# Yoyo Physics

I’ve found the physics I learned on high school and college quite useful for explaining the ways yoyos perform differently from each other. However, the conclusions I’ve drawn often seem to contradict the way I hear people explain different phenomenon. The post contains my understanding of yoyo physics, for a couple purposes:

• Hopefully it’ll help people make their own understanding more precise, both for curiousity’s sake and also to aid in deciding which yoyo to buy.
• I might be misunderstanding something myself! If you think so, I’d love to hear what you think I got wrong; I’m far from an expert physicist, so it wouldn’t surprise me too much if I get something wrong (although I am trying to stay within the bounds of my knowledge).

# What makes a yoyo spin for a long time?

## Moment of inertia

There are a lot of external factors that determine how long a yoyo spins, like how hard and straight it was thrown, or how good the bearing is. In other words, yoyos don’t spin long by themselves. However, there are parts of a yoyos design that make it more liable to spin for a long time.

The chief factor for spin times is something called the “moment of inertia” with respect to the spin axis. The higher the moment of inertia, the longer a yoyo will spin. To calculate the moment of inertia, split up a yoyo into a billion tiny pieces, and add together all the masses of each piece times the square of its distance from the axis of spin. You can calculate the moment of inertia for any axis you want, but for yoyos we only really care about the axis that the yoyo spins around.

Now, nobody wants to chop their yoyo up, so you can settle for the intuition that the more weight you put further from the axis of spin, the longer a yoyo will spin. I imagine someone designing a yoyo will go to more trouble to precisely calculate the moment of inertia for their design, which can be done iwth calculus.

This concept seems to be fairly well understood in the yoyo community, as far as I can tell. Often the way it is explained is that “more rim weight means a yoyo spins”. I think this is a good explanation, but I think I’ve seen one misinterpretation of it that sometimes pops up, which is that wide yoyos spin longer because they have more rim weight. In other words, there’s a confusion between weight being far from the gap of the yoyo and weight being far from the axle of the yoyo. The important thing is being from from the axle (or, rather, the infinite extension of the axle); being from from middle plane of the yoyo is not too important.

Have any of you had the above confusion; let me know if you have. I’m not asking to embarrass anyone - mostly just curious; have I imagined this being a confusion when actually everybody already understood it perfectly well?

## Smoothness

I believe smoothness does have an effect on spin times. The smoother a yoyo is, the longer it’ll spin. I believe the reason is that vibrations cause the yoyo to have to push air around every spin, thus slowing it down. I’m not too confident that this is the most significant reason for vibe reducing spin time, though.

# What makes a yoyo “fast”?

## Weight

For elements of a trick where the yoyo moves along a straight line, all that matters is weight. The lighter a yoyo is, the easier it is to accelerate it in a single direction.

## Moment of inertia

When the trick element is an arc, such as a pinwheel, or any sort of mount, the question is far more interesting. I’ll admit that when I first started writing this post, I was very skeptical that weight distribution played any significant role in yoyo speed; however, after going through some calculations (which I’ll
show you below), I don’t think it’s as preposterous of a notion. I do still think the difference too small to notice, and thus center weight vs rim weight doesn’t affect speed, but the numbers leave room for a bit of debate.

Since we’re talking about whether weight distribution is an important factor, we have to bring up the moment of inertia, once again. This time, however, the axis of the moment is the finger doing the pinwheel, rather than the axis of the yoyo itself. To simplify the question, let’s compare two extreme yoyos; both yoyos have the same specs (diameter is 57mm, weight is 70g), but have
vastly different weight distributions:

• Yoyo A has all of its mass at very center of the yoyo. This is, of course, impossible to manufacture, but it’s useful for the thought experiment; amusingly, it also has a moment of inertia of literally zero about its spin axis, which means it has the worst spin times imaginable. However, it will
have the lowest moment around the finger axis as well, which means it should be a bit faster moving through tricks.

• Yoyo B has all of its mass at the rim. Again, this is impossible to manufacture, since there isn’t any material to attach the rims together. It will have the best possible spin time, but also the highest moment about the finger axis, so it should be a bit slower moving through tricks.

