Question

Problem: A popular brain teaser is to consider the motion of a quarter when you roll it around another one; how many times will Washington’s head rotate while the quarter rolls once around the other fixed quarter. This problem is a generalization of this puzzle. Assume a system of units such that the radius of the quarter is equal to one unit. Your task is to determine a parameterized position vector describing the location of the white dot on the rim of the rolling quarter. As the quarter moves, this dot will sweep out a curve in space. Derive the parametric equations for the curve and plot it. Include a copy or sketch of your plot. Hint: The center of the rolling quarter travels along a circle of radius 2

Answer 1

Consider that the center of the stationary coin is the origin.

Suppose the white dot is the lowermost point when it touches the +x axis.

The white dot will move with the coin, and come back to the lowermost position only when it completes one revolution.

Assuming that the moving coin has no tilt,

the x and y co-ordinate of the coin's path is given by the equation

This is the position of the bottom of the coin

The center of the rotating coin is at a position given by

and z = 1

The angular position of the white dot from its center is equal to the angular position of the coin from origin.

Let (x',y',z') be the position of the white dot

From the figure,

For angle greater than 180 degree,

for angle less than 180 degree

and

Plotting this is difficult. So, I wrote a MATLAB code to plot this

Z axis is upward.