Energy loss and jerk on the loop-the-loop

Abstract

In its usual form, the loop-the-loop (LtL) problem involves a uniform solid sphere rolling from rest down a linear ramp that transitions into a circular loop. The task is to find the minimum height from which the ball must be released in order to roll completely around the loop without breaking contact. The answer, found using the conservation of mechanical energy and Newton's second law, is invariably less than the actual measured height. The difference, attributed to non-conservative forces, is consistently larger than the experimental uncertainty. To get a more detailed understanding of the effects of dissipative forces on the loop-the-loop, we made high speed video recordings of balls moving on the commercial LtL apparatus and used video analysis to study their motion in detail. We present our results along with a simple model to predict the motion of the ball on an LtL track taking energy losses into account. Calculations based on the model are in excellent agreement with our measurements.