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[Iacopo]

EDIT: split from How to measure the moment of inertia
================

Now I want to calculate how much energy I can put in my spinning tops.

One factor we need for calculating the kinetic energy in a spinning top is its angular velocity:
a laser tachometer gives data as RPMs.
We need to translate RPMs into radiants per second:
1 rad/sec = 9.55 RPMs
A top spinning at 1200 RPMs is spinning at 1200/9.55 = 125.6 radiants per second

The formula for the kinetic energy is:
1/2  I x S x S =  E rotational
"I"  is the moment of inertia, "S" is the speed in radiants per second. 
"E" is the rotational energy expressed in Joule, if  kilograms, metres and seconds have been used.
If instead the used measures are grams, millimeters and seconds, the energy will be expressed in billionths of joule.
 
Example:
Top Nr. 20  moment of inertia:  142,871 .

Top Nr. 20 top speed by a twirl of the fingers: 945 RPMs
945 RPMs/ 9.55 = 99 rad/sec

Energy of this top at 945 RPMs:
1/2 x 142,871 x 99 x 99 =  700,139,335
which, divided by one billion, is  0.7  joule.

So, 0.7 joule is the maximum energy I can put into this top with a single twirl of my fingers. 
I have various tops, with different shapes, dimensions and weights.
The maximum energy I can put into a top is different from top to top, here are some data:

Top Nr.     Moment of inertia     Maximum energy
3       l                      2,640                         0.09       
1                              5,729                         0.11
10     l                     64,197                         0.51         
8       l                     45,197                         0.52     
18   *                      63,858                         0.53   
13   *l                     68,046                         0.57
12   *l                     69,216                         0.59 
6       l                   304,361                         0.61     
15   *l                     76.487                         0.67
20   *l                   142,871                         0.70 
14   *l                     67,058                         0.71
 
* : knurled stem instead of a smooth one: it can be seen that knurled stems are more efficient.  The slightly conical shape of my more recent knurls (Nr. 14, 15, 18, 20) is more efficient than the more evident conical shape of the older Nr. 12 and 13.

l : long stem instead of a short one:  long stems are more efficient than short ones, because the top is more stable and can be spun more aggressively: the worst of the six knurled stems is the only one to be a short one. Anyway tops with short stems loose energy slower while spinning, so in the whole it is not so clear if long stems are better than short ones as for spin time.

Note how there is no way to put much energy into the lightest tops, with the lowest moment of inertia.  The Nr.1 (AMI, axial moment of inertia, 5,729, grams 19.5) can't receive more than 0.11 joule.  The Nr.3 (AMI 2,640, grams 23.3) no more than 0.09 joule.  So light spin tops cannot spin for long.  Since the moment of inertia is low, energy can come only in the form of higher rotational speed, but there is a physiological limit to speed for the fingers.  I have never been able to spin a top to more than 2400 RPMs (it is 40 rounds per second !) with just one twirl of the fingers, and at this so high speed the top has still only 0.09 joule, (it is the top Nr. 3).

More data would be needed, but it seems like the optimal moment of inertia for a spinning top to be spun with a single twirl of the fingers could be between about 50,000 and 100,000.  With heavier and larger still tops there is no further advantage in terms of received energy, but frictions become higher, and, consequently, spin times become shorter.

[Jack]

for me i feel the answer to this question is: alot @-@

[the Earl of Whirl]

I agree.  "A lot" describes it pretty well for me.

[Jeremy McCreary]

Yes, generally a lot, but a lot more into some tops than others when you can only use your fingers.

Iacopo: Excellent work, as usual!

Your observations and conclusions regarding the amount of spin kinetic energy one can transfer to a top with fingers alone match my own WRT my LEGO tops, which are all over the lot in terms of mass, AMI, size, shape, stem design, drag, and tip friction. I would only add 2 things.

First, you and I and Alan and others discussed the importance of stem diameter and taper at some length starting on the 2nd page of the thread at http://www.ta0.com/forum/index.php/topic,4441.0.html, and of course those influences still apply. In fact, I'd rank stem diameter as the 2nd most important factor limiting energy transfer -- right after AMI, and before stem grip.

Second, tip friction and aerodynamic drag can also limit energy transfer -- especially in very heavy tops or in very dirty ones (like many of mine). Of the two, drag is more likely to be the issue in my case.

