Why Doesn't a Spinning Top Fall Over?
― Spinning pushes the toppling force sideways
A top that isn't spinning falls the moment you let go. A spinning top can lean quite a bit and still stay up. The pull of weight hasn't gone away. When you push on something that is spinning, the push doesn't tip it over. It turns it sideways instead.
You spin a top for a child, and they ask, "Why doesn't it fall over?" You say, "Because it's spinning." They ask, "But why doesn't it fall when it's spinning?"
Look closely. The top leans a little, and its head slowly swings around in a circle. Still, it doesn't drop to the floor.
Then the spin slows. The head's swing suddenly gets faster and wider. In the end, the top wobbles and rolls onto the floor. There is a clear reason behind all of this.
There are only two reasons
When weight tries to tip a spinning top, the top doesn't fall. Its head swings sideways instead. The pull of weight is used up keeping the wobble going.
The faster a top spins, the less the same pull of weight can change its direction. As it slows, the wobble becomes faster and wider. Below a certain speed, the top can't hold itself up and falls.
Adults often stop at "because it's spinning". Let's go through the part they skip, step by step.
Why does weight push sideways?
First, think of a top that isn't spinning. Tilt it a little. Its weight acts at the middle, and the pointed tip is the pivot it tips around. If it isn't spinning, it simply falls.
A spinning top is different. Look at one small piece of its rim. Even if that piece is on top now, the spin carries it to the side almost at once. The effect of weight pushing "down" shows up where the pushed piece has moved to.
So the top as a whole doesn't fall. Its axis shifts sideways. Repeat this over and over, and the tip of the axis traces a circle. The dotted line at the top of Figure 1 is that path. This wobble is called "precession".
Why does it fall when the spin slows?
A top's "spinning momentum" grows with its spin speed. The pull of weight changes the axis's direction by about the same amount each second. If the momentum is large, that change is small by comparison, so the wobble is slow.
Friction with the floor and the air slow the spin, so the momentum drops. Then the same pull of weight moves the axis much more, and the wobble speeds up. In Figure 2, drag the slider to lower the spin speed and watch the time for one wobble get shorter.
For the model top in this article, below about 8 turns per second it can no longer keep a steady wobble. The head starts to swing wildly, and soon its edge hits the floor and it falls.
A top spun hard may start tilted, then after a while stand upright and spin quietly. This is explained by friction between the tip and the floor, which pushes the axis upright. The state where it stands straight and looks motionless has long been called a "sleeping top".
The Earth's rotation axis is pulled by the Moon and the Sun, and traces a circle about once every 26,000 years. That is why the star that serves as the pole star changes over the long run (the article on the pole star).
Summary
The pull of weight does act on a spinning top, trying to topple it. But because of the spin, that pull is spent on making the top wobble sideways instead. The faster the spin, the slower and smaller the wobble. When the spin drops, the wobble gets faster and wider, and finally the top falls.
A top doesn't avoid falling.
It keeps turning the direction of its fall sideways, through its spin.
A moving bicycle stays up for a slightly different reason from a top. You can read about it in the bicycle article. For another example of a spinning object changing direction, see the stone-skipping article.
- Spin a top with a slight tilt and time one full circle of its head with your phone's stopwatch.
- Wait until the spin has slowed, then time one circle again. The later time should be shorter.
- If you film it in slow motion, you can see the wobble suddenly get faster and wider just before the top falls.
Counting the spin speed is hard, so comparing "just after you start it" with "just before it stops" is enough. Choose a smooth, flat surface.
For those who want to know more ― terms, formulas and links to textbooksWe mark which level each part is at, from junior high science to university courses
- Junior highCovered in junior high school science
- High schoolCovered in high school physics
- High school+Advanced high school material, or treated as a textbook sidebar
- UniversityNot taught in high school; covered in university courses (mechanics, rigid-body mechanics)
- ResearchNot taught even at university as settled fact; researchers are still studying it
Junior highTerms: this phenomenon has a name
- Precession: the slow circling of the axis of a spinning object as it changes direction. It is the wobble of a top.
- Angular momentum: a quantity for the "momentum" of spinning. It is the product of how hard the object is to spin (moment of inertia) and how fast it spins.
- Moment of force (torque): how strongly a force tries to turn an object. It is the force multiplied by its distance from the pivot.
Junior highHigh schoolChecking with a formula: how many seconds is one wobble?
