Can a Stack of Humans Win the 50 m Freestyle?

Hat tip to twitter user grabe31 for sending me a link to this video. It is indeed awesome. Let's consider some questions regarding the video.

Could You Stack Humans That High?

Well, first - this is probably against the international rules for swimming races. There, I have that out of the way. Second, let me assume that this is a 50 meter pool lane. How many swimmers would it take to create a stack this tall? I will ignore the height of the starting block, just for simplicity.

I am going to say an average adult male is about 1.7 meters with a mass of about 65 kg (these swimmers look fairly thin). If they stand each the lower person's shoulders, this might make an effective height of just 1.5 meters. That means that to span a 50 meter length of the pool, there would need to be 50/1.5 = 33 humans.

How much weight would the bottom person need to support? Well, that is 32 humans each with an average mass of 65 kg. That gives a weight of (9.8 N/kg)(32)(65 kg) = 20,000 Newtons (or over 4,500 pounds).

Problems With a Falling Stack

Here is a screen shot from the video showing the falling stack of humans. I added a green line on top of the stack so you could see it better.

I hate to bring down the party, but this is NOT how a stack of humans would fall. I think this video might be fake.

Consider two rigid sticks that are mostly vertical, but falling over (with the base stationary). One stick has a length L₁ and mass m₁. The other stick has length L₂ and mass m₂.

Each stick has 3 forces acting on it. There are two forces from the ground and one from the gravitational force from the Earth. I didn't label these - maybe because I am lazy. Let me just look at the gravitational force. From a rigid-body view, this gravitational force will produce a torque about the pivot point (where the stick touches the ground). This torque will then change the angular momentum of the stick. If the first stick is at an angle θ₁, then it would have a torque from gravity with a magnitude of:

The angular momentum principle says that this torque would change the angular momentum (I already said that). It is sometimes easier to write this as:

I₁ is the moment of inertia of the first stick. What's that? I like to call the moment of inertia the "rotational mass". This is the property of the object that makes it difficult to change its rotational motion. The rotational mass depends on both the mass of the object as well as how this mass is located with respect to the point of rotation. For a straight stick with uniform density, the moment of inertia is:

Now, what about α? This is the angular acceleration of the object in radians per second. So, if I put this all together, I can get an expression for the angular acceleration of the first stick.

A few points about this angular acceleration.

• It has the right units. The gravitational field (or constant if you like) has units of N/kg which is the same as a m/s². If I divide this by the length, I get units of radians/s².
• It doesn't depend on the mass of the object. I guess that isn't completely unexpected.
• The angular acceleration DOES depend on the angle that the stick is at. The greater the angle, the greater the acceleration. This makes sense also. The closer the stick is to the horizontal position, the greater the torque.
• The angular acceleration depends on the length of the stick. This is the important part.

Since the angular acceleration depends on the length, a longer stick will have a lower angular acceleration than a shorter one.

Here is the key point. What about the linear acceleration at the end of these sticks? If I know the angular acceleration of an object, I can find its linear acceleration (magnitude) by:

Where r is the distance from the point of rotation to the part of the stick in question. The end of this falling stick would then have a linear acceleration of:

For some values of θ, the linear acceleration will be greater than the acceleration of a free falling object (g). So, in order for that end of the stick have a greater acceleration than g, gravity alone won't be enough to cause this. The other part of the stick has to pull down on it as well. If the forces acting in between pieces in the stack isn't large enough the stack doesn't stay in a straight line.

An Example

Since I don't have 33 people to create a human stack, I will just make a stack out of blocks. Here is a stack as it falls over. One important difference with the blocks is that it is only the gravitational force that holds separate block pieces together.

Ok, this isn't the best quality but here are two views of blocks falling over. The first one is at 240 frames per second and the second at 480 fps.

You can see that at some point, the blocks are no longer a straight object. Also, the blocks bend UP and not down like the people in the video further evidence that the video is fake. But where would this falling stack of blocks break from a straight line? For now, I will put this question on hold. Really, modeling a falling stack of blocks is going to be a fun project.

Could This Method Win?

Suppose you could stack 33 humans and they could stay in a straight line as they fell. How long would it take for a 50 meter tall stack to reach the other end of the pool? I assume that the human stack acts like a rigid rod of uniform density, I can use my estimation for the angular acceleration:

Since the angular acceleration is not constant (but depends on the angle), I can't use the kinematic equations to determine the time. Instead, I will write a simple numerical calculation that breaks the problem into a whole bunch of small time steps. During each small step, the angular acceleration is approximately constant - so I can do the problem.

Just to show you how simple this can be done in something like python, here is the entire program.

With a stack height of 50 meters, it would take 9.87 seconds to fall over and reach the end of the lane. This is significantly shorter than the world record of around 20 seconds. Oh, I should point out that the falling time depends on the starting angle. If the stack starts completely vertical, it will never fall over and if tipped a tiny bit it can spend a long time in a near-vertical position. Let's assume the humans can lean over a little bit. So, I guess it's a legit method.