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When we spin a top (give it kinetic energy $K_0$) and place it on a table, it starts precessing around the vertical axis.

Is the total energy of the spinning and precessing top equal to the initial spinning energy $K_0$?

If your answer is a Yes, please explain how the energy transfers from spinning axis to precession axis? i.e. How the spinning speed is reduced, while the moment of weight is perpendicular to the spinning axis, how can it reduce the amplitude (speed) of the spin?

If your answer is a No, please explain how the moment of weight can do any work, when there is no displacement? (angle between the top and vertical is constant).

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The mechanics of the onset of gyroscopic precession is discussed in a 2012 answer (submitted by me): gyroscopic precession

Indeed the total amount of energy is conserved. The only change of energy is due to friction.

Gyroscope

The above image is from the earlier answer I linked to. In the case depicted in the image the gyro wheel is gimbal mounted, so the center of mass of the blue gyro wheel cannot change height.

The brown cylinder represents a weight that has been attached to the fixed blue rod. Initially that weight is supported. (The support is not shown in the image).

When support of the brown weight is removed the immediate effect is pitching motion of the gyro wheel. In response to the pitching motion the gyro wheel starts to swivel.

In classroom demonstrations the demonstrator will invariably release the weight gingerly.

If the release is instantaneous the resulting motion is that the spin axis is rapidly sweeping out a narrow angle cone. That motion is called 'nutation' (after the latin word for 'nodding').

Such nodding motion makes the demonstration look messy, so demonstrators invariably develop the habit of suppressing the nutation as they release. (Most likely most demonstrators are unaware they are actually suppressing an intrinsic aspect of the gyroscope mechanics.)


Demo with fast spinning gyro wheel

When the gyro wheel is spinning very fast the nutation is accordingly fast, and the amplitude of the nutation is very small. The energy of the very fast nutation tends to dissipate rapidly in the bearings of the gyro wheel, leaving the gyroscopic precession. The gyroscopic precession tends to last a long time because it experiences comparitively little friction.

To the onlookers it appears as if the gyro wheel has started to precess instead of being pulled downward by gravity. That is in fact not the case: the gyro wheel does yield to gravity, but only initially.


The kinetic energy of the precessing motion is provided by conversion of gravitational potential energy.

In the case of the setup depicted in the image:
the brown weight sags.

In the case of a spinning top:
Assume the spinning top has an initial lean. Upon release that lean becomes a bit bigger, with a corresponding sag of the center of mass of the spinning top. That conversion of gravitational potential energy provides the kinetic energy of the precessing motion.


This gyroscopic precession mechanics has been corroborated in a 2010 tabletop experiment by Svilen Kostov and Daniel Hammer. It has to go down a little, in order to go around

Journal publication:
The Physics Teacher, Volume 49, Issue 4, pp. 216-219 (2011)

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