In a recent video, Celtics player Kyrie Irving stands on a basketball court, with his legs straight and feet planted together, and leeeeeeans forward. He looks like he’s about to topple. But he doesn’t. He calmly returns to a normal upright standing position, shakes out his arms and stretches his neck, and then leeeeeeans a ridiculous amount to the side. Again he holds the position in what looks like a brazen defiance of the laws of physics. So what’s going on?
To understand how to think about center of mass, we have to start with a few basics.
In physics courses, we often treat objects as “point masses.” A point mass has no dimensions. You can describe the location and orientation of a point mass with just three variables—its position in the x, y, and z directions. That’s it. This point mass approximation is very nice. It allows us to make a complicated problem just a little bit easier (and more manageable).
If you toss a tennis ball across the room, you can approximate this as a point mass. It doesn’t matter if the ball is rotating or not (at least for most cases). There is only one force acting on the ball (the gravitational force) and it doesn’t really matter WHERE this force acts on the ball. Anyway, it’s just a ball—it’s almost a point mass anyway.
Now consider something else. Suppose I place a pencil on a table (you can do this yourself). If I push the pencil near the eraser (or tip), the pencil will rotate. If I push the pencil at the middle or the other end, something different will happen. If you don’t have a pencil to try this yourself, this is what it would look like.
This pencil is NOT a point mass. Clearly the size of the object and the location of the applied force changes the result. Actually, we call this a “rigid object” since it has a shape but the shape doesn’t change (unlike something made out of jello or something like a human).
But what do rigid objects have to do with the center of mass? This post is supposed to be about the center of mass (and Kyrie Irving), not some silly rigid object. Right? Yes, but just be patient. There are a couple more things to talk about. Don’t worry, I am going to show you some awesome demos—it’s going to be great.
Force and Torque
If you exert a force on a point mass, that object will accelerate. That’s what forces do. But what happens when you apply a force to a rigid object? Sure, it can indeed accelerate, but it can also do more. An applied force can also cause a rigid object to have a rotational acceleration. The magnitude and direction of the rotational acceleration depends on the magnitude and direction of the force as well as the location that it is applied. We call that the torque. You can think of it as a rotational force.
The torque is calculated as the product of the force and the torque-arm where the torque-arm is the distance from some point (you get to pick the point) to the location that the force is applied. Note: torque is actually way more complicated than this, but this is enough for now.
OK, one more torque example to show why we need the idea of the center of mass. Take that pencil again and hold it horizontally. Put one finger pushing up and one finger pushing down with neither past the center of the pencil. Like this.
In order for the pencil to be in equilibrium (stay in place), two things have to be true. First, the total force has to add up to zero. That means the force from the finger pushing up must be equal to the downward gravitational force plus the downward pushing finger. With zero net force, the pencil’s acceleration will be zero. Second, the total torque (about any point) has to also be zero. Let’s just pick the left end of the pencil as the torque point. Since the gravitational force and the downward pushing finger would cause a clockwise rotation, we can call these negative torques. The upward pushing pencil is positive and the pencil is in equilibrium.
Center of Mass
I pulled a trick with the pencil in equilibrium and you might not even have noticed. It has to do with the gravitational force. I put the gravitational force as though it were pulling down on the pencil and acting at the center of the pencil. In fact, the gravitational force is an interaction between ALL parts of the pencil and Earth. Gravity doesn’t just act on the pencil at one point, but at all points. However, the physics works the same if I pretend the gravitational force is just applied at the center—the center of mass.
This is essentially the definition of the center of mass: the single location that one gravitational force can act on a rigid object. Oops: I lied again. Technically, that point where it seems like there is one gravitational force is called the center of gravity. But in a constant gravitational field (like on Earth), the center of mass and the center of gravity are in the same location.
Now for some cool physics demos. You’ve been patient so far, you deserve this.
Let’s start with this first demo. Get up. Stand up. Boom. That’s physics right there. Yes, simply standing up and not falling over is an example of the physics of the center of mass. How about a physics diagram to show how this works.
That’s a pretty boring diagram, I agree. But it shows something important. In order for a human to not rotate, the center of mass must be in between (or directly over) the contact point with the floor. For the situation above, two things are true. First, the total force is zero. This is because the downward pull of the gravitational force is equal to the sum of the upwards forces by the floor. Second, the total torque about some point is also zero. In this case, I am approximating the location of the center of mass for a human (the big red dot). Usually, it’s fine to estimate the location of this center of mass as somewhere near your belly button.
If the center of mass is not between those two upward-pushing forces, it doesn’t matter where you pick the point about which you calculate the torque. There will be no way to make all the torques add up to zero torque. With a non-zero torque, the human will have a changing rotational motion. The common term for this is “falling over.”
Ready for a better center of mass demo? This one is great for parties. Here’s what you do. Take a human and ask them to stand straight up. Now place some object on the floor in front of them—maybe about half a meter from the feet and ask the person to pick it up without moving the feet. Most humans can do this.
Here’s see what this looks like; I will be the human.
Now for the trick part. Ask if they can repeat the move (picking up the object) while standing with the heels of the feet up against a wall. For all but a few rare individuals, this is impossible. Again, I will demo this.
