How Olympic Gymnasts Use Physics to Pull Off Those Crazy Twists
Quite a few Olympic events involve flipping and twisting, including gymnastics, diving, and trampoline. You might ask, “What is the difference between a flip and a twist?” I am going to give my definition of these two terms and stick with it. Some people might use different words, but mine are the best hand-picked organic high quality words.
Flip: A human rotation about an axis that runs from left to right side and through the hip area. A front flip has the face moving in the direction of rotation (like a front tuck or front layout). A back flip has the face pointing in the opposite direction of rotation (back tuck or back layout).
They say a picture is worth a thousand words, but what about an animation? This is a front layout (yes, I created the human figure in VPython). The red arrow shows the direction of the angular velocity vector.
Now for my next definition.
Twist: A human rotation about an axis that runs the length of the body.
Again, here is an animation showing a twist along with the VPython code.
What’s the deal with the red arrows? They represent the vector quantity of angular velocity. Yes, angular velocity is a vector. The direction of rotation matters, just like the speed of rotation matters. Now, the convention is for this angular velocity to follow the axis of rotation—but which way? Here’s where you use the right-hand-rule: Let the fingers of your right hand curl in the direction of rotation. Your right thumb points in the direction of the angular velocity vector.
I know you want to get to twisting, but there’s just a little bit more physics. First, there is torque. This is sort of like a rotational force and it also is a vector. Instead of discussing torque too much (here is a nice post about it), I’d like to talk about what torque does. If you have a net torque on an object, it changes the angular momentum of that object. For a constant torque, this can be expressed with the Angular Momentum Principle:
This says that if there is no torque on the system (like after a gymnast leaves the ground), then the angular momentum must be constant.
But what the heck is angular momentum other than the vector L? Angular momentum is just like the rotational equivalent of linear momentum—but with a twist. See what I did there? Let me write an expression for the angular momentum:
This states that the angular momentum is the product of I (the moment of inertia tensor) and ω. The moment of inertia is indeed a tensor (don’t worry about that right now). If you like, you can think of this as the “rotational mass.” This moment of inertia is a “thing” that describes how the mass of the object is arranged. The only thing you need to know about the tensor-nature of I is that when you operate this tensor on the angular velocity vector, you get another vector. But here is the key: The angular momentum vector does not have to be in the same direction as the angular velocity vector. I know that sounds crazy, but that’s what tensors do.
Here is a quick demo you can do. Toss a block (or something similar) into the the air, causing it to flip. After leaving your hand, there is no torque so the angular momentum is constant. However, the angular velocity is not constant. Take a look at it in slow motion:
Notice how the white side of the block doesn’t always rotate the same way? OK. You have waited long enough and I think you are ready. Let’s discuss three ways you can twist in the air. I’ll start with the simplest.
One way to start a rotation about the axis through your feet and head (a twist, as I defined it) is to exert a torque in this same direction. But how can you exert a torque when you are in the air? You can’t. You must exert torque during the jump. It’s easy; even a blogger can do it. Just swing your arms and push forward with your left foot and backward with your right (or the other way around):
Yes, I admit that I only twisted, I didn’t twist plus flip. Sorry, it’s the best I could do. Just imagine I did twisting and flipping at the same time. This is what it would look like:
The code for this motion is a bit complicated—but here it is.
OK, I’m not going to show you a real torque twist jump. Sorry, but that’s not the best way to do it.
Constant Angular Momentum Twist
This is the real way to do a twist. Once a gymnast leaves the ground with some angular momentum (such as in a layout), a twist can be initiated without any extra torque and keeping the angular momentum constant. Yes, it’s true. How does it work? The key is the moment of inertia tensor.
For any rigid object there are three axes about which the object can rotate with a constant angular velocity vector in the same direction as the angular momentum. They’re called the principle axes, and they are found from the moment of inertia tensor. But what if it’s not a rigid object (like a gymnast)? What if during the flight, the gymnast creates a change in the moment of inertia tensor? She can do this by placing her arms into a non-symmetrical arrangement, like one over her head and another over her chest. The angular momentum will no longer be in the same direction as the angular velocity vector and the angular velocity will not be constant. The result is a spin in both the “flip” and the “twist” directions.
Now for an example. This is one of my daughters (obviously she practices gymnastics).
OK, seeing this, there might be a bit of both torque and non-torque twisting. Notice how she moves her arms before leaving the beam? That might start the twisting rotation, but it can’t be a significant torque because she only has one foot on the beam such that the torque arm would be small. After leaving the beam she continues to put her body in a non-symmetrical position—even if just slightly. This is enough to produce a non-constant angular velocity that gives both a twist and a flip. Don’t ask me how you know where or when to land. I can’t do this stuff.
Quick quiz. For the flip-twist seen above, identify the direction of the angular momentum vector. I’m not going to tell you the answer. It’s a quiz, remember.
If you still don’t like the no-torque twist, take a look at this epic flip-twist.
In this example, a SkyLab astronaut starts off flipping and transitions to twisting by changing his body position. I think it’s clear that there is no external torque in this case.
Zero Angular Momentum Twist
There is one more special case: What if you start with no rotation at all? If you aren’t rotating, then a change in your moment of inertia tensor changes nothing (since you aren’t rotating). However, there is a trick to getting yourself to rotate—but it works best if you are a cat. This is how cat can fall upside down but still land on their feet.
The key to this is to rotate part of the body in one direction and part of the body in the opposite direction (so angular momentum is still zero). However, by extending legs on the back and retracting on the front the cat can achieve a rotation that results in a new downward position. This has nothing to do with the Olympics, so I will let Destin from Smarter Every Day provide a complete cat-drop explanation.
If you still want to look at angular momentum, here are a couple of resources:
- “Do springboard divers violate angular momentum conservation?“, Cliff Frohlich. Am. J. Phys 47, 583 (1979)
- “An insight into Biomechanics of Twisting,” Hardy Fink
- “Twisting Somersault,” Holger R. Dullin, William Tong. arXiv 2015.(Lots of maths in this one.)
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