The loop-the-loop can be found in many physics classrooms. In the classic demonstration, which dates to the mid 1800s, a ball or cart starts from rest at a certain height, travels down a track, and completes a vertical loop. Students apply their knowledge of conservation of energy and kinematics to predict the minimum height needed to complete the loop. But as students and teachers often discover, the apparatuses need larger starting heights than predicted by those simple methods. Somewhere, the ball or cart loses energy. The loss is usually dismissed as arising from “nonconservative forces,” says Karl Mamola of Appalachian State University in North Carolina.
That hand-wavy explanation wasn’t good enough for Mamola and his collaborator, Toby Dittrich of Portland Community College in Oregon, both of whom have taught physics for more than 30 years. “We wanted to learn exactly what’s going on with the nonconservative forces,” Mamola says.
In a recently published paper in the American Journal of Physics, the pair used high-speed video analysis to reveal finer details of a ball’s motion through a loop. They found that most of the energy loss occurred as the ball transitioned from the linear portion of the track to the loop.
The forces that govern the loop-the-loop are, in theory, straightforward. Gravity pulls down, the normal force pushes up from the track, and friction opposes the motion. Combined, however, they form a complicated system with many possible sources of energy loss.
Dittrich initially focused only on geometric considerations. His track was made of a long, flat strip of aluminum folded at a right angle to form a V-shaped trough. The resulting track was then bent into the loop-the-loop shape. “It occurred to me that the effective radius for rolling is less than the radius of the ball,” he says. With a smaller effective radius, the ball needs a higher start to traverse the loop.
But even that adjusted height didn’t match up with experimental observations. Without accounting for energy loss, students would find that the ball should be released at a height of 2.9 times the radius of the loop; for Dittrich and Mamola’s track, the release height would need to be 58 cm. But the researchers observed that the initial height had to be 91 cm for the ball to complete the loop while maintaining contact with the track. “You find that there is about a 50% energy loss someplace,” Mamola said. “We got to thinking, how can we understand that energy loss?”
To do that, Mamola recorded the ball’s motion and analyzed the video to gather information about its position. The position-versus-time data were used to calculate velocity and total energy, which Dittrich and Mamola then compared with the output of a model they created of the ball’s motion.
As expected, some energy was lost between the initial drop and the start of the loop, primarily because the ball was rolling and slipping. But Dittrich and Mamola were surprised to find that the largest decrease in total energy occurred at a specific location on the track: the start of the loop. The sudden transition between linear and circular motion cost the ball one-fifth of its initial energy.
The researchers’ model shows that the energy loss is mainly due to the ball’s large rate of change of acceleration, or jerk. “It turns out that the normal force on the ball changes by almost a factor of 10 at that point, and the change is almost instantaneous,” Mamola says. “The reaction to that force causes the track to deform somewhat at the expense of the energy of the ball.” Jerk can be minimized with a transition curve that creates a gradual change from linear to circular motion. Therefore, different tracks should have different levels of energy loss.
In addition to providing a better understanding of the physics involved, Dittrich and Mamola want their work to have a positive impact on physics education. Their paper uses language that is accessible to introductory physics students.
The researchers also hope that someone will pick up their work where they left off. Undergraduate researchers could investigate open questions about rolling with slipping and track deformation during loop-the-loop motion. “This study is a good example of how teachers can get their undergraduate students involved in more realistic research projects,” Dittrich says. “That training is vital for the success of those students in reaching graduate school.”
