*In which we use math and physics to show that the pilots of my flight from Toronto to Albany this past weekend were full of crap.*

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As previously noted, I was in Waterloo, Ontario this past weekend for the Open House at the University of Waterloo’s new Quantum Nano Center. My talk went very well, save for some new-building technical glitches, and video of both the talk and the panel discussion should be posted to their web page soonish– I’ll post a link when it goes up.

The trip back from Toronto was slightly marred by the fact that the gates from which the tiny little prop planes they fly on the Albany-Toronto route are probably the worst of any airport I’ve flown through: cramped, confusing, understaffed, and with less charm than the average bus station. This did, however, provide material for this blog post.

At one point, when I went to the counter to ask when they’d be boarding, the agent at the gate told me they’d be boarding “ladies first,” for reasons having to do with “weight and balance.” That was a little surprising, to say the least, and I was somewhat relieved when they didn’t actually ask us to sort by gender before boarding the plane. When we got to the plane itself, however, the pilots told us to disregard our boarding passes, and had all the women sit toward the back of the plane, and the men sit toward the front. This, again, was explained as something that “helps us balance the aircraft.” This sort of statement obviously triggered my physicist instincts, making me wonder just how much of a difference sorting passengers by gender could possibly make.

So, how could you quantify the effect on “balance” of the gender-split seating arrangement? Well, the tiny little prop plane in question was a Beechcraft 1900, with the seating diagram seen at the top of this post, and also below:

The passenger cabin contains 9 rows of seats, eight of them in pairs, and the ninth with a third seat in the middle. This gives it a capacity of 19 passengers, and the cabin length is 10 meters.

The way to quantify issues of balance is through considering the center of mass of the object. The center of mass is not a physical thing, but is sort of the “average” position of the mass of an object, where we can imagine all the mass of the object being concentrated, when we do the physicist thing of treating it as a spherical featureless point mass. If we’re thinking about trying to balance an object, what we usually think about is making sure that the center of mass is located directly above or below the position where the object is supported.

Mathematically, we find the position of the center of mass of a composite object by taking the position of each individual component, multiplying by the mass of that component, and adding that product for each of the pieces making up the object. We then divide the sum by the total mass, which leaves us with the position of the center of mass. In equation form, it looks like this:

So, if we distribute a bunch of passengers through that seating chart, we can find where their center of mass is, and see how much it moves when we sort the passengers by gender. For the sake of mathematical simplicity, we’ll consider 18 passengers to start, so the whole thing is symmetric– leaving the middle seat in the last row empty.

With a 10-meter cabin containing 9 rows of seats, each row gets 1.1 m of space. We’ll take the origin to be at the position of the middle row (5th from the front), so there are two seats each at 0, ±1.1m, ±2.2m, ±3.3mn, and ±4.4m. If the passengers all had the same average mass, with the third rear seat empty, the center of mass of the passengers as a group would be located right at the zero point, in the fifth row of seats.

Of course, the passengers don’t all have the same mass, which was the point of the pilots sorting us by gender. So, how much of a difference does the mass make? According to Wikipedia (which is plenty good enough for this sort of back-of-the-envelope calculation), the average American man has a mass of 88.6kg, while the average American woman has a mass of 77.2 kg. To account for personal belongings and make the mass a little easier, I’m going to round these to 90kg and 79kg for men and women respectively. We’ll further assume an equal split: 9 men and 9 women on the plane.

The odd number for each gender leaves us with four rows of two men, a row with one man and one woman, and then four rows of two women. These situations are exactly symmetric, and plugging in numbers for the position along the length of the plane, we find that the men as a group have their center of mass at -2.58m, and the women as a group are at +2.58m. We can use those positions and the total masses of each gender to find the position of the center of mass of the passengers as a whole, and we do, indeed, find that it is shifted forward. By 0.168m, or about 6.6 inches.

(Leaving the last seat empty doesn’t make much difference, by the way. If you fill it with a woman (giving 10 women and 9 men), it shifts the center of mass from 17cm forward of the middle seats to about 6cm behind them. Even if you add another man, making it 10-9 M-F, you still fill the last seat with a woman, and the center of mass is still about 6cm behind the middle seats.)

So, how much of an effect would you expect this to have on the plane? I’m not a pilot, but based on the stats on that aircraft site, the total shift is about 1/10th of the width of the wings of the Beechcraft. That’s not a huge difference.

And even that overstates the case, because the plane itself has a large mass– about 4900kg empty. If we assume that the center of mass of the plane is near the middle (the wing-mounted engines presumably being the heaviest parts), moving the passenger center of mass forward by 17cm shifts the center of mass of the loaded plane forward by only about 4cm, less than two inches. The passengers don’t really contribute very much to the calculation at all.

So, why were our pilots so insistent on separating passengers by gender? Most likely superstition, rather than physics. They might very well believe that putting the lighter women toward the back changes the balance in some significant way, in which case doing so makes them feel better about flying.

Personally, I believe in math and physics. And I just don’t see anything in the numbers suggesting that re-sorting the passengers would make any significant difference. And if the tiny little shifts that we see from the above calculation *are* important to the safe operation of the aircraft, well, I don’t think that’s a plane I want to be flying on.

(Not that I’m much of a fan of the Beechcraft in the first place– the seats are tiny, and the “headrest” doesn’t even reach the top of my shoulders. The only reason I took that flight (which I’ve done twice before) was that it’s the only direct flight from Albany to Toronto, and the alternative was taking *two* flights on tiny uncomfortable planes, with a layover in Chicago in the middle.)