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

In summary, the problem is to find the relationship between the semi-axes of an ellipse, a and b, and the amplitude of a sine curve, c, when the ellipse rolls without slipping on the curve y = c sin(x/a) completing one revolution per period of the sine curve. The correct relationship is b^2 = a^2 + c^2, determined by requiring equal arc lengths for one period. The approach of setting a = c and b = πa/2 does not work because an ellipse does not fit perfectly into a sine curve.
  • #1
ehrenfest
2,020
1

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

 
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  • #2
do people understand the problem?
 
  • #3
should I draw a picture?
 
  • #4
An ellipse does not fit perfectly into a sine curve. I don't know what you are talking about.
 
  • #5
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.
 
  • #6
Because they don't fit. The profile of an ellipse only resembles a sine curve. It's not an exact match.
 

1. What is an ellipse?

An ellipse is a type of geometric shape that is formed by the intersection of a cone and a plane. It is a closed curve with a central point called the center, and two focal points which are equidistant from the center. It looks like a flattened circle.

2. What is a sine curve?

A sine curve is a mathematical function that represents a smooth repetitive oscillation. It is a graph of the sine function, which is a ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. It appears as a smooth, wavy line.

3. What is the relationship between an ellipse and a sine curve?

An ellipse can be described as a stretched or compressed circle, and the shape of a circle can be represented by a sine curve. The equations for an ellipse and a sine curve are also similar, with the only difference being a scaling factor. Therefore, an ellipse and a sine curve are closely related and can be seen as different representations of the same underlying concept.

4. What are the properties of an ellipse?

An ellipse has several important properties, including the major axis (the longest diameter across the center), the minor axis (the shortest diameter across the center), the eccentricity (a measure of how elongated the ellipse is), and the foci (the two points that determine the shape of the ellipse). It also has a perimeter and area that can be calculated using specific equations.

5. How are ellipses and sine curves used in real life?

Ellipses and sine curves have many practical applications in various fields, such as engineering, physics, astronomy, and architecture. In engineering, they are used to design gears, pulleys, and other mechanical components. In physics, they are used to describe the motion of objects in circular or oscillatory motion. In astronomy, they are used to predict the orbits of celestial bodies. In architecture, they are used to create aesthetically pleasing designs, such as arches and domes.

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