Suppose a disc-shaped top, 6 cm across and weighing 50 g, spins at 20 turns per second. Its centre of weight is 2 cm above the tip. Let's work out the time for one wobble.
| In symbols | Ω ＝ m × g × r ÷ ( I × ω ), time for one wobble T ＝ 2π ÷ Ω |
| In words | Wobble speed ＝ turning effect of weight ÷ spinning momentum |
| Where it comes from | It comes from the law of rotational motion: "the moment of force equals how much the spinning momentum (angular momentum) changes each second". When the turning effect of weight keeps changing only the direction of that momentum, the result takes this form. |
| Symbol | Meaning and unit |
| m | Mass of the top (kg) |
| g | Gravitational acceleration (m/s²) |
| r | Distance from the tip to the centre of weight (m) |
| I | How hard it is to spin about the axis = moment of inertia (kg·m²) |
| ω | Angular velocity of the spin (angle turned per second, radians per second) |
| Ω | Angular velocity of the wobble (radians per second) |
| Mass of the top | 0.05 kg |
| Radius of the disc | 0.03 m |
| Height of the centre of weight | 0.02 m |
| Gravitational acceleration | 9.8 m/s² |
| Spin speed | 20 turns per second |
| Moment of inertia of a disc | taken to be half of mass times radius squared |
| Radius squared | 0.03 × 0.03 ＝ 0.0009 |
| Multiply by mass | 0.05 × 0.0009 ＝ 0.000045 |
| Halve it for the moment of inertia | 0.000045 ÷ 2 ＝ 0.0000225 |
| Angular velocity (one turn is about 6.28 radians) | 20 × 6.28 ＝ 125.6 |
| Spinning momentum (angular momentum) | 0.0000225 × 125.6 ≒ 0.002826 |
| Weight (newtons) | 0.05 × 9.8 ＝ 0.49 |
| Turning effect of weight | 0.49 × 0.02 ＝ 0.0098 |
| Angular velocity of the wobble | 0.0098 ÷ 0.002826 ≒ 3.47 |
| Time for one wobble (s) | 6.28 ÷ 3.47 ≒ 1.81 |
One wobble takes about 1.8 seconds. That matches the starting position of the white dot in Figure 2. If the spin dropped to 10 turns per second, the momentum would halve and the top would wobble twice as fast, taking about 0.9 seconds per circle. This proportion is why a real top's wobble speeds up as it is about to stop.
High schoolHigh school+Seeing why force acts "sideways" with arrows
High schoolIn high school physics, the turning effect of a force is taught as the "moment of force". In a tilted top, weight and the support from the floor create a moment in the direction that topples it.
High school+Spinning momentum can be drawn as an arrow pointing along the axis. If you draw the moment of force as an arrow too, it points horizontally, at right angles to the momentum arrow. Keep adding right-angled arrows and the momentum arrow keeps its length but turns. That is what the wobble really is.
UniversityRigid-body rotation and Euler's equations of motion
In university mechanics, a top is treated as a "rigid body in rotation". Its exact motion is solved with Euler's equations of motion, or with the Lagrangian form using Euler angles. The formula above is an approximation for a fast enough spin (the gyroscopic approximation). The boundary of about 8 turns per second at which this model falls was estimated from an inequality called the "sleeping top stability condition". The small up-and-down nodding of the axis is called "nutation".
📖 Derivations and further reading: Precession (Wikipedia, in Japanese) ／ Angular momentum (Wikipedia, in Japanese)
ResearchWhat is still not well understood
- Exactly how to treat friction at the tip How sliding and rolling mix between the tip and the floor changes a lot with conditions. Predicting exactly how fast a top rights itself is still said to be difficult.
- The flipping top (tippe top) Some tops turn upside down when spun. The broad mechanism is explained by friction, but the fine details of the motion in between are still being studied.
- Behaviour just before stopping Just before the spin dies out, the contact with the floor becomes intermittent. This motion is said to be too complex for simple formulas.
So even this article gives "the explanation as far as we understand it today". The wobble itself was solved long ago, but some points involving friction with the floor remain unresolved.
Links to textbooks (by level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Junior high | Science: how forces act | Weight acts downward at the middle |
| High school | Physics: balance of rigid bodies (moment of force) | The turning effect that tries to topple a tilted top |
| High school+ | Advanced physics: angular momentum | Only the direction of the momentum arrow changes |
| University | Mechanics: rigid-body rotation | Euler's equations of motion, the sleeping-top stability condition |
| Research | Nonlinear dynamics, mechanics of friction | Righting itself, the flipping top, motion just before stopping |
| ― | Links to daily life | Playing with tops, the slow wobble of the Earth's axis |
- Landau and Lifshitz, Mechanics (Japanese edition: ランダウ・リフシッツ『力学』, Tokyo Tosho), chapter on rigid-body motion
- Morikazu Toda, The Science of Spinning Tops (戸田盛和『コマの科学』, Iwanami Shoten)
- Feynman, Leighton and Sands, The Feynman Lectures on Physics I: Mechanics (Japanese edition: ファインマン・レイトン・サンズ『ファインマン物理学 I 力学』, Iwanami Shoten), chapter on rotation
- Precession - Wikipedia (in Japanese)
※This article is a science explainer for general readers. The numbers given are rough guides to help you understand how things work. The wobble speed and the spin speed at which a top falls vary with the top's size and shape and with the floor material.