Act Four: The moment (of inertia) of turning

Have you ever noticed that doing a pirouette in attitude takes much more energy than doing the same pirouette in passé? Also, in a completely related segue—I promise—have you ever noticed that it is much harder to swing a heavy bat than it is to swing a lighter one?

If you have, then congratulations! You have experienced first hand the effects of moment of inertia!

What the heck is moment of inertia?

Simply put, moment of inertia[1], or rotational inertia, is how hard something is to turn. You may have heard of Newton’s second law, which is force equals mass times acceleration. This law describes things that are moving in a straight line. Whenever we start doing turns, however, we turn towards the rotational version of it, which is, torque equals moment of inertia times angular acceleration.

Where π is torque, I is moment of inertia, and α is angular acceleration. 

Torque is the tendency of a force to turn an object. If you want to know more about it, you can check this article about balancing where I explain it more deeply.

Angular acceleration [2] is the same as regular acceleration, but angular. It is how fast the angle of something changes. So if you start facing forwards, and half a second later you are facing backwards, your angle has changed, and so you have experienced angular acceleration.

Now, moment of inertia is analogous with mass. Just like a heavier object is harder to move, an object with a higher moment of inertia is harder to turn.

But a heavier object is also harder to turn, you say! That is correct, because moment of inertia does depend on mass. So the heavier something is, the harder it is to turn. However, moment of inertia also depends on how far away you are from the axis of rotation. So, depending on how you choose to turn the same object, it can be much easier or much harder.

For example, it is much easier to rotate the bat along its long axis (with the circular part of it facing you) rather than to rotate it along its short axis (as in, you swing the bat).

Figure 1: the rotation of a bat along its long axis (1) and along its short axis (2)

Because how far away you are from the axis of rotation influences the moment of inertia, the shape of something matters too. It is easier to swing a shorter bat rather than a longer bat with the same mass. This is why turning in an attitude takes up more energy than a passé, because even though your body is the same, the shape of your body is different, and thus, your moment of inertia is also different.

The moment of inertia for one single point can be calculated as

where I is moment of inertia, m is the mass of that point, and r is the distance between that point and the axis of rotation [1].

However, most things are not a single point, but made up of so many different points added up together. The good news is that we can just add moment of inertias together, just like we add up mass!

Therefore, the more of your mass that is further away from your axis of rotation, the higher your moment of inertia is, and the harder it is to turn your body! You can experiment with this, and check how much energy you needed to turn in a passé, in a small attitude, in a large attitude, and in à la seconde. (Spoiler: those are in order of how much mass is away of your axis of rotation!) You can also experiment with your arms, and see what the difference is between having your arm in first, in second, in fourth, and in fifth position is!

Moment of inertia is only the first of the many, many different components of rotational motion, and thus, of turning! Stay tuned for the breakdown of the rest of the parts!

References

Guest Posts, The Physics of BalletBelinda Kusumaphysics of dance, physics of ballet, physics, applied physics, dance, ballet mechanics, ballet world