Gymnastics is an extremely difficult sport, and not just extreme for Olympic athletes like five-time (so far) medalist Simone Biles. The physics is also quite challenging. Let’s consider something seemingly simple, like a flip.
There will be some version of a flip in all four of the women's gymnastics events: floor, bars, vault, and beam. It’s one of the two types of rotations a gymnast can make midair. In physics terms, a flip is a head-to-foot rotation about an imaginary axis that runs through the gymnast’s hips. For the second type of rotation, a twist, imagine an axis that runs from their head to their feet.
Maybe it's easier just to see them. These two animations were created in Python. (You can see the code here and here.)
A gymnast can actually perform both of these types of rotation at the same time—that's what makes the sport so interesting to watch. In physics, we would call this type of movement a “rigid body rotation.” But, clearly, humans aren't rigid, so the mathematics to describe rotations like this can be quite complicated. For the sake of brevity, let's limit our discussion just to flips.
There are three kinds of flips. There is a layout, in which the gymnast keeps their body in a straight position. There is a pike, in which they bend at about a 90-degree angle at the hips. Finally, there is a tuck, with the knees pulled up towards the chest.
What's the difference, in terms of physics?
Rotations and the Moment of Inertia
If you want to understand the physics of a rotation, you need to consider the moment of inertia. I know that's a strange-sounding term. Let's start with an example involving boats. (Yes, boats.)
Suppose you’re standing on a dock next to a small boat that’s just floating there, and isn’t tied up. If you put your foot onto the boat and push it, what happens? Yes, the boat moves away—but it does something else. The boat also speeds up as it moves away. This change in speed is an acceleration.
Now imagine that you move along the dock and pick a much larger boat, like a yacht. If you put your foot on it and push it, using the same force for the same amount of time as you did for the smaller boat, does it move? Yes, it does. However, it doesn't increase in speed as much as the smaller boat because it has a larger mass.
The key property in this example is the boat’s mass. With more mass, it’s more difficult to change an object’s motion. Sometimes we call this property of objects the inertia (which is not to be confused with the moment of inertia—we will get to that soon).
When you push on the boat, we can describe this force-motion interaction with a form of Newton's Second Law. It looks like this:
(Note: I am using the one-dimensional scalar version of this equation just to make it simpler. In fact, forces and momentum are actually vectors.)
This says that a net force (F) changes the momentum (p) of an object, and momentum is defined as the product of mass (m) and velocity (v). Since both the large and the small boat have the same force, they have the same change in momentum. But with the larger mass of the yacht, you get a smaller increase in velocity.
Now let's consider rotational motion. We have a very similar expression for rotations, which looks like this. (Again, these are scalar versions of the real vector equations.)
This time, there is a bunch of new stuff here—so let's go over it. First, there is the torque (the Greek letter τ). The torque on an object is the result of a force applied at a particular point on it. You can think of this as a type of rotational force. Just like a force changes the momentum of an object, a torque changes the angular momentum (L). Angular momentum is the product of the angular velocity and the moment of inertia.
But what the heck is the moment of inertia? By looking at torque and angular momentum, you can sort of see what the moment of inertia does. Just like the mass of an object is its resistance to changes in linear motion, the moment of inertia is an object’s resistance to changes in angular motion. You could call it the “angular mass” if you wanted to—and I would be cool with that.
We can also put it this way: If you apply the same torque to two objects with different moments of inertia, the one with the lower moment of inertia will have a greater increase in angular velocity.
But what does the moment of inertia depend on? It's not just how much mass an object has, but also where that mass is located. The moment of inertia is a measurement of both the amount of mass and how far it is from the axis of rotation. This has a very interesting consequence: You can actually change the moment of inertia without changing the mass.
Here's an at-home experiment you can try. Sit on a spinning stool or office chair and push with your foot to get yourself rotating. (Your foot exerts a torque that changes your angular momentum.) Try spinning with your arms held out from your body, and then another time with your arms tucked in. With your arms spread out, you’ll have a larger moment of inertia and your rotation rate will be lower than with your arms tucked in. Notice that you aren't changing your mass, just the way it is distributed.
