I can't help myself. When I'm out in the real world and I see something cool, I have to turn it into a physics problem. It's just what I do. In this case, I was changing planes in the Atlanta airport. Like many other airports, Atlanta has a mini subway to take you between terminals. You walk in, the doors shut and then it accelerates up to some traveling speed. At some point it slows down and stops so that you can get off and catch your plane.
But what about the physics? The first thing to do is to collect some data. This isn't too difficult to do in the world of smartphones. Just about every phone has an accelerometer that lets you record acceleration in three different directions. There are many apps out there that record this data, but right now I like phyphox—it runs on both Android and iOS.
So here's what I did. In between terminals, I put the phone down on a rail so that it would be stationary. Then I recorded the acceleration. Here is the data for the acceleration in the direction of the subway (the other two directions are mostly boring).
From this data, you can see that at the start, the subway has a maximum acceleration of about 1.2 m/s2. During the stopping motion, the subway has a maximum acceleration of about -1.5 m/s2. Now for some physics questions.
How difficult is it to hold on during acceleration?
These subways only have a few seats. Just about everyone stands up and holds on to a pole or a loop hanging from the ceiling. Suppose that you just grab the vertical pole and hold on during the acceleration. How much grip strength do you need? If you want to stay on the car, you have to have the same acceleration as the subway. In order to accelerate a human, there must be a force on the human in the direction of the acceleration. The magnitude of this force will be equal to the product of the mass of the human and the acceleration. For a 70 kg human, this would be a force of 84 Newtons. Just for comparison, the weight of that same human is 686 Newtons. That means the force the human exerts on the pole (and the pole exerts on the human) would be 12.2 percent of the weight. That's not too tough.
What if someone doesn't hold onto the pole?
Some people think they are just too cool for hand rails. Instead of using a pole to exert a force to accelerate, they will just use friction. The magnitude of this friction force would have to be the same as the force from a pole, if that is what was being used. But would this floor be able to provide a large enough frictional force? Although friction can be pretty complicated, it can be modeled as a force with a maximum value proportional to the amount the floor pushes up on an object. As an equation, it looks like this:
In this expression, μs is the coefficient of static friction—a value that depends on the two types of surfaces rubbing together (in this case, a shoe and the floor). The N is called the normal force. This is the magnitude of how hard the floor pushes perpendicular to its surface. Since the human is standing on a flat surface and not accelerating vertically, the magnitude of the normal force is just equal to the person's weight (the human's mass multiplied by the gravitational field with a value of g = 9.8 N/kg).
Since this frictional force must be equal to the product of the mass and acceleration (from the car), I can solve for the minimum coefficient of friction.
Using an acceleration of 1.2 m/s2, the minimum coefficient would be 0.122. With a stopping acceleration of 1.5 m/s2, you would need a minimum coefficient of friction with a value of 0.153. That shouldn't be too difficult to get coefficients of friction higher than that. Even leather on wood has a coefficient of 0.3 (according to this table). So, yes. You could possibly just stand there and pretend you are cool.
Will you tip over?
Yes. If you aren't careful and you are just standing there in a subway you could easily fall over. Why? Suppose you are in an accelerating car while standing up straight and not holding on to the rail. Here is what a force diagram would look like in that case.
Everything looks fine, right? No, not right. Here's the problem. Suppose you put a pencil on a flat table and then push the eraser at an angle perpendicular to the pencil. The pencil will do two things: It will accelerate in the direction of the push (at least a little bit) and it will start to rotate. That's exactly what happens here with the person in the elevator. The frictional force at the feet will exert a net torque on the person and cause a rotation. We typically call this kind of rotation "falling over."
Of course there are two ways to not fall over. Both of these methods need to do something about the total torque on the human. The first method is fairly simple—stand sideways and stand with feet wide apart. In this case, the back foot can exert a greater upward force than the front foot. This means the back foot also produces more torque about the center of mass than the front foot and this torque should be enough to counter the torque from friction. Yes, you won't look as cool.
The second method involves leaning in the direction of the acceleration. If the car is accelerating, it's almost as if there is a fake force pushing you in the opposite direction (yes, this is a fake force). In this reference frame of the accelerating subway, the frictional force does not exert a torque since it's applied at the point of rotation (the feet). However, the fake force does exert a torque. If you lean in the direction of the acceleration, you can create another torque from the gravitational force. When the torque from the fake acceleration force and the gravitational force are equal, you don't tip over.
Yes, this method also has a problem. The acceleration isn't constant so you have to keep changing your lean angle. But in the end, you will either look cool, or fall over.
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