How high do you have to start to loop the loop? | Marble Tracks & Loop-the-Loop

Bruce has built dozens of marble tracks for his classroom, from a Galileo track to racing tracks to a loop-de-loop, all to show how height turns into speed and back again. You'll build a loop-the-loop track from foam pipe insulation, then find the lowest start that still gets a marble all the way around. Most people guess you only need to start as high as the top of the loop. You need more, and finding out how much is the fun part. Bruce bends his tracks from metal shelf track on wooden frames; detailed build plans for two of his classroom tracks are on the site.

Safety. A grown-up should do the long cuts in the foam tube. Marbles are a choking hazard for small children, so keep them away from little brothers and sisters and put them back in a cup when you're done. Marbles on the floor are easy to slip on. Measure heights while standing on the floor, never by climbing on the table.

“Now that we have the track done, it would be an interesting investigation to compare the height of the loop to the minimum height that we could release the ball and still make it.”
Bruce Yeany, in the video (6:14)

Step by step

Build it

Observe

Try it & notice

Predict first: measure how high the top of your loop is above the floor. What is the lowest flag a marble can start from and still go all the way around? Lower than the top of the loop, the same, or higher? Write your guess down.

Start a marble from a low flag. How high does it climb inside the loop before it falls off or rolls back? Does it ever get higher than the spot where it started?

Move up one flag at a time. Find the lowest flag where the marble makes it all the way around three times in a row. How many inches higher is it than the top of the loop?

Watch closely at the top of the loop on a run that just makes it and on one that just fails. Where is the marble moving slowest? Where does it fall off?

Try a different ball, like a steel ball bearing or a bigger marble. Does the lowest starting flag change?

Explain

What's going on?

For young scientists

A marble held up high has stored energy, called potential energy. As it rolls down, that stored energy turns into speed, called kinetic energy. When it rolls uphill again, the speed turns back into height. A marble can never climb higher than where it started, and because rubbing (friction) steals a little energy, it always ends up a bit lower. Galileo noticed this hundreds of years ago, and Bruce's Galileo track shows it.

At the top of the loop the marble is upside down. It only stays on the track if it is still moving fast enough to keep pressing against the track as it curves around. If it gets to the top too slowly, gravity pulls it down and it falls.

So starting level with the top of the loop is not enough: the marble would arrive with no speed left. It needs extra height to still be moving at the top, a little more to make it spin as it rolls, and a little more to make up for friction. That is why the lowest start that works is quite a bit higher than the loop.

For grown-ups & teachers

This is conservation of energy plus circular motion. To stay on the track at the top of the loop, the ball needs enough speed that gravity alone can't pull it inward faster than the circle curves: v^2 >= g r at the top, where r is the radius of the path of the ball's center. Energy from release height h (measured from the bottom of the loop) is shared between the height of the top (2r), forward motion and spin. For a solid ball rolling without slipping, total kinetic energy is 7/10 m v^2, so the minimum release height is h = 2.7 r, about 1.35 times the loop's height. A block sliding without friction would need 2.5 r. Bruce's part 2 video ends with exactly this comparison of a measured minimum height against the calculated one.

Expect your measured minimum to be higher than 1.35 times the loop height. Foam has much more rolling resistance than metal track, tape joints and kinks cost energy, and a homemade loop is rarely a perfect circle (a tight spot at the top needs more speed). The channel itself matters too: the ball rides on the two edges of the channel rather than on its bottom, so it rolls on a smaller radius and spins faster for the same forward speed. More of its energy goes into spin, leaving less for forward speed. Bruce's track notes raise the same question about how fast balls rotate on rails of different widths.

Common misconceptions: that starting at the loop's height is enough (the ball must still be moving at the top), and that 'centrifugal force' holds the ball up. From the ground, nothing pushes the ball outward: the track pushes it inward, and the ball stays on as long as it is moving fast enough that it would otherwise fly off in a straight line (inertia). Roller-coaster loops are teardrop shaped, tighter at the top, so riders can take them at lower speed with gentler forces at the bottom.