[Iacopo]

the importance of stem diameter and taper

Of course stem diameter and taper shape are fundamental.
I don't know which could be the best diameters and tapers, probably they change depending on the AMI of the top and also on the hand of the spinner, so there could be many combinations.
Your findings were a bit different from the mine.
My little contribution here is that, for tops with an AMI between 63,000 and 67,000, (which I know better) and for my hand, my best actual stem (Nr. 14) is knurled and slightly tapered, with diameters from 3 (upside) to 5 (downside) millimeters.  Larger diameters and less acute tapers demonstrated to be worse in this my case. 

   
Nr. 12   mm 3-7                 Nr.14   mm 3-5

The data about energy have been interesting for me, for judging the stems.  I already had the sensation that the new ones are better, but numbers tell this with more clarity.

[Jeremy McCreary]

I don't know which could be the best diameters and tapers, probably they change depending on the AMI of the top and also on the hand of the spinner, so there could be many combinations.

Yes, way too many influences to sort out easily. Regarding the hand of the spinner, I get to watch people of all ages and backgrounds twirl my tops at LEGO shows. Two very strong influences stand out WRT release speed: (i) The spinner's hand strength and coordination, and (ii) the amount of practice he or she has had with a particular top. The greater the AMI, and the lower the rotor to the ground, the greater the range of release speeds without practice. But the range narrows quite a bit with even a little practice. Keeping one's eye on the hand throughout the twirl also helps a lot.

My little contribution here is that, for tops with an AMI between 63,000 and 67,000, (which I know better) and for my hand, my best actual stem (Nr. 14) is knurled and slightly tapered, with diameters from 3 (upside) to 5 (downside) millimeters.  Larger diameters and less acute tapers demonstrated to be worse in this my case. 

The data about energy have been interesting for me, for judging the stems.  I already had the sensation the the new ones are better, but numbers tell this with more clarity.

Yes, your Nos. 14, 15, and 20 with the more gradual 5-to3 mm tapers definitely stand out in the data.

Hmmm, wonder if the optimal stem geometry might also depend on whether you spin up the top with just one twirl, as I usually do, or with many twirls, as you often do? As mentioned earlier, I tend to get higher release speeds with 4.8 mm stems, or with stems that taper from 6 to 4.8 mm. But I can see how extending the taper to 3 mm might improve release speed after multiple twirls.

When you spin up with multiple twirls, do you find yourself starting later twirls higher on the stem?

[the Earl of Whirl]

So, Jeremy McCreary and Iacopa.....where do you live?  Is it too far to ever see your Lego collections?  How can I get the right pieces to make some of your Lego top creations?  Would you ever consider selling me some of your tops and mailing them to me so I can assemble them?  Better yet, would you consider selling some tops to me and sending them asap to the Renaissance Hotel so we could assemble them at worlds?

[Jeremy McCreary]

I live in Denver, Mike. I'd be happy to show my collection to you or any other forum member when you're in town. Meanwhile, some 15-20% of my tops are online at http://www.moc-pages.com/folder.php/188421.

Very flattered by the interest in my tops in connection with Worlds. Getting the necessary parts in time would be a long shot at this late date, but if you see some tops of interest at the link above, let me know. If I have the parts on hand, it might just work. And if the parts aren't too rare or expensive, I'd probably be willing to send you some tops gratis. They'd arrive assembled, but you could always take them apart and tinker with them.

[Jeremy McCreary]

How can I get the right pieces to make some of your Lego top creations?

I've made tops from just about every kind of LEGO part there is, but some parts are more useful than others, and there are only a few that make good tips and stems.

You pretty much have to cobble LEGO tops together from parts and your own imagination, as the LEGO folks have never offered a top-making set to my knowledge. If you have access to any parts at all, and I know you have kids and grandkids, chances are good that you can turn them into some decent tops. You wont need any instructions. Just start putting pieces together with what you already know about tops in mind, and see where that leads.
 
No question, the place to buy LEGO parts of all kinds is http://www.bricklink.com. I'll try to think of a way to communicate a list of parts that would make a good starter kit.

[Iacopo]

wonder if the optimal stem geometry might also depend on whether you spin up the top with just one twirl, as I usually do, or with many twirls, as you often do? As mentioned earlier, I tend to get higher release speeds with 4.8 mm stems, or with stems that taper from 6 to 4.8 mm.

All the data I have written above are intended for a single twirl of the fingers.

With the multi-twirl technique, things are a bit different.  The ideal stem is narrower than that for the single twirl, but less than mm 3 seems less effective; I could experiment with mm 3 cylindrical. At the highest speed I twirl only the uppest and most narrow part of the stem. 

