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