College football is all about traditions, and most schools have some signature thing they do at games. Mississippi State has the headache-inducing din of cowbells. Arkansas fans summon their team to the field with a hog call. “Woooo Pig Soooie!”
The Oklahoma Sooners have the Sooner Schooner. It's a little covered wagon pulled by a pair of enthusiastic ponies—you know, a prairie schooner—that careens onto the field whenever the home team scores. It’s pretty exciting.
Until something bad happens. During a touchdown celebration this past weekend, the Sooner Schooner crashed (video here), throwing its spirit squad riders to the turf. Fortunately, neither humans nor horses were injured. But everyone wants to know why it crashed—so it doesn’t happen again.
Really, it all comes down to two key physics ideas: the acceleration of an object moving in a circle, and the effect of torque on a rigid object. Let’s get to it.
Suppose you were looking down on the field from a blimp. Let’s start with the simplest case, where the wagon starts from a resting position (1) and speeds up as it moves in a straight line. So, after some short amount of time (Δt), it's in a new location (2) with a new velocity (v).
Since the wagon’s velocity has increased, it has an acceleration. Acceleration is simply change in velocity over change in time, as shown below. (The arrows indicate that these are vector quantities, meaning they have not only magnitude but also a specific direction. That’s going to be important in a moment!)
For instance, if the magnitude of velocity rises from 0 to 6 meters per second in 3 seconds, that would be an acceleration of 3 m/s2. So, that's your basic linear acceleration.
But wait! There’s another way to accelerate. Since velocity is a vector, if the wagon changes direction—e.g., if it follows a circular path—that will also change its velocity. So you again have an acceleration, even if the speed of the wagon stays the same.
The magnitude of acceleration in this case depends on both the speed (v) of the wagon and the radius (R) of its circular path. You know all about that—you can feel it when you drive your car around a curve. The faster you drive, or the tighter you turn, the greater the acceleration.
So the magnitude of acceleration for a turning object is:
Again, that’s the magnitude. But since acceleration is also a vector, it needs a direction. For an object moving in a circle, the direction of the acceleration vector (a) always points toward the center of the circle. (That is why some call it centripetal acceleration, which means "center pointing.")
So the Sooner Schooner was indeed accelerating, simply because it was turning. Also, you might notice that right before the crash, the horses seem to take a sharper turn. That reduces the radius of curvature and increases the centripetal acceleration. But why did it tip over? Torque!
Physicists like to simplify things as much as possible. So for an accelerating wagon, it's easier to think of the wagon as just a point with no dimensions, instead of an extended object. In that case the acceleration is just one vector, and it doesn’t matter where the forces are applied to the object.
But if the wagon is just a point, it can't flip over. So clearly we can’t use that assumption here! The next level of approximation is to treat the Sooner Schooner as a rigid body—like a box. A rigid body has size and can rotate, but it doesn't deform. Obviously, a real wagon would have some type of deformation, but this model should work for now.
When you have an object with size, the location of the forces on it matter a whole lot. If you push on something, that force will cause it to accelerate. If the force doesn't pass through the center of mass, the force will also exert a torque on the object, causing it to rotate.
Torque can be a little confusing, so how about a quick demo to show the difference between force and torque? Place a pencil (a good rigid object) on a table and push it with your finger. If you push (exert a force) in the middle, it will slide but not turn. If you push near the end, there will be torque, causing the pencil to rotate. Forces causes objects to accelerate, but a torque causes an object to change its rotational motion.
The amount of torque depends on two things: how hard you push and where you push. A larger distance from the center of mass produces a larger torque. That’s why the pencil above will rotate more if you apply force farther away from its center. We call that distance the torque arm.
Now for a more useful example. What happens when you accelerate a block by pushing on it from the bottom? In this case, I have two blocks on a platform. (OK, it’s a Lego baseplate.) The platform accelerates to the right. Since there is a frictional force between the blocks and the platform, there is a force pushing to the right on the bottom of the blocks. For comparison, I have one block standing up and one lying down. Here is what it looks like in slow motion:
For the standing-up block, the frictional force has a much larger torque arm than on the other block. This produces more torque—enough to tip it over.
Now imagine that you accelerate the platform by moving it in a circle. The same thing would happen: There would be a frictional force pushing toward the center of the circle now. If that force were large enough or the torque arm were long enough, the block would tip over to the outside.
So, what can the Sooners do about their Schooner? Well, several options. First, they could reduce the acceleration. According to the equations above, that means either (1) drive slower, or (2) don't make such sharp turns. I know that’s not as exciting, but falling down and limping off the field doesn't convey the image you’re after, either.
Second, they could shorten the torque arm. If the wagon’s center of mass were closer to the ground, the frictional force on the wheels would produce less torque and it would be more stable. So, lowrider covered wagons. Why not? The real ones needed high clearance to get over boulders and brambles—not really an issue here—and speed wasn’t a design goal back then.
They could also place the wheels farther apart—sort of a sports-schooner look. That wouldn't reduce the torque, but the wagon would be able to handle more torque before reaching the tipping point.
Finally, it's possible to make a “leaning” schooner. If the vehicle leaned into the turn (like a motorcycle rider), the gravitational force would produce a countervailing torque to help keep the thing upright. Some high-speed trains have systems like that.
I know, that might sound a little high-tech for a covered wagon, but the original Sooners of Oklahoma were a resourceful bunch—I think they would have gone for something like that.
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