Nib asked
Why does a spinning top not fall over?
Gravity does not push a spinning top down, it pushes its angular momentum sideways, so the top circles instead of falling, until friction slows the spin.
7 steps, drawn by Nib. Read it here, or watch him draw it. Updated .
1the puzzle
Balance a top on its point without spinning and it falls over at once. Spin it fast and it stays upright, even though gravity is pulling on it the whole time. Spinning is not a trick that switches gravity off. It changes how the top responds to gravity's pull.
The spin is what changes how gravity acts on a top.
a common mix-upGravity does not stop acting on a spinning top. The top responds to it differently.
Is gravity weaker on a spinning top?
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No. Gravity pulls exactly as hard. The spin changes the response, not the pull.
2one point
A tilted top touches the floor at a single point, its tip. Gravity acts on the top's centre of mass, which is off to one side of that point. A force acting off to the side of a pivot is a torque, a twisting push. Whatever the top does next, torque has to explain it.
A tilted top feels a torque from gravity about its tip.
3spin is L
A spinning object has angular momentum L. It is a vector that points along the axis of the spin, by the right-hand rule. For a top, that is along its axle, and its size is the moment of inertia times the spin rate, L equals I times omega. A fast spin means a large L.
L points along the axle, and its size is I times omega.
4the torque
Gravity pulls down at the centre of mass, which sits to one side of the tip. Torque is the position vector crossed with the force, and that cross product points sideways: horizontal, and perpendicular to both the tilt and gravity. Here it points straight into the page.
Torque equals r cross F, so it points sideways, not down.
Which way does the torque from gravity point on a tilted top?
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Horizontally, perpendicular to both the tilt direction and gravity.
5precession
Newton's second law for rotation says the rate of change of L equals the torque. So L changes in the direction of the torque, which is sideways. L's size barely changes, only its direction, so the tip of L traces a circle and the whole top swings slowly around the vertical. That slow circling is called precession. The top is never pushed down, because the push is always at right angles to where L already points.
Torque equals the rate of change of L, so L turns sideways instead of falling.
a common mix-upGravity does not push the top down. It turns the top's angular momentum around the vertical.
Why does the top circle instead of tipping?
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The torque is perpendicular to L, so it turns L's direction rather than tipping it over.
6how fast
Set the torque, m g r sin theta, equal to the rate of change of L, which is L sin theta times the precession rate. The sines cancel, leaving the precession rate equal to m g r divided by L, where r is the distance from the tip to the centre of mass. For a toy top spinning about 20 turns a second this gives roughly one lap every second or so; halve the spin and it laps twice as fast.
Precession rate equals m g r divided by L, so faster spin means slower precession.
A top's spin halves. What happens to its precession rate?
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It doubles, because the rate is inversely proportional to L.
7why it falls
Friction at the tip and air drag slowly rob the top of spin, so L shrinks. Since the precession rate is m g r over L, the circling speeds up and the wobble widens. Below a minimum spin the top can no longer hold itself up, and it tips over. A spinning top is not defying gravity, only delaying it.
Friction shrinks L, so the precession speeds up until the top falls.
a common mix-upA spinning top never fights gravity. It only delays falling until friction wins.
A spinning top is never defying gravity: its spin turns the push sideways, until friction slowly takes the spin away.