Why Do a and b Need to be Related to c When an Ellipse Rolls on a Sine Curve?

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Homework Help Overview

The problem involves an ellipse with semi-axes of lengths a and b rolling without slipping on the curve defined by y = c sin(x/a). The original poster seeks to understand the relationship between a, b, and c, particularly why the relationship b² = a² + c² is necessary, and questions the validity of simpler assumptions regarding the dimensions of the ellipse.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the original poster's assumptions about the dimensions of the ellipse and whether they can be simplified. Questions arise about the fitting of the ellipse to the sine curve and the implications of different approaches to defining the ellipse's dimensions.

Discussion Status

The discussion is ongoing, with participants exploring various interpretations of the problem. Some have offered insights into the mismatch between the ellipse and the sine curve, while others are questioning the original assumptions and seeking clarification on the relationship between the parameters.

Contextual Notes

There is a mention of the ellipse completing one revolution per period of the sine curve, which may influence the relationship being sought. Additionally, the original poster's assumptions about fitting the ellipse into the sine curve are under scrutiny.

ehrenfest
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Homework Statement


an ellipse whose semi axes have lengths a and b rolls without slipping on the curve y =c sin (x/a), find the relationship between a, b, and c. Assume that the ellipse completes one revolution per period of the sine curve.

The answer is b^2 = a^2 + c^2 and you find it by requiring that the arclengths be the same for one period.

Why is it wrong to just require that a = c and b = pi a /2 ? That would seem natural to me because then one half of the ellipse would fit perfectly into one "hump" of the sine curve?

Homework Equations


The Attempt at a Solution

 
Last edited:
Physics news on Phys.org
do people understand the problem?
 
should I draw a picture?
 
An ellipse does not fit perfectly into a sine curve. I don't know what you are talking about.
 
My approach was to make the ellipse have minor axis equal to half the period of the sine curve and a semi-major axis equal to the amplitude of sine curve. All I want to know is why that approach produces ellipses that are different from the ones in the answer.
 
Because they don't fit. The profile of an ellipse only resembles a sine curve. It's not an exact match.
 

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