The Physics Behind Last Week's Metro-North Derailment

A great entry point to the physics of trains is right near my house. For just a mile-long stretch, the highway and railroad share basically the same right-of-way on a ledge of dry land between the Columbia River and the rock wall of said river’s namesake gorge. Anecdotally, there’s about a 30 percent chance of being on that ledge together with a freight train, possibly driving in the same direction and about the same speed. You’ll have the experience of steel train wheels mating with iron at a distance of what feels like an arm’s length from your ear and it will be excruciating and awesome, a screech in the style of demons.

It’s the sound of inertia being tortured actually, as this stretch lies along a broad curve. It’s the sound of many millions of kilograms of mass pushing in the wrong direction, at odd angles from the train’s actual direction of travel. The curve will eventually straighten up and if you’re still next to the thing, you can almost feel the air relax around the train as its momentum assumes a more proper direction in relation to the tracks beneath.

A second lesson in the physical enormity of trains should take place next to an actual train. This lesson also has to do with mass and momentum. Part of it is just being there and touching things, but also the very long walk that a freight train’s length might entail. Ideally, you’re next to the train when it begins moving. You’ll hear the rushing whoosh of air pressure releasing the train’s brake system; then, the creaks and metallic pops of energy being transferred; finally, the cascade of louder and louder booms as the train begins to move forward and take up slack. Then, the smooth anticlimax of the train cars moving forward at first nearly silently, but rising to a clacking din as the engines pull the mile or so of cars (much less on a passenger train, of course) up to speed.

When engineer William Rockefeller took his Metro-North Hudson Line train into its terminal curve on Sunday, that speed was 82 mph. 700 feet later, when its lead cars left the track, it probably wasn’t going all that much slower. That train would have taken about a full mile to stop completely; after 700 feet, it was just getting started. Of course, Rockefeller wouldn’t have needed to stop completely to prevent the accident, just get the train below a certain very defined threshold beneath which inertial forces keep the train upright rather than off the tracks.

The physics equation works like this. A train on a curve is subject to two opposing forces: there is, first, centripetal force which, if our train was a planet in orbit around some star, the source for this would be the star, always pulling the planet inward. The train on the curve is more similar than you think. It too is being pulled toward the center of the imaginary circle that the curved track is a part of. Take a ball in motion and imagine trying to make it go around some object in a circular path by tapping on it with a pencil. To keep it in orbit, you need to tap the ball directly at the center of its orbit. In the train’s case, that tapping is the curving rails, pulling our train inward instead of allowing it to proceed in the straight path it would prefer under Newton’s laws of rest and motion.

The pencil tapping example, provided in George Bibel’s book Train Wreck, the Forensics of Rail Disasters, is more helpful in imagining centripetal forces than you might think at first go-round. Just think of a tennis ball rolling forward, that wants to continue moving forward, and how you can change that path into a circle around a point by applying acceleration in a direction. Now, understand that Newton’s laws of motion provide an opposing force to this, the famed centrifugal force. This is just inertia—not a new introduced force—the mass of a body resisting acceleration in some direction. Along a curve, there is acceleration in the direction of the center of the rail curve’s imaginary circle-home. The center of this imaginary circle is a new direction for the formerly straight-traveling train and, crucially, it’s a new direction. So as the train moves through the curve, it’s accelerating toward the center of the curve.

Make sense? If not, go back to the tennis ball. It’s going straight and it’s your task to make it going in a circle, making use of the ball’s forward velocity and just taps. To get that tennis ball to curve, the only way is to accelerate it in the direction of the center of the curve (or the circle the curve is a part of). Now, the train wants to stay at rest relative to this new direction. That’s just how the universe is, always conserving. So as the train is pulled toward the center of the curve, there’s centrifugal force on hand to resist that acceleration.

Here’s where it gets interesting and potentially ugly for the train. All of these forces are intimately tied to the mass of the train, the velocity of the train, and the radius of the curve. A huge, gradual curve is going to generate less centrifugal force—specifically a force known as centrifugal inertial loading—because the acceleration toward the center of the curve will be less. That is, the train's velocity toward the center of the curve increases slower.

