[Univ of Cambridge] [Dept of Engineering]

A spinning top

Dr Hugh Hunt
October 2009

For fun stuff on spinning things go to my
Dynamics movies page
What stops a spinning top from falling over? Gyroscopic effects are what it's all about. Here are two explanations, including one that doesn't rely on any knowledge of the gyroscopic effect. See which you prefer. One thing's for sure, anything to do with the gyroscopic effect is counterintuitive!

Explanation A:

What stops a spinning top from falling over?
1. A spinning top is governed by the gyroscopic effect.
2. A gyroscope will spin about a constant axis unless acted on by a couple* - eg the Earth's axis is at a constant 23.5 degrees, kept stable by the spin of the Earth.
3. The faster a gyroscope spins, the bigger the gyroscopic effect - ie the more resistant the gyroscope is to any disturbing couple.
4. For the top, gravity acts down through its centre (blue arrow in the picture above), and an equal force acts up where the tip sits on the table (black arrow)
5. If the top is tilted then these two forces are not opposite each other, as in the picture, so generating a couple.
6. This couple is fixed, independent of how fast the top spins so with paragraph 3. the top is more stable the faster it spins because gyroscopic effects dominate.

(* the word "couple" is synonymous with the words "torque" and "moment" - I like the word "couple" because it conjures up the idea of two forces, like the ones shown in the picture. A couple is just any pair of equal-and-nearly-opposite forces that act to twist something around.)

Gyroscopic effect:

7. A spinning rotor has an axis of spin.
8. A couple acting about this axis can only ever change the spinning speed
9. To change the direction of the axis of spin the only remaining possibility is to apply a couple at right angles to the spinning axis.
10. The axis of spin will deviate so as to direct its spin in the direction of the applied couple. This is called Gyroscopic Precession".

11. More precisely, "the angular momentum changes in the direction of the applied couple". Angular momentum is a measure of how fast the rotor is spinning. With reference to the diagram, the spin axis (angular momentum) is the black arrow and it responds to the applied couple (green arrow) by changing in that direction (red arrow) to give a new spin axis (blue), and this change in direction is called "precession" (orange).
These "arrows" are called "vectors", but never mind that. To figure out the arrow relates to the direction of spin just point along the arrow and then wiggle your finger clockwise.

There's an important result from all of this - you get slower precession if the spin is fast or if the couple is small. This is because the smaller the couple, or the larger the spin (angular momentum) then the change in spin direction (red arrow) is smaller.
see animations.

[note: this is exactly analogous to the way velocity changes in circular motion: a force in the direction of motion will cause an object to speed up or to slow down, while a force at right angles to the direction of travel will change the direction of travel. The moon, for example, goes around the Earth at constant speed, but its' direction of travel is changing continuously due to the force of the Earth's gravity.]

There's quite a nice video here showing how a gyro always points in a constant direction. See other gyro links here.

Explanation B (less technical):

What stops a spinning top from falling over?
All we need to show is that the force of gravity is insufficient to cause the top to fall.
1. The only force acting to push the top over is gravity.
2. First think what happens when the top is not spinning. It falls over and we get a feel for how long it takes for the top to fall and how fast it is going when it falls.
3. This gives us a feel for how much angular motion that gravity alone can give to the top if it were to fall over. Let's call this angular motion "spin", but note that gravity can only create "spin" about a horizontal axis.
4. Now when we get the top going normally (spinning about a vertical axis) we can tell that we are giving the top much more "spin" about the vertical axis than gravity managed to achieve about a horizontal axis when the top fell over.
5. With the top spinning really fast, let's suppose the top were to manage somehow to fall over, with its axis horizontal. Then we would see that the top possesses a large "spin" about a horizontal axis. The "spin" we'd then have is far greater than anything gravity alone can produce. Since there are no other forces acting other than gravity, the top can't have fallen over (if not gravity, what else is there to generate the required "spin" we need about a horizontal axis?).
6. Things are different if the top is spinning quite slowly, roughly the same sort of speed as the top got to in 2. Then we see the top is verging on toppling. This is because gravity is now capable of producing the "spin" required about the horizontal axis as observed when the top falls over.
7. If the top is spinning very fast then it is ultra stable because the effect that gravity can produce is way too small.
[note: to make this read more technically correctly, replace the word "spin" when it occurs with the words "angular momentum"]

The minimum spin speed (in revs per second) for a typical top to be stable is roughly equal to √g/a where g=9.81m/s2 and a is the radius of the top. This is about 10 revs per second for a typical "pump-up" toy top, like the one shown in the picture at the top of the page.

(For other gyroscopic things: go to Gyro and boomerang page

and for all sorts of video clips on spinning things go to: Dynamics movies page

and for other stuff: go to Hugh Hunt's Cambridge University home page )