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

[ehrenfest]

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

 

[ehrenfest]

do people understand the problem?

 

[ehrenfest]

should I draw a picture?

 

[Dick]

An ellipse does not fit perfectly into a sine curve. I don't know what you are talking about.

 

[ehrenfest]

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.

 

[Dick]

Because they don't fit. The profile of an ellipse only resembles a sine curve. It's not an exact match.