We can get a value for the force on the train acting against this inward acceleration induced by the curve—acting outward, in other words—fairly easily. (This is again borrowed from Bibel.)

The weight I came up with for a Superliner class railcar is 148,000 lbs. We’re going to covert that to mass, multiply it by the velocity of train squared, and divide it all by the radius of the curve. We’ll need to find that last bit. Just by taking the Google Maps scale guide from my homebrew graphic above (and a gum wrapper folded to be equivalent to 500 ft.), I come up with a radius of 850 feet. There are better ways to find that radius not involving a gum wrapper and laptop screen, but it's late.

The whole formula looks like this:

And the result: 108,176 lbs of inertial loading force. There’s a bit more. What’s twisting the car outward and off the tracks is torque, so we take that number above and multiply it by the length of our “wrench,” which is the distance from the ground to the train car’s center of gravity. I’ll go with Bibel’s 80 inches, used for a steam locomotive in 1947, but probably not too far off of a semi-modern commuter railcar.

108,176 lbs times 80 inches is 8,654,080 inch lbs of torque, all of it twisting the train car away from the curve like a giant hand turning a screw. This is counterbalanced by the weight of the car, 148,000 lbs. In torque terms, it works out to be 4,144,000 inch lbs twisting against our centrifugal inertial loading, which overpowers the train car’s weight easily, two times over in fact. The funny (not really funny) thing is this: Sunday’s Metro-North train might have been able to take the curve at 70 mph. It’s within 100,000 inch lbs or so inch lbs between the inertial loading at 70 mph and the train car’s mass resistance. About four million inch pounds of torque in either direction.

There’s a lot of fudging in my physics mess above, such as my guesstimates of the train car’s actual weight, the curve's radius, the actual speed at the derailment, the angle (or "cant") of the railbed. (Railbeds are always built at angles on curves to fight inertial loading, so that force helps push the train into the rails and not off of them.) The closeness is real however and it's not unreasonable to think that the engineer Rockefeller could have unrolled a cot in his cab and slept, just as long as the throttle kept the train at the pre-curve speed limit of 70 mph. Unfortunately, he was half-awake and doing 82.

I doubled checked my figures above with Bibel himself last week and he seemed to think they were at least “approximately” correct for a close guess—taking the same center of gravity for an old locomotive and a passenger cab car is particularly specious—and he noted that there are Federal Railroad Administration regulations in place that mandate that curves should be safe with a considerable speed buffer in place. Planners wanted to make this kind of accident really, really difficult to happen. For the most part they’ve been pretty successful.

Yet here we are. When I talked to MTA spokesperson Marjorie Anders early in the week, she seemed rather incredulous about my incredulity over the lack of buffer between the 70 mph and 30 mph speed zones—as you typically see on roads—so the overcaution in the curve speed limit might account for some of that. When asked about curve warnings and speed zone warnings, Adler responded, (to paraphrase) that operators just know it’s coming. They can see it and, what’s more, that sharp curve inland away from the Hudson River at Spuyten Duyvil station is, frankly, the defining feature of the Metro-North Hudson Line. Snoozing at that curve is a bit like forgetting to turn into your own driveway.

Bibel, over the course of several email conversations, noted that last Sunday’s derailment was still even more unlikely in the grand scheme of derailments. "Trains most commonly derail on curves at the speed limit or slightly over," he said. "Trains rarely overturn. As the centrifugal loads increase, the weight on the wheels decrease; and something else occurs to shift the wheel off the rail." For example, as the force increases around a curve on the outer rail, it becomes more likely that rail will "roll," or flop down on its side. With more force applied to one rail, it's also possible to widen the gap between rails beyond a certain point, essentially dropping half of the train. Nonetheless, in this case: "[The train] was going 82 in a 30. It overturned."

@everydayelk