Gymnasts Make the Wolf Turn Look Easy. Physics Shows It’s Not

Yes, this is just an image—but it’s really balanced. If you run the code, you can see that it is indeed stationary and doesn’t tip over. It seems pretty clear that it should work—I mean, we humans do this all the time in order to stay upright.

Spinning About an Axis of Rotation

If the wolf turn was just about balancing on one foot, it probably wouldn’t be in an Olympic-level beam routine. It’s the spin that really makes this thing so difficult.

The great thing about building my three-mass human model is that I can also make it spin. If you take a hard object (like your phone or a wrench) and toss it into the air, it’s going to tumble. We call this a rigid-body rotation, and as I mentioned, the physics gets super complicated. But if you want just a tiny taste of the awesome stuff, here’s a blog post with all the details—have fun with it.

However, with the mass-spring model, the same calculations for the balancing will work just fine. So here is a diagram of a rotating object with two equal masses evenly spaced. I added a vertical line to represent the axis of rotation and to show that it passes right through the balance point—the foot.

Illustration: Rhett Allain

Again, I really don’t think there are any surprises here. Everything is symmetrical, it’s balanced in the middle, and it rotates about an axis that goes down the middle.

But wait! What if we rotate the non-symmetrical case? Let’s see what happens. (I should mention that I added a sideways force on the bottom pivoting mass so that it wouldn’t “fall off” the support point: Check it out.)

Illustration: Rhett Allain

Just in case it’s not clear, this object is balanced at the pivot point but won’t rotate about a fixed axis. If you wanted to force it to rotate around that vertical axis, you would need to either exert an external torque on the object or change the position of the masses. (Like I said, rigid-body rotations can be really complicated.)

There’s actually another real-life situation that’s just like this—the balancing of the wheels on your car. Even if the center of mass for a car wheel is right on the axis of rotation (its actual axle, in this case), the wheel can still try to wobble while spinning. The solution is to add some extra small masses to the rim of the wheel until its axis of rotation is in the same direction as the axle.

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