Here is a picture from the filming of Iron Man 3 (from shockya.com - or maybe it looks like Star News Online was the first to get the pictures). I have no idea what is supposed to be going on here, but I can guess. It looks like Iron Man is pulling some people somewhere. Probably to safety, but I am just guessing.
When I see that picture, I wonder how hard it would be for the top people in the human chain. Let me first look at a simpler case, several people hanging in a stationary and vertical human chain. If you look at the bottom person, this would be the force diagram.
I have labeled the two forces on the person as mg for the gravitational force and Fh for the upward force. I guess the "h" stands for "human" since this is the force the human above this person pulls up. If this person is at rest, then the net force must be zero. This means that in the y-direction:
If a force of magnitude mg is required to hold the person up, the person must also pull with a force of mg. This isn't as easy as it seems. Sure, many people can hang on a pull up bar - but most adults can only do it for a few seconds. What about with one hand? Forget about it. Your average person just can't do this for any significant time interval.
What about a two-human chain? Here is the force diagram.
Sorry, but I changed my force labels. Here I am calling F3-2 the force from human 3 (or whatever is at the top) on human 2. Likewise, the 1-2 is the force that human 1 exerts on human 2. There are two important points to remember. First, when I am drawing the forces on human 2, all the forces should end with "2". I shouldn't draw a the weight of the gravitational force on human 1 in the force diagram for human 2. Second, all forces are an interaction between two objects. This means that the magnitude of the force from human 1 on human 2 is the same magnitude but opposite direction as the force of human 2 on human 1. Yes, you would call this Newton's third law if you were into those labels. Just don't say "equal and opposite reactions" - that just sounds silly.
If I set up the y-direction forces for human number 2, I get:
If each human has the same mass (m), then as you go up the human chain, the force needed to hold on would get greater. You know I love plots, so here you go.
The sixth person in the human chain (the top person in a chain of six) would have to hold up a force of 660 pounds. This assumes an average human-chain person weight of 130 pounds, which might be low. But 660 pounds is not low. Not at all. Imagine hanging upside down with a 660-pound weight in your hand. How long could you hold this? For me, the answer would be around 0.005 seconds.
The only way this Iron Man human chain could work is if the human chain was constructed from other superhumans.
Hanging Angle
Look back at the picture of Iron Man carrying the human chain. Now look back at me. What did you notice? The human chain is hanging at an angle. Why would it be like this? There are two plausible reasons (or a combination of the two).
Acceleration. If Iron Man is accelerating while pulling the people, then the chain would indeed swing back. Let me draw just the forces on the bottom human in the case of an accelerating human chain.
For this case, I will assume the vertical (y-direction) acceleration is zero m/s2 and the horizontal acceleration is to the right wit a value a. With this, I can write the following two equations for the forces:
Here I can solve the y-direction equation for the magnitude of the force holding up this human and substitute this into the horizontal equation. That will leave me with an expression with just θ in it. Like this.
Looking back at the photo from the filming of Iron Man 3, I can get an estimate of the angle, θ. This gives a value of about 54°. Putting this into my expression above gives an acceleration of 13.5 m/s2. FYI, that's seems pretty high. If Iron Man and his human chain started from rest and had this acceleration for just 3 seconds, they would be going 40 m/s (90 mph). Also, notice that the force (F2-1) is now greater than the weight of the person. In this case, a 130 pound person would have to pull with a force of 221 pounds. (I switched to pounds to better relate to the average person - well the average U.S. person.)
What about the second person to the bottom in an accelerating human chain? Let me skip the diagram and just write the two force equations for this second person.
If you solve for the magnitude of the force F3-2 (assuming F1-2 is 221 pounds) you get 442 pounds. The fifth person in the chain would have to support a weight of 1,100 pounds. No small feat.
Air Resistance. What about another way to make this angle? What if Iron Man isn't accelerating, but rather moving at a constant speed? In this case, there could be an air drag force on the people that would cause this kind of angle. Here is the force diagram on the bottom person for this case.
Here I can use the typical model for the magnitude of the air drag force:
Where:
- ρ is the density of air.
- A is the cross sectional area of the object.
- C is the drag coefficient - a parameter that depends on the shape of the object.
- v is the velocity of the object with respect to the air.
I agree that finding some of these parameter for a person hanging from another person while being carried through the air would be difficult. But guessing is pretty easy. If I assume these people are similar to sky diving humans, I know that at terminal velocity the drag force is equal in magnitude to the weight. If terminal velocity is around 120 mph, then I can write the air drag as:
Where the K constant would be 0.23 N*s2/m2. Using this force, let me re-write the x- and y- equations for the force. Remember in this case both the x- and the y-accelerations are zero.
In this case, I can solve for the magnitude of the force F2-1 from the vertical equation and then plug that into the x-equation to solve for the velocity.
Let me estimate the mass of the bottom person as just 130 pounds. This would put the velocity of Iron Man at 58 m/s or 130 mph. Yes, that seems excessively fast. Slow down Iron Man.
One other thing. If the people are angled due to air resistance, the human chain wouldn't be in a straight line. The bottom person would have a greater angle than the top person. This is because the top person has the same air drag force horizontally, but a much greater vertical component of force. I could model this, but I think I have spent way too much time on this picture. The movie isn't even out yet.
