It’s the last flight of the day and the plane is pretty empty. So you’re thinking, maaaybe you’ll just move up a few rows, where there’s a nice window seat with a view that’s not obstructed by the wing.

Not so fast, buster. The flight attendant says that’s a no-go. You have to stay in your assigned seat or you’ll mess up the weight distribution of the plane. Really? Would moving one normal-size human make a difference? Yeah, you know where this is going: Answering this question requires a bunch of awesome physics. So let’s get to it!

Center of Mass

People often say an object’s center of mass is the location at which all of the gravitational force acts. That’s a fair working definition, and you can use it to solve many physics problems, but it’s not really true. In fact the gravitational force pulls on all parts of an object, not just one point.

(A quick side note: We’re actually going to look at the center of gravity, not the center of mass, but in a constant gravitational field like here on Earth, they’re the same.)

If you really want to understand the center of mass, you need to think about torque. Looking back at Newton’s second law, it says that a net force changes the motion of an object (F_net = mass × acceleration). So if the net force is zero, the motion of an object won’t change. If it’s moving with a certain velocity, it will keep doing so. If it’s at rest, it will keep on resting.

Here’s a little experiment: Place a pencil on a flat table and then, taking your two index fingers, push from opposite sides, right at the center. It just stays there, right? Because you’re applying equal and opposite forces, the net force is zero. But what if you push on it like this:

The pushing forces are measured in newtons (N). The net force is still zero—1 newton down plus 1 newton up equals zero newtons. But the pencil doesn’t remain still—it rotates. That’s because the net torque is not zero. Yes, torque is like the rotational cousin of forces. Torque (τ) depends not just on how hard you push but also where you push. We can define it like this:

Here F is the applied force, r is the distance from the point of rotation at which the force is applied, and θ is the angle between r and F. You can see that if you really want to rotate something, you don’t necessarily need to push harder, you can just push on a point farther away (larger r). That’s why the handle on a door is on the far side opposite the hinges. If it were in the middle of the door, it would be much harder to open.

Just out of convention, we say that a force that makes an object rotate counterclockwise is a positive torque, and a clockwise-pushing force produces a negative torque.

Now consider this situation: Suppose I have four balls on a very low-mass rod (one could even say a massless rod). We’ll consider this the x-axis, with a fixed point of rotation at the origin (O). They all have different masses, and they’re at different distances (x) from the pivot point.

For each ball, there is a downward-pulling gravitational force equal to m × g, where m is mass and g is the gravitational field. All of these forces exert a clockwise (negative) torque, so we can just add them together to get the net gravitational torque.

(Since m₁ is at the pivot point, where x = 0, it contributes nothing to the torque.) If I want this stick thing to be in rotational equilibrium, I have to push up with a force equal to the total gravitational force (the sum of the weights). However, this force has to be applied in exactly the right spot—one that produces a positive torque equal to the negative torque from the balls. Well, that’s not hard. It would be at the center of mass (x_cm), which we can calculate like this:

So this is essentially just a weighted average. If we have many more masses than just four, we can write this center of mass like this:

Now maybe we can state a better definition of the center of mass: It’s the point where you could apply a force to counteract the gravitational force. If the gravitational force acted only at that point, it would give you same thing.

Do you see where we’re going?

Lift and Stability

Imagine a plane flying horizontally with a constant velocity. If it has a constant velocity, the net force has to be zero. We can consider the interactions between the plane and air and Earth as four forces: weight, lift, thrust, and air drag. Here’s a diagram:

Let’s just focus on lift and weight. Since we can approximate the gravitational force as acting at a single point, we can do the same thing for the lift force. In reality, all parts of the plane interact with the air and can produce an upward-pushing force. However, it’s easier to just find the spot where we could apply a single force to get the same effect. This point is often called the center of lift. (I mean, that makes it just like the center of mass.)

With the gravitational force and lift at the same location, the net torque due to these two forces is zero and the plane won’t rotate. But what if we moved the plane’s center of mass forward?

Now the net torque is not zero. The plane would rotate nose-down, which is generally a bad thing in air travel. If the center of mass is behind the wing (behind the center of lift) it would cause the plane to rotate nose-up. Wait. Does that mean the center of mass has to be exactly at the location of the center of lift? Well, there’s another part of the plane that can produce lift and torque. It’s that small wing on the tail called a stabilizer. So the center of mass can shift around a little bit and the plane can still fly.

Estimating the Change in Center of Mass

How about a quick estimation? Suppose you move from the rear of the plane to the front. How much would the center of mass change? I’m going to assume this is a Boeing 737 with a length of 37 meters and a mass of 65,000 kilograms. Let’s use our x-axis again, with the origin at the plane’s original center of mass. Say a human with a mass of 75 kilograms starts at x₁ = –15 meters, and they want to move up to a seat at location x₂ = 10 meters (i.e., a difference of 25 meters).

If you have an empty plane and put that human in the back, the center of mass for the combo (plane plus human) moves back 1.7 centimeters. That’s not much—which is what you’d expect. It’s a 75-kg human on a 65,000-kg plane! OK, now that person moves up near first class (but not in first class, because we have to have maintain social distinctions). The center of mass would move 2.9 centimeters from its previous position. For you imperial people, that’s a smidge over 1 inch. It’s not a big deal.

So, it’s clear that the movement of one person doesn’t make much of a difference. But if you let one person move, you have to let everyone move—that’s only fair. If 10 humans make the same move, the center of mass would shift 28.5 centimeters. Maybe that’s too much to handle. I guess the best option is to just have a rule: No one gets to move.