The Moment of Inertia of Juggling Clubs
Moment of Inertia
The Moment of Inertia of some solid body is a measure of its tendancy to keep spinning around some axis, just as the Mass is a measure of its tendancy to keep moving in a straight line. Things can have high Mass but very low Moment of Inertia, if all the mass is concentrated really close to the axis of rotation; or, they can have low Mass and high Moment of Inertia, if most of the mass is situated a long way from the axis of rotation. However, for things of a given size and shape, the Moment of Inertia is proportional to the Mass.
Jugglers are concerned with Moment of Inertia because they want to know how "fast-spinning" or "slow-spinning" a particular type of club is. I think it's fair to say that most people's "Ideal Juggling Club" would have a Moment of Inertia around .0045 Kilogram.Meter.Meters, but clubs between about .003 to .007 still remain juggleable.
It seems strange to me, with all the discussion about fast-spinning and slow-spinning clubs, that no manufacturer bothers to quote the Moment of Inertia, even though they do specify the Mass. But juggling-club Moments of Inertia cover quite a wide range, and makers really should specify them, so that people know what kind of club they're ordering.
If you double the mass, the Moment of Inertia doubles;
if you expand the length by 10% then the Moment of Inertia increases by 21%.
The Moment of Inertia M of a uniform rod of mass m
and length L is 
Measurements of some Common Clubs
So here are some measurements of the clubs I have around the house. All these figures are in MKS units. The human tibia are the plastic things they sell to medical students; these are the male adult ones. They impress audiences, they're very hard and Really Hurt if they hit your knuckles. They're probably where club-juggling was at in the stone age :-)
The "as compared to a rod" column compares the club's Moment of Inertia with that of a uniform rod of the same length and weight. If it's greater than 1 then the club is slower-spinning than a uniform rod, meaning that its mass is more distributed out towards the ends (e.g. the knob). Conversely, <1 means it's faster-spinning than a rod would be, because its mass is clustered closer to the Centre of Gravity. So the Beard Beach is particularly slow-spinning for its size and weight. Tibia are >1 because of the knobbly bits of bone at each end; in their case that's just as well, because they're so short that otherwise they'd be impossibly fast-spinning.
Here's the numbers, arranged in order of Fast- to Slow-Spinning. I reckon the measurement error at about 2%..3% . . .
| Mass (Kg) | Length (Meters) | CoG distance from knob | as a proportion of length | Moment of Inertia (Kg.Meter.Meters) | as compared to a rod | |
|---|---|---|---|---|---|---|
| Human Tibia | .245 | .358 | .220 | .615 | .00324 | 1.24 |
| Radical Fish | .195 | .520 | .297 | .571 | .00419 | 0.96 |
| Beard Circus Special | .221 | .517 | .305 | .590 | .00439 | 0.89 |
| Henrys Pirouette | .223 | .518 | .298 | .575 | .00446 | 0.89 |
| PX3 Vegas | .220 | .519 | .291 | .561 | .00495 | 0.98 |
| Beard Beach | .256 | .523 | .287 | .549 | .00647 | 1.11 |
| Dubé Knives | .355 | .527 | .271 | .514 | .00675 | 0.82 |
No surprises there, and it's nice that the numbers confirm how they feel in the hands. If other people could measure other popular types of club, I'd be delighted to incorporate their results into the table . . .
Measuring Moment of Inertia
I measured the Moment of Inertia (M) by attaching two parallel and equal-length (l) strings to the club at equal distances (d) either side of its Center of Gravity, and timing the club as it rotates to and fro. If phi, the angle of the club from its equilibrium, is not too large, then :
 | The angle of each string to the vertical is |  |
| The tension in each string is |  |
| The moment exterted by one string is |  |
| and by both strings is |  |
| The law of motion is |  |
| so the angular frequency will be |  |
| and the Moment of Inertia is given by |  |
| Now |  |
| So that |  |
where m, d, l and f can all be measured. First you locate the Center of Gravity by balancing the (horizontal) club on something. The strings have to be light, and flexible (e.g. fishing-line or cotton). Make sure the strings are attached equal distances d from the Center of Gravity. The club should rotate back and forth without the Center of Gravity moving; I set the club swivelling by hand, and then damped out the motion of the Center of Gravity by touching lightly underneath it. You can measure the frequency f by counting with a stopwatch, but I used an electronic metronome, adjusting till it clicked twice per oscillation. Once the club is prepared, it's worth doing several different measurements with different string-lengths l, so as to take the average.
Moving up the price-range somewhat, equipment for measuring Moment of Inertia can be purchased from idicb.com, or from space-electronics.com. Club makers could use these during development, or both vendors and makers could use them to offer accurately matched sets. Space-electronics.com also provide measurement services, which would allow Club makers to specify Moment of Inertia in their catalogues.
This page descends from some postings to the rec.juggling newsgroup.
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