My hypothesis had been that the moment of inertia at a reasonable distance from your finger would be within about 1% if you compared the two. However it turned out to be a bit more. Assuming the distance between your finger and the yoyo is 70mm, which would mean there is 35mm of space between the rim and your finger, the difference between the two moments is 16.58%. That’s no nothing to turn
your nose at, but keep in mind that we’re comparing the most extreme yoyos possible, when the yoyo is quite close to the finger. Thus, although this particular comparison seems to make a difference, I still hold that practical yoyos that can be machined will be only less than 3% different, which sounds unnoticeable to me. Here’s a few other data points, just to help you get an intuition for how the different variables interact:

1. Yoyo A has all its weight at 54mm, yoyo B has all its weight at 57mm, distance from the finger is still 70mm. The difference between moments is 1.48%.
2. As initially, Yoyo A is has all its weight at the center, and yoyo B has all its weight at 57mm, but this time the finger is slightly further at 100mm. The difference between moments is down to 8.12%.
3. Yoyo A has all its weight at 54mm, yoyo B has all its weight at 57mm, and the distance from the figner is at 100mm. The difference between moments is at 0.78%.

I believe the weight distribution becomes even less of a factor when the trick element is more elliptical and less circular. Thus, speed combos where the trick is mostly side-to-side elements are almost entirely unaffected by weight distribution.

So that the programmers/physicists can check my work and try out other combinations of variables, here is the python code I used to calculate these numbers (I hope I don’t make some dumb mistake that invalidates everything I just said):

``````import math

def moment(yoyo_diameter, yoyo_mass, arm_distance):
total = 0
point_mass = yoyo_mass / 360
for i in range(0, 360):
point_distance_squared = yoyo_radius ** 2 + arm_distance ** 2 - 2 * yoyo_radius * arm_distance * math.cos(angle_radians)
total += point_mass * point_distance_squared

def moment_ratio(a_diameter, b_diameter, yoyo_mass, arm_distance):
return moment(a_diameter, yoyo_mass, arm_distance) / moment(b_diameter, yoyo_mass, arm_distance)
``````

One last thing is worth mentioning. The moment of inertia merely tells you how much effort you have to put into bring the yoyo up to a certain speed. You can always make it go faster by using more force, so technically all yoyos can do combos at any speed.

# What makes a yoyo “floaty”?

I really have no clue what “floaty” means, so I’m not gonna try very hard with this point. My best guess is that a yoyo feels floaty when unwinds really quickly and easily on the initial throw, and then people keep that impression around in their head during the rest of the throw, even though it doesn’t have any affect. Unsurprisingly, the moment of inertia is again the determining factor for how quickly a yoyo unwinds.

Note that ease of unwinding is reversely correlated with spin times. This makes perfect sense, since if it is easy to get a yoyo started spinning, then it will be just as easy to make it stop spinning.

Anyway, I could be totally wrong about what floatiness means, so please let me know what you think it means!

# In conclusion…

Thanks for reading this far! I would love to hear what you think about it all. I welcome both avid disagreement and any other sort of discussion.

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EDIT: This post reads a lil weird and I don’t really like it, but it’s just trying to use the same terms as the OP. When I’m taking about higher MOI, this is me saying “a yoyo trying to concentrate the highest possible percentage of its mass as far away from the center as possible.” eg “rimweighted” Just saying higher/lower MOI is a little weird because this isn’t taking various overall specs into account. A yoyo design can have a higher MOI than another design while being “floatier” than another design with specs that just don’t allow for it to reach that same MOI, despite this hypothetical yoyo two having a higher overall percentage of its mass in the outer rims. Yoyos are a little awkward to compare with generalizations.

Yoyos with a higher MOI tend to have a more “dense” feel on the string. What some people reference to as “presence on the string.” A yoyo with a lower MOI will tend to have a “lighter” feel/presence on the string. This to me is “floaty.” A yoyo is made “floaty” by intentionally choosing to have a lower MOI by either taking away mass from the rims, spreading out the overall mass over a greater amount of space (which is also taking away mass from the rims assuming the target weight for the design doesn’t change), or intentionally putting more mass in the central hub of the yoyo.

I think the term just gets weird because people conflate having a lower overall weight as being “floaty”, and a higher overall weight as being “dense.” When in reality the weight distribution has significantly more to do with this feeling than the overall weight of the yoyo. I think this is also why people are so apprehensive to designs over 70g. Because people aren’t considering that how a yoyo feels on the string has more to do with weight distribution than overall mass.

