# Day 43: Elliptical Conundrum

For their capstone, a pair of students wanted to do a video analysis of a marble going around the inside of a circular hoop and an elliptical hoop. They wanted to see how the speed changed as the marble traveled. They thought the marble would travel at fairly constant speed in the circular hoop, but would gain speed as it whipped around the tightening curve of the ellipse.

Although I knew that wouldn’t be the case, what they found surprised us all: the marble in the elliptical hoop decelerated much more rapidly than in the circular hoop.

Why?

(The photo shows us trying to reason it out with geometry. Does the elliptical wall’s normal force not act at a right angle to the marble’s instantaneous velocity?)

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In both cases what makes the ball slow down is the force of rolling friction. This force is directed opposite to the ball’s direction of motion and has a magnitude f that is proportional to the normal force. The normal force N exerted by the curved hoop provides the ball’s centripetal acceleration, so N = mv^2/r, where r is the radius of curvature of the hoop. For a circular hoop r has the same value at all points around the hoop, but for the elliptical hoop r is smaller — and hence the normal force N and rolling friction force f are both larger — when the ball is near the ends of the major axis of the ellipse. Depending on the relative sizes of the circular hoop and elliptical hoop and the eccentricity of the elliptical hoop, there could be a substantially larger friction force for the elliptical case.