A Comment on Stuart Boersma's article
"A Mathematician's Look at Foucault's Pendulum"
by Greg Johnson
Rancho Palos Verdes, CA
Stuart Boersma's delightful article on Foucault's Pendulum presents a calculus-based analysis of the apparent rotation of a pendulum as the earth rotates under it. There is a pleasing elementary geometrical argument that gives additional intuition into the problem. The quantity being considered, the rate of rotation of a plane tangent to the earth as the earth rotates on its axis, does not vary with time. This leads to the intuition that a precalculus argument may be possible.
The approach will be explained in a moment, but first let me slightly reframe the question. Imagine you are in a car, and inside the car you have with you a swinging pendulum. You drive the car around the perimeter of a circle on a plane. From the time you start until you stop, what is the total amount of turning you will have completed? Obviously enough, if you drive all the way around, you will have turned a total of 360 degrees, half way around will get a total of 180 degrees, etc. In each
case, from inside the car it will appear that the pendulum's direction of swing has turned the given number of degrees in the opposite direction.
Now, what happens if you drive around a circle inscribed on the outside of a cone? (Say the side of the cone forms an angle theta from the vertical.) This is equivalent to driving all the way around the earth at latitude theta, and we are interested in finding how much the direction of swing of the pendulum on our dashboard will appear to have rotated.
If you drive all the way around and end up where you started, a simple way to determine the total amount of turning you will have accomplished is to slit the cone along a straight line up to its point, and then flatten it out, creating a circular disk with a pie slice missing. From start to stop, you will have turned the fraction of 360 degrees that corresponds to the fraction of the disk that remains after the pie slice was removed. A bit of simple geometry shows that this is exactly 360 degrees times sin(), the quantity derived by Boersma.
To see this, assume for simplicity that the disk comes from a circle of radius one. The perimeter of the circle is 2, and the perimeter of the disk is 2f, where f is the fraction of the circle that remains after the pie slice was removed. Folding the disk back into a cone, the circle at the base of the cone has perimeter 2f, and therefore radius f. Form a right triangle by dropping a vertical line from the point of the cone to the plane, then distance f to the base of the cone, and back to the point on a line up the side of the cone. f is the length of the side opposite of a right triangle whose hypotenuse is 1, so f is sin(), the angle between the vertical line and the side of the cone.