Why loops need a high start
Why a roller coaster has to climb so high before it loops the loop, and the 2.7 rule that predicts it.
A loop-the-loop looks like magic. A marble, or a roller coaster, goes upside down and does not fall off. What keeps it on the track?
The bucket trick
Fill a bucket with water. Swing it quickly in a big circle, all the way over your head. The water stays in!
Now swing it slowly. At the top, the water falls out on you.
(Try it outside with a bucket you don't mind getting wet.)
What is going on? At the top of the circle, the bucket is upside down. Gravity pulls the water down, out of the bucket. But the bucket is also moving in a circle, and the faster it goes, the harder it pushes the water into its base. If the bucket is fast enough, the push is stronger than gravity and the water stays put.
A marble in a loop is the same. The track is the bucket. At the top, gravity tries to pull the marble off the track. The marble stays on only if it is going fast enough.
Try it
Pick a start height and press Release marble. The marble rolls down the ramp and into a loop with a 1 metre radius. Which start heights make it all the way round?
Did you find the line? Below some height the marble always falls off at the top. Above it, the marble gets round every time.
The 2.7 rule
The smallest start that works has a simple rule. Measured from the bottom of the loop, the marble has to start at least 2.7 times the radius of the loop.
| Loop radius | Start height needed |
|---|---|
| 0.5 m (small loop) | 1.35 m |
| 1 m (big loop) | 2.7 m |
| 2 m | 5.4 m |
Bigger loops need much higher starts. This is exactly why Marble Machine Mission 4 gives you only 1.6 m: the big loop is out of reach, but the small one is easy.
Where does 2.7 come from?
Think about the marble's energy, from what you know about stored energy and motion energy.
- To get to the top of the loop, the marble must be lifted by two radii, so that costs 2 × the radius of height.
- At the top, the marble still needs to be going fast enough to stay on the track. A marble rolling just fast enough needs the equivalent of 0.7 radii of height to pay for its speed. (A rolling marble spins too, which is why it is 0.7 and not 0.5.)
- Add them: 2 + 0.7 = 2.7 radii.
To stay on the track at the top, the marble needs speed² ≥ g × radius. Its motion energy is 0.7 × speed², so at least 0.7 × g × radius. Add the energy needed to climb 2 × radius, and the start must have g × height ≥ 2 × g × radius + 0.7 × g × radius. Divide by g: height ≥ 2.7 × radius.
Why do real roller coasters have so much extra height?
Friction takes a little energy, so real coasters start even higher than the minimum. They also don't use perfect circles. Real loops are shaped like a teardrop, narrow at the top. A tighter curve at the top means the rider stays pressed into the seat with less speed, and a gentler curve at the bottom means the riders don't feel squashed.
In the game
- Loop the loop: build a track that sends the marble round the big loop. What happens if you add lots of flat pieces? (More track means more friction, so there's less energy left.)
- Small but mighty: you only have 1.6 m. Which loop can you use?