# A Tale of Friction

Roller coasters projects are frequently used in middle and high school physics classes to illustrate the principle of conservation of mechanical energy. Potential energy transforms to kinetic energy and vice versa, with gravity being the driving force during the entire process. Even though friction force is mentioned, it is rarely considered in the velocity calculations along the coasters’ paths. In this high school lesson, the friction force is considered in the process. Using basic calculus and the work-energy theorem for non-conservative forces, the friction along a curved path is quantified, and the cart’s velocity along this path is predicted. This activity and its associated lesson are designed for AP Calculus. Practice problems/answers, a PowerPoint® presentation and student notes are provided. The starting point in this analysis is the solution found using the work-energy theorem to the problem of a spherical body rolling on an incline when friction is present. This approach is extended to a spherical body rolling on a curved path. Assuming that a curved path can be approximated by a sequence of many very short inclines, the problem is approached as a body rolling on this sequence of inclines, solving each with the work-energy theorem. Defining the curved path as a differentiable function, the slope of each incline is obtained through the function derivative. Formulas for the coefficient of static friction, friction force and velocity are found and through them, values of these properties along the curved path can be determined. Students use these equations in the associated activity to design and construct simple roller coasters that consider the friction present, using a flexible material like foam pipe insulation as the coaster’s path and a marble as the cart.