When people refer to how a yoyo unwinds, that generally brings up a term you didn’t mention, “kickback.” When a yoyo doesn’t smoothly unwind all the way, that’s kickback. More narrow gap widths and wider response diameter will lessen kickback, more rimweighted yoyos with a higher MOI will also have more kickback.

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EDIT: This post reads a lil weird and I don’t really like it, but it’s just trying to use the same terms as the OP…

Got it. Maybe I’m nitpicking too much below then, but I’m not quite sure we’re on the same page…

I think the term just gets weird because people conflate having a lower overall weight as being “floaty”, and a higher overall weight as being “dense.” When in reality the weight distribution has significantly more to do with this feeling than the overall weight of the yoyo.

Interesting; I’m interested to hear a justification, if you’re willing. My thinking here is actually the opposite, which is that weight distribution has very little to do with the feeling of the yoyo during tricks, since the axis of the MOI is usually far enough away that differences in the MOI are insignificant.

Ahh, kickback is the word I should have used; I forgot about that term. Anyway, your point about narrow gaps is really good; I imagine it has as much to do with kickback as weight distribution. The response diameter point I’m more skeptical about though; I can imagine a large response diameter allows for more slippage and thus perhaps less kickback at the end of the throw, but other than that I wouldn’t expect it to matter too much.

Just look at how people describe yoyos more often. People think anything over 70g will be a brick just because it’s “heavy.” Lower target weights are chosen when people want to design “floaty” yoyos. I think this is just a weird misconception and entirely the wrong way to view overall weights of yoyos.

Beater is a good example of a yoyo using weight distro to intentionally move mass away from the outermost rims to provide a lighter presence on the string. They wanted the Beater to feel “floaty” while having a higher total mass.

Japan Technology started pushing weight away from the outer rims of their 4A yoyos because they wanted to achieve a lighter and more comfortable feeling while maintaining a higher overall mass. The Chief is a legendary “floaty” yoyo because it’s using the double rims and large nipple to push a bigger percentage of its mass away from the outermost rims.

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One of the motivations I had for writing this up was that my understanding of physics stands in stark contrast to the way people talk about yoyo designs. The designs you bring up are good examples of this disparity; I’m sure people call the Chief float because of something real about its design, but I have trouble believing it is because of the weight distribution. Do you have any physics reasons for thinking that weight distribution changes the way yoyos feel during tricks?

Here’s a weird thought: would a yoyo feel different if you could do tricks without it spinning? Like, what if it could stay upright and aligned in the same direction without spinning; would it feel more or less floaty. I believe the answer is that it would feel exactly the same, which means that the MOI about the spinning axis isn’t the important calculation - the MOI with respect to the finger touching the string is more crucial to feel of a yoyo, but as the original post describes, I don’t think it is different enough to matter.

I can’t really make any physics claims, it’s all a bit over my head honestly. I don’t do yoyo design myself, I just really love yoyos and being able to understand why my favorite yoyos play the way they do got me very interested in yoyo design. I’m an English major who only works with visual arts now, STEM stuff is so far beyond me lol.

I can’t really make any claims beyond what I feel when I play, and playing prototypes and comparing yoyos and really trying to understand what their shifts in weight distros have to do with how they play.

Weight distribution is the main thing that makes yoyos unique. It’s why a yoyo with an identical profile, identical weight, identical diameter and mass, but having a modified cup will make them play different. I think the OD Benchmark series tried to get this point across, but OD did it in a pretty bad way imo. People hyper focused on just the silhouette instead of considering the inner rim rips of each yoyo were really different, and those were a big driving factor for what makes each Benchmark yoyo play in a unique way.

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For some people, floaty means hang time, that is how long the yoyo “floats” before it went down because of gravity…

… which is total nonsense because gravity is constant.

I used to use this term to describe yoyos with less rim weight, however I realize some others use this to describe light yoyos even with more rim weight. This is total backwards to what I was thinking.

There used to be a long discussion about “floaty” in a forum (either here or Yoyonation) that I take part of. Basically the conclusion is, don’t use that word.

Nowadays I just use more rim weighted or less rim weighted and I have never been happier.