So, what’s the deal? You should totally try this yourself before making someone else do it. But why can’t I pick up the ball while standing against the wall? Let’s start with the pick-up without the wall. Look at it again. Notice that as I lean over and pick up the ball, my rear end (butt) moves back. By moving my rear back, my center of mass stays over my feet and I don’t fall over.
Now look at the case against the wall. With the wall right behind me, my butt can’t move back. As I lean over and pick up the ball, my center of mass starts to move forward past the front of my toes. If I didn’t move one of my feet forward, I would fall. But like I said, there are a few rare humans who somehow manage to pick up the ball without falling over. They are probably mutants.
Here’s another simple center of mass demo—the hanging mobile. You can find them in all sorts of places and you can make one yourself. Here’s one I made with some materials in the physics lab. It’s a physics hanging mobile.
The key to making a mobile is to hang each piece from the center of mass for that piece. Let’s take a rigid bar (or stick) with two different masses on the end. Since the stick is both stationary and non-rotating, both the total force and the total torque must be zero. Here is a diagram.
Notice that the mass on the left is larger and has a greater gravitational force pulling down. If I pick the point to calculate the torque as the location of the string pulling up, then this string must be closer to that mass such that it produces the same torque as the smaller force from mass 2. Oh, and the stick itself has a gravitational force pulling at the center. Really, this whole piece can be treated as just one point mass at the location of the string. Now when I hang this from another stick, all these masses are just like one single mass (with regards to the same calculation for the next stick). You can keep adding more and more layers until you run out of things to add.
Balance Bird Toy
I’m not sure of the actual name of the toy, but I call it a balancing bird. It’s basically a small plastic bird with its wing spread out. If you place the bird’s beak on some small object, it will balance. It balances in a way that makes it look impossible, but it’s not impossible—it’s just physics.
The best way to understand this balance bird is to build one yourself. It’s not hard. You can do it with a bit of stiff wire and some small weights (I use hex nuts). This is what it looks like.
It looks cooler in real life. But how does it work? The mass of the wire is pretty low compared to the two hex nuts. Also, you can bend the wire so that the two nuts are slightly lower than the balance point. The result is a center of mass for the whole “bird” that is directly below the point where the wire touches the support. Now we have a situation in which the center of mass is below a support. In just about ever case, this makes a super stable situation. It is essentially the same as hanging a mass from a string. If the center of mass moves so that it is no longer directly under the support, the object will just swing until it is again under the support. It looks cool too.
Hammer and a Ruler
Here is another variation of the same thing as the balance bird. Take a hammer, a ruler, and a string. If you put it together in a particular way, you can make something that looks impossible. Here is the result.
Hopefully it should be clear that the center of mass for the hammer plus ruler is directly below the support point. But why does the hammer stay connected to the ruler? If you think of the forces acting on just the hammer, there is the gravitational force pulling down, the string pushing up and the contact point with the ruler pushing down. In a sense, this is the exact same situation as the case with the two fingers holding the pencil (see above). But again, it looks super cool.
Most people do this with the ruler sitting on the edge of a table—I used this rod support so you can see what’s going on a little better.
A Trick to Find the Center of Mass
Suppose you have some irregularly shape object. How can you find the center of mass? There is one method that involves hanging it from different points. Let’s start with a simple cardboard shape that I cut out. You can make one too—just make some silly shape. Next hang the shape from a point on the edge of the shape. The center of mass will have to be somewhere directly below this hanging point. Maybe you should draw a vertical line from the hanging point straight down. Now hang it from another point. Repeat this as many times as you like. Here is what I get.
The point where the blue lines cross has to be the center of mass. Technically, you only need two hanging points but I did three for fun. But is this really the center of mass? Yup. What if I support this object from a small holder placed at the location of this center of mass. If it’s really the center of mass, it should be balanced.
Check that out. Physics works.
Leaning Human Trick
Now we get to the best center-of-mass demo—the video of Kyrie Irving.
What the heck is going on? It seems impossible, right? OK, it’s essentially impossible. I don’t know exactly what’s going on but it has to be some type of trick. If a human leans over this far to the side, the center of mass for the human will be past the support of the feet and the human will fall over. It doesn’t matter how strong or athletic you are, you can’t stop physics.
Then how does he pull off this stunt? One way would be to use the same trick that Michael Jackson used in the music video for Smooth Criminal. In the video, Jackson does this cool lean move that appears to defy gravity. Of course he doesn’t really cheat physics, he used physics. The trick was a special shoe with a small clip on the floor. When he wanted to do the magic lean, he would clip his shoe into the floor and lean away.
How does the shoe-floor clip allow someone to break the “center of mass over the feet” rule? The reason that the center of mass has to be between the feet is that that is the only way to have a net torque of zero—except it isn’t. There is another way to get zero torque. If the floor could pull DOWN on one of the feet instead of pushing UP, you can get this to work. Here is a force diagram that might help.
That outer foot has to be pulled down. It’s the only way for this to work. Normally floors don’t pull on feet—unless there is that super special floor clip. I’m not certain that’s what happened in this Kyrie video, but it’s a good guess.
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