Try it again with your arms tucked in, but this time get yourself spinning and pull your feet off the ground. In this case, you are rotating with zero torque (after the initial push). Now try starting with your arms outstretched and then pull them in mid-spin. This is what it will look like:
You can clearly see that, with your arms pulled in, the rotation rate increases. That's all because of a change in the moment of inertia.
Changing Moment of Inertia to Change Angular Velocity
Now for some fun. Let's look at the angular velocity of Simone Biles as she changes her body position. In particular, I'm going to analyze her Yurchenko double pike vault.
She also does a double layout (with a twist) and a double tuck (with a triple twist—her famous triple-double move) in her floor routine. However, both of these tumbling passes have a twisting motion that makes them more difficult to analyze.
In the Yurchenko double pike, she starts off running toward the vault table. Before the actual vault, she completes two roundoffs—one onto the springboard and then one from the springboard to the vault table. (This is the Yurchenko part.) In this initial motion, she rotates in a mostly straight position. Once she leaves the vault table, she bends at the waist into a pike position. This change in position changes her mass distribution and therefore changes her moment of inertia.
For the double pike portion of the vault, the only force acting on her is the downward-pulling gravitational force. Since this force acts at the center of an object’s mass, it doesn't exert any torque on her. That means that her angular momentum must remain constant. But by changing her moment of inertia, her angular velocity will also change. So, the product of the moment of inertia and angular velocity before hitting the vault should be equal to the product after leaving the vault table.
If I can get a measurement of her angular velocity during the Yurchenko and the pike parts of the motion, I can use that to see how her body position changes her angular velocity. That's where I turn to my favorite video analysis tool—Tracker Video Analysis. By looking at the position of Biles’ head relative to her waist in each frame of a video, I can get her angular position. If I do this for multiple frames, I can also get a measurement of time. Since the video plays at 30 frames per second, each frame is 1/30th of a second. Then I can get the angular velocity from the slope of the angle vs. time plot.
Actually, finding the angular position is easier than measuring Biles’ actual position and velocity. In order to do that, I would need to somehow measure her actual position in each frame of the video. Normally, you would do this by using the size of a known object. But in videos like this, the camera both pans and zooms, making the whole thing complicated. Finding the angular position ignores all these problems.
OK, now for the graph. Here is the angular position of Biles both during her Yurchenko and double pike.
From this, it looks like there are three different phases with three different angular velocities. For the roundoff, the slope of the angle-time plot is 12.0 radians per second. That's cool. But then, when she makes the transition from the ground to the vault table (while still rotating) she has a lower angular velocity of 6.72 rad/s. I'm not really sure why she slows down here. Perhaps when she hits the springboard, there is a torque which decreases her angular momentum (and thus her angular velocity). Finally, in the air (during the pike) she has an increased angular velocity with a value of 15.49 rad/sec.
I'll assume that Simone's angular velocity in the pre-pike position is 6.72 rad/s and then in the pike it's 15.49. From this, we can calculate the ratio of moments of inertia for the straight and pike positions.
Of course, this only gives us the ratio of moments of inertia (from straight to pike positions). However, you can see that the pike position has a lower moment of inertia than in the straight position. Why does this matter? It's a big deal, because when a gymnast is in the air, she wants to have an angular velocity high enough that, by the time she returns to the ground, she has rotated so her feet are pointed toward it. Without that, you can't land on your feet.
If you want to complete a double flip, the tuck position is going to have the lowest moment of inertia and give you the greatest rotational speed. This means that you don't have to jump as high and be off the ground as long as for a complete rotation.
The next hardest flip is the pike, since it has a higher moment of inertia. That leaves the layout with the largest moment of inertia and the lowest angular velocity.
Although the layout is the hardest, in the near future, I think someone in the women’s competition is going to land at least a triple pike. It's already happened in men's gymnastics with Russian gymnast Nikita Nagornyy, so I really wouldn't be surprised to see this show up in the women’s competition soon.