# Spinning Cylinder

A piece of pipe with a mark at each end is set rotating and spinning at the same time. In the blur of the moving cylinder, one of the marks appears three times, forming a stationary triangle.

- Cut a piece of PVC pipe that is three times as long as its diameter. (Err on the long side; this experiment will still work with tubes up to 3.15 diameters long.)
- Make a simple mark near one end of the pipe and a different mark at the other end. For example, in the image below, the pipe is marked with an
*X*and an*O*; in the photos at the top of this page, it's marked with a plus and a minus sign—it doesn't matter what the marks are, so long as they are different. - Cut a few more lengths of pipe so that you have a set of pipes with lengths that are two, three, four, and five times their diameters. After the first experiment, you'll need these additional pieces of pipe.

Take the piece of pipe that is three times as long as its diameter. Place your finger on the *X* and in one quick motion, push down while at the same time pulling the pipe toward you to make the pipe spin. (See the animation below.)

The pipe cylinder will spin and rotate, making a blurred circle in which you should be able to see *X*s. Notice that as the spinning cylinder stabilizes, you can see three *X*s that mark the vertices of a triangle. Notice that the *O* does not appear.

Next place your finger on the *O* and spin the cylinder. Notice that as the motion stabilizes, three *O*s appear, each at the vertex of a triangle. Notice that the *X* does not appear.

Try some experiments to figure out what's going on. Here are some suggestions:

- Make several markings on one end.
- Look at the spinning cylinder from underneath, through a transparent table.
- Look at the spinning cylinder in sunlight (which doesn't strobe on and off the way fluorescent lights do).
- Look at the spinning cylinder with a stroboscope (a handmade stroboscope in which slits are cut into the edges of a spinning disk works just fine).
- Draw a line down the side of the cylinder. Make one half of the line red and the other half blue.
- Try cylinders of different lengths. Notice the different stable patterns.
- Notice the cylinder rotates with one end in contact with the table surface.
- Notice that the center of the spinning cylinder is above the table.

*Note: To get the most out of this activity, don't read this section until you've done some experiments yourself!*

When you launch the cylinder, it spins around its long axis and rotates around a line perpendicular to this axis. As it rotates about its center, the cylinder forms a blurry circle on the table top. As the cylinder spins, the top of one end moves in the same direction as the end that is rotating, while the top of the other end moves opposite the rotation. The arrows in the diagram below, showing the pipe as seen from above, illustrate these relationships.

The two arrows within the cylinder show how it spins. The two arrows outside the cylinder show how it rotates. On the right end, the two motions cancel each other, and when the mark on the spinning cylinder is at the top, it actually stops momentarily (see the Going Further section below for the mathematical explanation). On the left end, the two motions add together, and when the mark on the spinning cylinder comes to the top, it moves twice as fast as it would with either motion alone (again, see the Going Further section below for the mathematical explanation).

Human eyes can see the stopped mark easily, while the extra-fast moving mark is a blur. Thus, only the mark on one end is visible. Since we see three marks around the blurred circle, we know that the cylinder is making three spins for every rotation (see the Going Further section below for the mathematical explanation). Cylinders that are cut so that their lengths are four diameters have a stable square with four markings. Those cut to two diameters create a stable pattern of two marks.

At first, using the cylinder that is three diameters long, the marks on one end appear but they do not form a stable pattern. After a few seconds, however, the marks settle into a stable triangular pattern which persists until the cylinder slows to a stop. To understand this behavior, notice that the cylinder spins and rotates with one end on the table and one end in the air. The cylinder makes a stable pattern when the end touching the table rolls without slipping. Usually the cylinder is launched so that it is spinning faster than it is rotating. This means that the end touching the table rubs against the table, dissipating energy and slowing down until it reaches a speed where it rolls without slipping. This is why the pattern is not stable at first, but then stabilizes.

The discussion below gives the "Math Root" of some of the behaviors noted in the previous section. References to the left and right ends of the cylinder refer to the following diagram, which shows the pipe as seen from above:

The circumference of the cylinder’s rotation is *pi* times its length:

*C = πL *

The circumference of the cylinder itself is *pi* times its diameter:

*c = πd*

So the number of times that the cylinder spins in one rotation is *C/c = 3*. This is why there are three markings!

*C= 3πd = 3c*

The linear speed due to rotation at either end of the cylinder is:

*V ^{rotation} = distance/time = 3πd/T*

where *T* is the time for one whole rotation.

The linear speed due to spin at either end of the cylinder is:

*V ^{spin} = distance/time = πd/t*

where *t* is the time for one whole spin. But there are three spins for every rotation, so *T = 3t*, and therefore:

*V ^{rotation} = 3πd/T = 3πd/3t = πd/t = V^{spin}*

This shows that the two speeds are equal (see the arrows in the diagram at the beginning of this section). This means that for the instant that the marks are facing upward, the mark at the right end actually comes to a stop, since the speeds are in opposite directions, and the mark at the left end moves doubly fast, since the speeds are in the same direction.