In his upcoming movie, Captain America stops a helicopter from taking of, by grabbing the helicopters skid and a piece of railing.
Is this feasible, and how much strength would be required to ground a helicopter like this?
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In this exact way? Less than it seems, to be honest.
I'll explain with minimal physics, and overly generic simplifications, just for some basic reasoning. This is an answer for a layman.
A helicopter works by generating lift. This lift kinda cancels out the gravity effects, and thus keeps the vehicle aloft. The more lift a helicopter generates, further it raises from the ground. That's easy.
So, to hold a helicopter directly to the ground, you need to pull it down with equivalent forces.
Captain is holding the Helicopter by the side of the landing gear. This means that, should the pilot speed up the Helicopter, it would start tilting to the side, probably making the vehicle crash. This is called a Dynamic Rollover, and can be caused by pretty much a single steel cable still attached to the vehicle. Heck, it can be caused by anything at all, if the pilot is not careful or skilled enough. Helicopters are finicky machines, and prone to disasters if you don't take care.
So, the pilot here is not really pushing Cap - if he does, and Cap happens to be strong as a steel cable, the helicopter would probably start rolling over, crash, and kill everyone on board.
That said, this scene can't really be used to estimate how strong Cap is, since we don't know how much force he is putting here. A steel tie-down cable can withstand up to 4kN (around my region) and is enough to cause a rollover. An Olympic Grade gymnast withstand around 3kN during competitions, so it's not a stretch to assume that Cap is stronger than that.
I think this is a Bell 206, you can't see all of the copter, and there may be another model that looks about the same.
If it is a Bell 206, the dry weight is 2331 pounds and the max. takeoff weight is 3200 pounds. If you allow 250 pounds for pilot, fuel and misc. you have a max. net thrust of about 620 pounds. The amount Captain America is using is almost certainly less than that, as he is not vertically "stretched" the side thrust is quite limited as the copter is not obviously tilted at steep angle. So in this shot it is likely considerably less than 200 pounds. I.e., you could actually make this shot without having the strength of Captain America, just a normal stunt double, or even Chris Evans himself - the shot does not look particularly dangerous if wearing a wire to catch him in case of a fall.
For a movie shot, the studio would very likely select a helicopter with limited power as a simply safety measure (as well as being a cheaper model that the more powerful ones). What is important to the studio is that a shot looks impressive, not that it is actually impressive.
You need to apply a force equal to the excess force applied by the rotors above what's needed to keep the helicopter hovering plus the weight of the person holding it. I'll leave someone else to put some numbers to that, but it doesn't strictly need to be very much, just enough to destabilise the chopper should it try to move away.