If you roll a circle down a parabola, small circles can roll all the way down and up, whereas large circles get stuck between the sides at some point. What is the transition radius? In other words, what’s the maximum circle that keeps on rolling?

Scroll down for a solution to this problem.

## Solution

The maximum rolling circle has radius 1/2.

We will calculate the x-coordinate of the touch point of a larger circle. Call it a (the other touch point will have -a). Now we can draw the two triangles from the touch point using the radius and the tangent as shown.

The angle α has a tangent of 2a, being the derivative of x^{2} at x=a. From rotational symmetry, the angle at the circle centre is also α. Therefore we know that the side along the y-axis has length 1/2.

Applying the Pythagorean theorem we know that a^{2}=r^{2}-1/4. So, at r=1/2 the two touch points coincide. This is the maximum rolling radius.

## Poem

The Rolling Stone

That is the Sisyphus story

Pushing up to the mountain

A great stone from the earth shining beauty

Rolling it ceaselessly

From the bottom to the top

For humankind absurdity

This myth is an allegory

For our day to day activity

Please don’t forget to be happy

## One reply on “The rolling stones”

A second way:

We start with a circle k touching the parabola in the origin:

k: x² + (y – r)² = r²

p: y = x²

To calculate the points of intersection

of k and p we solve the equation

x² + (x² – r²) = r²

and we get

x²·(x² – (2·r – 1)) = 0

The solutions are:

x_1/2 = 0 and (if r ≥ ½) x_3/4 = ± √(2·r – 1)

We get the maximum circle that rolls down to the origin with r = ½.

If r is greater than ½ the circle will get stuck between the

points P_3 and P_4 with P_3/4(± √(2·r – 1)| 2·r – 1).