation. This method isuseful for studying rolling systems such as trains and cars.

Analysis of Rotational Motion based on Rolling Friction Torque