If one spins a spinning top, it'll stay up for a while before falling down. However, if one spins a pencil, it falls down immediately, just like if you didn't spin it at all. My question is why there is a difference, and what equations govern this motion. I understand gyroscopic precession, but I can't find a connection between not falling down and precession. The equations for precession say that the pencil should just have a relativley high precession rate (very low moment of inertia), but it doesn't say anything about it falling down.
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2$\begingroup$ Think or the moment of inertia. With a pencil, the mass is very close to the axis of rotation. A top is often shaped so that most of its mass is at the outer edge of its radius. Think too of the rotational speed: very low for a pencil, but a top can be accelerated with a string or a pump action. With its low mass and rotation a pencil isn't really a gyroscope. There is not enough inertia to prevent it toppling immediately. $\endgroup$– Weather VaneCommented Oct 26, 2022 at 16:32
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$\begingroup$ You have answered half of the question. If you give any gyroscope a very low angular velocity, it won't precess. It will just fall. For a pencil, you will have to give a much higher initial angular velocity for it to start precessing. $\endgroup$– Ashmit DuttaCommented Oct 26, 2022 at 16:50
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1$\begingroup$ As well as its small radius, a pencil doesn't even have much mass anyway. $\endgroup$– Weather VaneCommented Oct 26, 2022 at 17:09
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$\begingroup$ How come it will just fall? I understand intuitively why that is, but that's not what the equations say. $\endgroup$– Eli YablonCommented Oct 26, 2022 at 17:51
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2 Answers
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You write:
"The equations for precession say that the pencil should just have a relativley high precession rate (very low moment of inertia), but it doesn't say anything about it falling down."
Well, the equations that are usually presented are a simplified form, a simplified form that only works well for high spin rates, and high moment of inertia.
The simplified form suggests that the a spinning object will start precessing instead of falling over.
I get the impression that you are suggesting: "Sure, the pencil is spinning slowly, but if any spinning object will always precess instead of falling over that shouldn't matter."
For fast spinning object the simplified form is an acceptable approximation. I assume that is why in physics textbooks only the simplified form is given
It is essential, however, to not think of onset of precessing motion as happening instantaneously. It is a process that takes a certain amount of time, and the slower the spin rate the more time it takes.
For a discussion of gyroscopic precession, and the process of onset of gyroscopic precession, see my 2012 answer about gyroscopic precession.
No matter how fast an object spins, it is never quite the case that the gyro wheel start precessing instead of dropping down a little. A gyro wheel that is released always drops a little. (But yeah: the faster the spin rate the more it looks as if the transition to precessing motion is instantaneous.)
This property has been experimentally verified, with a table-top experiment:
Svilen Kostov and Daniel Hammer, 2010
'It Has to Go Down A Little, In Order to Go Around'- Following Feynman on the Gyroscope
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A top spins around its axis of maximum inertia. That means it has the minimum energy for a given angular momentum. Thus, friction does not destabilize the rotation. Your pencil is spinning around its axis of minimum inertia. That means it has the maximum energy for its angular momentum. Friction thus destabilizes the rotation, causing it to evolve toward a lower energy spin around a different axis.
The Explorer 1 satellite was supposed to be spin-stabilized to rotate like your pencil, but the tiny energy dissipation due to structural flexing caused the spin axis to settle down perpendicular to the intended axis.