Some data for the multi-twirl technique:

Top Nr.     Max.RPMs    Stem diameter     AMI         Joule
   6               1793           6.0 - 7.0        304,361     5.29
  10              2370           4.5 - 7.2          64,197     1.97
   8               2453           3.2 - 6.3          45,197     1.49
  12              2453           3.2 - 7.0          69,216     2.28
  13              2659           3.2 - 7.0          68,046     2.63
  18              2725           3.2 - 4.9          63.858     2.59
  14              2760           3.2 - 4.8          67,058     2.80
  20              3038           3.2 - 4.8        142,871     7.22

Here the narrowest stems perform at their best.  Large stems, still competitive for heavy tops with a single twirl, are no good anymore.  Highest speed is obtained by the narrowest stems in spite of the value of AMI: the  heaviest top is the worst, but the second heaviest is the best. In fact by many twirls there is no problem to accelerate also larger and heavier tops, the main limit to top speed becomes diameter and shape of the stem.  Still knurled and long stems perform better.

[Iacopo]

So, Jeremy McCreary and Iacopa.....where do you live?

Hi Earl of Whirl, 
I live in Italy, in Massa Carrara, where the famous marble comes from.

[Jeremy McCreary]

Here the narrowest stems perform at their best.  Large stems, still competitive for heavy tops with a single twirl, are no good anymore.  Highest speed is obtained by the narrowest stems in spite of the value of AMI: the  heaviest top is the worst, but the second heaviest is the best. In fact by many twirls there is no problem to accelerate also larger and heavier tops, the main limit becomes diameter and shape of the stem.  Still knurled and long stems perform better.

That makes a lot of sense, Iacopo.

Agree, longer stems usually result in higher release speeds in my experience as well, even with single twirls, as they help me keep my twirls under control. They also tend to promote sleeping, but only to a point. If they add too much TMI, they can promote wobbling instead, and that can eat into spin time.

Wish I could stretch out and smooth out my tapers as you do. With available parts, stepped-down stems are the best I can do. I have some high-AMI tops amenable to multi-twirl spin-up, currently with stems with single 6-4.8 mm step-downs. You've inspired me to test a 6-3 mm step-down. There can only be a single step, and there won't be any knurling, but you never know with tops until you try.

[Iacopo]

Geremy,
the formula about torque I suppose it could be written also:

Q = I x dW /dt

I have calculated the torque due to frictions of my Nr.20 at its multi-twirl top speed:
in one minute speed decreases from 3038 to 2813 RPMs.

Q = 0.000143 x 2148/60 = 0.0044 N m

This is the torque my fingers have to fight against when the top is at its highest speed. I really can't make it spin faster than so.  I was thinking that you obtained higher RPMs in spite of larger stem diameter and simple single twirls. Maybe you have a better hand, but could it be that you have a lower torque resistance in those tops ?

[Jeremy McCreary]

Iacopo: You're definitely on the right track, but I should have made clear that the formula

Q = I₃ dw/dt

is best applied over a short time interval dt. (A "short" interval here is one that sees a change in speed that's a small fraction of the speeds at either end.)

Graphically, the formula says that the torque is proportional to the slope of the speed vs. time curve at the time of interest. That slope was greater when you quit twirling than it was at the end of the first minute. Put another way, applying the formula over a minute's time gives only the average braking torque over that interval. At some point during that first minute, the torque really did equal that value, but the torque at the time of release was greater than that.

But if the braking torque really had been 0.0044 N m at maximum speed, then yes, that's what your fingers would have been fighting at the time.

After making well over 600 LEGO tops and testing and tweaking most of them many times, I probably do have a pretty good hand. But my fastest twirls were all with low- to medium-AMI tops by LEGO standards. My largest AMIs are still a good bit smaller than yours. After all, your brass flywheels are ~8.5 times denser than my plastic, and most of my parts are much less dense than the plastic they're made of due to all the air spaces they contain.

Most of my tops generate far more drag that yours. (Have you ever heard one your tops hiss?) The tops involved in the speeds I reported were certainly cleaner than my average top. Whether they generate less drag than your tops is hard to say, but I rather doubt it.

[Iacopo]

At some point during that first minute, the torque really did equal that value, but the torque at the time of release was greater than that.

You are right, Geremy. 
If I calculate torque for the first three minutes, I have:
1  0.0051  (not 0.0044, it was wrong)
2  0.0046
3  0.0041
0.0051 as average torque for the first minute could so mean something about 0.00535 as for torque at the time of release, (if the rate of decreasing torque is constant).  Or, as you say, I could measure RPMs in a littler time interval.