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I suspect that the physics take a back seat to perception, and therein lies the magic of good yoyo design which combines size, shape and weight distribution to not only achieve specific objective qualities, but to enhance them with (often counterintuitive) subjective qualities. E.g., using weight distribution to create a yoyo that plays lighter than it looks like it should will yield something perceived to have surprising “float” for its size, weight, spin power etc. The result could be yoyos with very different weight distributions being described as “floaty”, not because they are similar to each other but because similarly defy expectations of themselves.

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Great Analysis. Its so great to read the possible ways design can influence the yoyo and the moment of inertia.

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Great calculation with the Python above. May be better if you include input and output as well. Something I would also include is the aerodynamics, I would love how this might compute inside the physics of yoyo.
This assumption is from a hypothesis that V shape tends to move better because of less aerodynamic drag and O shape and H shape has more of air resistance to it due to more surface interaction and this tends to be more “floaty” if rim weight is not added.

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Did any of you know of these?

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Have you ever played with a Frisbee?

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Yes.

I know what you’re getting at but no, wind only have negligible effect on yoyos. Frisbees are lightweight and are basically circular wings.

If you drop two spinning yoyos vertically, it will land on the ground at almost the same time. One will not fall longer than the other significantly enough that any normal human being can see the difference.

If normal full sized yoyos are less than 5 grams then sure you can see the difference, or if yoyos are 65 grams but is significantly enlarged then also yes, air drag might be significant enough to make it fall slower or faster.

But a full sized yoyo at 65 grams? No way.

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I ran your program with an arm length of 200mm (a short pinwheel), mass of 65, and various diameters. Here’s a table of the results:

Ring Diameter MMOI vs. baseline (0mm)
60mm 2.65850E+06 2.25%
55mm 2.64916E+06 1.89%
50mm 2.64063E+06 1.56%
45mm 2.63291E+06 1.27%
40mm 2.62600E+06 1.00%
35mm 2.61991E+06 0.77%
30mm 2.61463E+06 0.56%
25mm 2.61016E+06 0.39%
20mm 2.60650E+06 0.25%
15mm 2.60366E+06 0.14%
10mm 2.60163E+06 0.06%
5mm 2.60041E+06 0.02%
0mm 2.60000E+06 0.00%

You can also calculate this more simply with the Parallel Axis Theorem, which is to add the body’s moment of inertia around an axis through its center of mass, to the mass times distance squared of a parallel axis that you offset it by. This second component ends up dominating the resulting value at even pretty small pinwheel sizes, but the smaller the pinwheel, the more it’s affected by the yo-yo’s moment around its axle.

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Regarding Aerodynamics. I had a random thought that I hope adds to conversation not distract. It’s way out there with potentially show-stopping major flaws (machining, vibe, …) but humor me.

In early days of ball golf, it was observed that the “caddies” hit the ball noticeably further than the “golfers”. Early golf balls were smooth (used by golfers) When they got nicked up or worn they were discarded (used by caddies). Some kind of aerodynamic thing.

Hurrying to punchline … this is where dimples were added to manufactured golf balls.

Way crazy idea - intentionally add “yo-yo dimples”. What would that do to “feel” elements that players experience?

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This is a great thread. As a newbie to “modern” throws, I find the discussion very helpful.

For example: my LaGrange Ti Mini is quick to to throw and spins very fast. Binds are like a jack rabbit up the string.

A larger bi-metal throw is slower to throw and spins much slower. Binds are slow to climb the string.

A have found a mid size throw like the Ditch to be a good balance for my low skill level.

???

PS. I do have degrees in chemistry and physics but they are very long in the tooth (like me). I have no doubt that yoyos follow classical physics but that is sometime counter intuitive. Hence, the lay persons thoughts above.

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Are these texts still available? I would love to purchase copies.

I wonder if the speed yoyos travel if aerodynamics plays a significant role?

(A question not a criticism.)

I think the spin and airspeed of a golf ball are way higher than that of a yoyo. You can how the aerodynamics of a golf ball trajectory changes as it slows down and no longer has lift. Baseball throws follow the same pattern but are more easily manipulated by the direction of spin or lack thereof imparted by the pitcher. (Baseballs have threads that behave much like dimples.)

I do not think so. I bought them back then on a BST.

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