A Tale of Friction
Designed by engineers, roller coasters are the main attraction in theme parks around the world. No matter their scary steepness and radical movements, gravity is the force that makes them all work. One key design consideration is friction—the force that opposes movement and eventually brings the coaster cars to a stop.Copyright 2012 Alex Brogan, Wikimedia Commons CC BY-SA 3.0 https://commons.wikimedia.org/wiki/File:Takabisha_roller_coaster.jpg
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.
With the costs to create roller coasters ranging from $20 to 100 million (R1), engineers must be scrupulous in their designs and consider myriad details before construction begins because mistakes have high costs, or even worse, terrible accident consequences (R2). But no matter a coaster’s complexity and cost, all have something in common—gravity and inertia propels the carts along the entire path. Potential energy transforms into kinetic energy and vice versa. Another key force that affects roller coasters is friction—the force that gradually decelerates the carts until it practically stops them. Engineers who design roller coasters aim to optimize the kinetic-potential energies and minimize friction effects.
In this lesson, potential-kinetic energies and friction force are quantified along a simple curved path. Using basic calculus and the work-energy theorem for non-conservative forces, a formula to find the velocity of a spherical body rolling along a curved path is determined. This basic equation enables students to design simple roller coasters and know before construction whether their designs will work or not—just like roller coaster engineers do in real projects.
After this lesson, students should be able to:
- Estimate the effect of friction for a spherical body rolling on an incline.
- Use free-body diagrams to analyze the friction force acting on a body rolling on an incline.
- Use the work-energy theorem for non-conservative systems to quantify the work done by friction forces.
- Quantify the coefficient of static friction for a spherical body rolling along a variable slope path.
- Quantify the static friction for a spherical body rolling along a variable slope path.
- Determine the linear velocity of a spherical body rolling along a variable slope path with friction.