# Center of Mass

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Standing on your tiptoes is hard enough, but imagine trying to do it while staying balanced on top of someone else's head. Or, just as difficult, imagine staying upright with someone standing on your head. To keep balanced, the Nanjing acrobats must be aware of their centers of mass, and the various forces—called torques—that might cause them to rotate and fall out of balance. In this unit, you'll learn how these two concepts make it possible for the acrobats to achieve their amazing acts of balance.

## Watch the Video

## Circus Physics: Center of Mass

Watch Jessica and Anan perform an acrobatic form of ballet. Such feats would be impossible without a strong understanding of center of mass.

### Questions to Consider While Watching the Video

- How is the woman able to balance on the man?
- How is her balancing like a seesaw?
- Where is her center of mass as she balances?
- What happens when she becomes unbalanced?
- Why does the man move beneath her?

## Digging Deeper

Here's a diagram of Jessica and Anan balancing:

To stay balanced, as in figure A, both acrobats must keep their centers of mass—CoM—above their points of support. For Jessica, the point of support is Anan's head. In Figure B, you can see what happens if her center of mass is not over Anan's head...she falls!

Figure C shows what happens if Anan's center of mass is no longer over his own point of support, his foot on the ground. If this happens, both acrobats fall.

When the center of mass is not over the point of support, a torque results. You can think of torque as similar to force, except that instead of causing linear acceleration, it causes rotational acceleration. In other words, torques cause objects to turn.

Just like for normal forces, there's a simple law that tells you how much effect a torque will have on an object's rotation:

*T *= *Iα*

*T* is the torque, *I* is the object's moment of inertia, a quantity that incorporates both mass, and distance from the pivot point. Finally, *α* is the rotational acceleration that results from the torque.

The reason Jessica keeps her arms outstretched and leg kicked back is that these extended limbs act as lever arms, generating balancing torques that keep her from tipping over. A good way to visualize how this works is to think about a seesaw.

Torque in these scenarios can be found by multiplying:

mass x gravity x distance to the pivot

In scenario A, the torque on both sides is *mgL*, so the people are balanced. In scenario B, the mass on the left side is the same as the right side but it is at half the distance, therefore the torque is greater on the right side and it rotates in that direction. In scenario C, the torque on the left side is 2*mgL* and *mgL *on the right. Now the torques are unbalanced in the opposite direction, so the see saw rotates the other way. In D, the mass on the left side is again doubled, but is only half as far from the pivot, so the torque is half that in C, thus the seesaw balances with no rotation.

## Your Turn

Jessica and Anan sit on opposite sides of a 3 meter-long seesaw. If Jessica sits on the end of the seesaw and her mass is 55 kg, where must Anan, whose mass is 70 kg, sit to perfectly balance Jessica?

##### Answer: Anan must sit ≈1.18 meters from the pivot

Jessica creates a torque of 55kg x 1.5 meters—half the length of the seesaw. To balance, Anan must create an equivalent torque, but since he weighs more, he must sit at some distance, *d*, that is closer than 1.5 meters from the pivot.

Setting these two torques equal to each other gives:

(55kg) x (1.5m) = (70kg) x (*d*)

Solving for *d*, we get:

*d* = (55kg) x (1.5m) / (70kg) ≈ 1.18 meters)

## Further Exploration

If you'd like to learn more about balancing acts and center of mass, check out these links:

- scienceworld.wolfrom.com: Center of Mass
- Circopedia.org: Chinese Acrobatic Theater
- Circopedia.org: Nanjing Troup