For a long time, I have believed that most introductory physics 롤대리 courses and textbooks do not cover rolling motion adequately. I feel this way because so many of the important principles of mechanics must be used when analyzing the motion of a rolling body. For example, to calculate the acceleration of a ball rolling down an incline we must use Newton’s second law and the relationship between torque and angular acceleration (I will call this the “Rotational Equation of Motion”); we must decide whether the ball is subject to kinetic or static friction; and we have to relate the acceleration of the center of mass to the angular acceleration around the center of mass. Also, when there is no slipping, the conservation of mechanical energy is often applied. This requires knowing how to express the kinetic energy of a system of particles and how to relate the velocity of the center of mass to the angular velocity around the center of mass. Basically, all of the important ideas are used when solving physics problems related to rolling motion.
There’s so much physics in this subject that I will devote four articles to it. The first will cover a standard physics problem solution for rolling without slipping; the second will describe how slipping affects the motion of a rolling body; the third will describe how the motion of a body rolling without slipping can be analyzed in terms of mechanical energy conservation; and the fourth will demonstrate how seemingly difficult rolling motion problems can be solved fairly easily when the basic principles are carefully applied. Now on to the first of these four articles.
The text editor does not accept many standard mathematical symbols. As a result, I have had to use some rather unusual mathematical notation in my articles. All of that notation is described in the Ezine article “Teaching Rotational Dynamics”.
Problem. A uniform solid sphere rolls down a plane inclined at an angle th without slipping. The body’s mass, radius, and moment of inertia around the center of mass are M, R, and Icm = 2(MR**2)/5, respectively. (a) What is the sphere’s acceleration? (b) What condition must the coefficient of static friction satisfy if there is no slipping?
Solution.(a) A sphere is touching the surface of the incline, which exerts a normal force N and a frictional force f on it. The weight of the sphere has components along the incline (MGsin(th)) and perpendicular to the incline (MGcos(th)). The motion of the sphere is analyzed using an inertial coordinate system whose x axis is parallel to and directed down the incline and a y axis directed normal to and away from the incline. With the help of a free-body diagram, we have