Why do the two sliders never crash? | Do-Nothing Machine

You'll build a cardboard and foam-board version of the trammel of Archimedes, a gadget people also call the do-nothing machine. Two sliders run in slots that cross in a plus sign, both pinned to one handle, and however fast you crank, they take turns through the middle without ever colliding. The end of the handle also draws a perfect oval. Bruce's wooden version needs power tools, so it has its own detailed build plan; this one needs only a craft knife and glue.

Safety. A grown-up should do the craft-knife cutting (use a ruler and a cutting mat, and cut away from your fingers) and drill or poke the holes in the paint stick. The points of brass fasteners are a little sharp, so press them with a fingertip, not a palm.

“The word trammel means that these pieces are confined to a particular path that they can travel.”
Bruce Yeany, in the video (0:36)

Step by step

Build it

Observe

Try it & notice

Predict first: if you crank the handle really fast, will the two sliders ever crash in the middle? Try it slowly, then fast.

Stop when one slider is right in the middle. Where is the other slider? Now stop when the other one is in the middle.

Set the base on the big sheet of paper, hold a marker against the end of the handle, and turn it all the way around once. Is the path a circle or an oval?

Stop the handle when one slider has just reached the end of its trip and is about to turn back. Is that slider acting like a pivot (fulcrum)? Is it at the end of the stick or in the middle?

If your sliders stick or wobble, look at where they rub. Rub a candle along the slots and try again.

Explain

What's going on?

For young scientists

Each slider is trapped in its own straight track. A trap like that is called a trammel. The stick holds the two sliders exactly 3 inches apart, and the crossing in the middle is only about 1 inch wide. So whenever one slider is in the crossing, the other one has to be 3 inches away, out in its own track. They can only take turns.

The end of the handle draws an oval, which mathematicians call an ellipse. It is long one way because the handle end is farther from one slider than from the other. That is why the machine is also called an ellipsograph: an ellipse drawer.

The stick is also a lever. When one slider stops for an instant at the end of its trip, it works as the fulcrum (pivot point). If the stopped slider is the one at the end of the stick, the other slider is the load in the middle, like a wheelbarrow: a second-class lever. If the stopped slider is the middle one, the pivot is in the middle, like a seesaw: a first-class lever. As you crank, the stick keeps switching between the two.

For grown-ups & teachers

A trammel is any mechanism whose parts are confined to fixed paths. Here two sliders (Bruce calls them shuttles) are constrained to perpendicular lines and joined by a rigid link of fixed length d (3 inches on this build). Their positions are x = d cos θ and y = d sin θ, so when one sits at the crossing, the other is a full distance d away. With a 1-inch crossing and 2-inch sliders, any spacing above about 2.2 inches guarantees they never meet, whatever the speed. Bruce notes he has never found any reference connecting Archimedes to the device; the classical name for the same idea is the trammel or ellipsograph.

Any point P on the link, or on its extension (the handle), traces an ellipse whose semi-axes equal P's distances to the two pivots. On this build the handle end is about 11 inches from the end pivot and about 8 inches from the middle pivot, so it draws an oval roughly 22 x 16 inches; the long and short widths always differ by twice the pivot spacing. The point exactly midway between the pivots traces a circle of radius d/2. Woodworkers use the same geometry in oval-cutting router jigs. Bruce also shows the string-and-two-pins method, which works for a different reason: the sum of the distances from any point on an ellipse to the two foci is constant.

The lever interpretation, Bruce's classroom framing, comes from the link's instantaneous center of rotation, which lies at the corner of the rectangle formed by the two sliders and the crossing. When a slider is at the end of its stroke and momentarily at rest, the instant center sits on that slider, so it is the fulcrum. If that is the slider at the end of the stick, the load (the other slider) is between fulcrum and effort: second class. If it is the middle slider, the fulcrum is between load and effort: first class. In between, the instant center is off the stick entirely, which is a nice discussion point for older students.

Misconceptions: kids often assume the sliders dodge each other by luck or timing, so cranking faster should cause a crash; the fixed link makes collision geometrically impossible. Extensions: change the pivot spacing and measure how the oval changes; compare the trammel's ellipse with a string-and-pins ellipse; or look at Bruce's three-shuttle version, where the tracks are 120 degrees apart and the result is the same ellipse.

NGSS performance expectations

3-PS2-2 Make observations and/or measurements of an object's motion to provide evidence that a pattern can be used to predict future motion.
3-5-ETS1-2 Generate and compare multiple possible solutions to a problem based on how well each is likely to meet the criteria and constraints of the problem.

Science words

trammela part that can only move along one fixed path, like a slider in a track

ellipsean oval shape, like a stretched circle

levera stiff bar that turns around a pivot point to move a load

fulcrumthe pivot point a lever turns around

pivota point that something turns around (here, each brass fastener)

From Bruce's notebook

Tips from 41 years of teaching it

Bruce first wrote up his trammel in 1982 as a classroom puzzle called 'What class lever?' Here's what his notes say.

Bruce used the trammel as a lever puzzle more than a toy: stop the handle at different spots and ask which class of lever it is right now. In his notes it alternates between first class and second class.
The shape of the slide is what holds it in. On his wooden version the slides taper from about 1 inch at the bottom to about 1/2 inch at the top, and the plates are cut at the same angle. Your cereal-box overhang does the same job.
Bruce drilled several holes in his handle, because the machine works at more than one spacing and that leaves room to experiment.
He put a metal washer between each wooden slide and the handle, so the moving parts don't rub wood on wood.
Size isn't critical: his base is about 10 1/2 inches square, but it can be made larger or smaller to whatever size is convenient.

When Bruce wrote up the trammel, he admitted he had no idea what its purpose was. He decided it had a certain amusement factor, worked as a lever lesson, and drew an ellipse as a nice math bonus. In his notes, fast or slow, the one bar never collides with the other.
From Bruce's classroom

More to try

Bruce's 7th graders designed and built a compound machine: a toy or device using at least three simple machines to do a useful task, with one mechanical advantage they could calculate. Can you turn your do-nothing machine into a do-something machine?

Go further

Make it a challenge

Change the spacing

Ask a grown-up to add another hole 1/2 inch past the middle hole and move one fastener there. Draw the oval again on the same paper. Is it fatter or thinner?

String-and-pins ellipse

Bruce's other way to draw an oval: push two push pins into paper on a corkboard or thick cardboard, drop a loop of string around them, and pull a marker around inside the loop. Move the pins farther apart and the oval gets long and narrow.

Find the major axis

In part 2, Bruce shows a trammel with three shuttles on tracks 120 degrees apart and asks which one moves along the major (long) axis. Watch it slowly and see if you can tell.