Bruce's part 1 video shows the other energy ideas: on his Galileo track (one side three times longer than the other) the ball returns to nearly its starting height whichever side it starts on, Galileo's route to the idea of inertia. On the identical curved tracks, balls released from different points on matching circular curves reach the bottom together, just as pendulums of equal length keep the same period at different swing sizes. His 'stringless pendulum' tracks are curved like pendulums with periods of 1, 2 and 3 seconds, showing why longer pendulums swing more slowly. His high road, low road tracks show a longer path through a dip winning the race.

NGSS performance expectations

4-PS3-1 Use evidence to construct an explanation relating the speed of an object to the energy of that object.
MS-PS3-2 Develop a model to describe that when the arrangement of objects interacting at a distance changes, different amounts of potential energy are stored in the system.
MS-PS3-5 Construct, use, and present arguments to support the claim that when the kinetic energy of an object changes, energy is transferred to or from the object.

Science words

potential energystored energy; a marble held up high has it because it can roll down

kinetic energythe energy something has because it is moving

frictiona rubbing force that slows things down and steals a little energy

gravitythe pull that makes things fall toward the ground

inertiathe way a moving thing keeps going in a straight line unless something pushes or pulls it

minimumthe smallest amount that still works

From Bruce's notebook

Tips from 41 years of teaching it

Bruce built marble tracks for his 7th and 8th graders over four decades. Here's what his notes add.

Bruce opened his 8th-grade energy unit with the Galileo track and one question: why won't the ball roll off the other side? The lesson he drew from it: you can never get more energy out of a machine than you put in.
In his notes on sharing track plans, Bruce says tracks can be made higher and steeper, but not so steep that a ball slips as it rolls uphill. The balls rely on friction to keep turning as they roll.
His 'stringless pendulum' tracks are curved like pendulums 24.9 cm, 99 cm and 224 cm long, with periods of 1, 2 and 3 seconds. The lengths go up 4 times and 9 times (2 squared and 3 squared), and he adds 0.5 cm to each to allow for the thickness of the metal track.
To draw a pendulum curve full size, Bruce anchors a long string under a heavy weight on the floor, wraps the other end around a pencil, and swings the pencil across paper taped to the floor.
Bruce points out that ball-on-track 'pendulums' run slower than real pendulums of the same length, because of friction and because some energy goes into making the ball spin.
His race-track notes say balls of different sizes run down a track together, as long as each one sits on its rails at the same ratio of ball size to rail spacing. He leaves 'why?' as a question for students.

In 1987 Bruce wrote a joke instruction manual for a ball-on-a-track 'Multi-Functional Potential Kinetic Random Result Generator'. Students could supposedly use it to pick multiple-choice answers, and teachers to hand out grades, propping up one end to improve the odds. The real use comes at the end: release the ball from the same spot several times, watch it stop in different places, and have students list every small thing that might change the result.
From Bruce's classroom

More to try

From Bruce's race-track notes: race balls made of different materials, such as brass against aluminum, on the same track and see whether there is a difference.

Go further

Make it a challenge

Galileo's valley

Bruce's Galileo track starts at the same height on both sides, but one side is three times longer. Tape a foam channel into a U with one short steep side and one long gentle side, roll a marble from one end, and put a tape mark where it turns back. Does it ever roll out the other end?

Bigger loop, bigger start?

Re-curl your loop larger or smaller and find the lowest starting flag again. Divide the lowest start height by the loop height each time. Does the answer stay about the same?

Bowl race

Bruce's identical curved tracks show that balls released from different spots on the same curve reach the bottom together. Release two marbles at the same moment from different heights on opposite sides of a wide, shallow bowl. Where do they meet?

Prediction track

Bruce's joke 'prediction tracks' let a ball roll back and forth until it stops on a label: 'Yes', 'Chances are slim' or 'No way'. Let a foam channel sag in a gentle valley between two book stacks, label the bottom, and release a marble from the same spot five times. Does it stop in the same place every time?

Long way or short way

Bruce's favorite tracks race a ball along a straight road against one that dips down low. Use your second channel to try it; there's a whole page on the high road, low road race.

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