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This should probably be solvable using Lagrangian mechanics, but I haven't learned that yet and so I would appreciate an explanation of what happens without referring to it, if possible, and ideally some sort of intuitive simplified reason why it happens.

My question: What would be a basic model for what happens in a relatively simple movement like the barbell bench press? The force acting on the humerus/upper arm from the chest muscles is easy to understand (especially if forearm is perpendicular to the ground), but what about the contraction of the triceps muscles? How exactly does it help in producing vertical force on the barbell?

I would really appreciate if someone took a look at this and tried to give me a convincing description of what happens, or maybe at least explain why the article I've linked below is wrong, so I don't feel so dumb that I can't understand it. I've really tried a ton of things and have not been able to solve this problem.

Below, I have written a simplified model of how we can think about the problem, what the triceps muscle contraction corresponds to, and some observations I've made, but feel free to ignore the below, and just skip to answering my question if the problem is clear to you and/or you know how to solve it.

diagram

As a simplified model of the problem I've posted the diagram above (the picture on the left) - assume we have three objects $A,B,C,D$ with their respective positions $(0,0),(1,0),(1,0.5)$, (this is in some sense inspired by $A$ being the right shoulder, $B$ the right elbow, and $C$ is the right hand), where $A$ and $B$ are connected by a massless rod, and so are $B$ and $D$ and $D$ and $C$. Additionally, we assume that the position of $A$ is fixed, $C$ can only move vertically and the angle between $BD$ and DC is constant 180 (we can assume that everything is symmetrical, i.e. the left side of the body is mirrored, and so this simplification should be reasonable. The vertical restriction corresponds to the fact that a barbell cannot be stretched or shortened). I think that for the purpose of understanding what happens, we can assume that all the objects are of the same weight.

Now the question is, if a force $F$ is acting diagonally on the middle of the rod between $B$ and $C$, in the direction $(1,-1)$, will this force somehow cause vertical movement of the barbell?

A more precise model would perhaps be that there is an object $D$ at $(1.1,-0.1)$ (picture on the right in the diagram to which I posted above) connected to both $B$ and $C$ by massless rods (this corresponds to the bony lump to which the triceps muscle attaches), and the force is acting on it, in the direction somewhere towards $A$ (so around $(-1,0)$).

Additionally, what would happen in a different "starting position", where $C$ (the hand) starts at say $(1,0)$ (and thus the elbow at $(-\sqrt{2},-\sqrt{2})/2$ or say $C=(1,1.5)$ ?

Some additional facts:

  1. We can think of the fact that the triceps might be able to essentially hold the angle in the elbow still. If it can do that, then the upper arm and forearm are essentially a rigid body that can only rotate around the shoulder, but because the the barbell can't stretch to the side, there will be no movement.

  2. There is empirical evidence that the triceps helps. I think intuitively, we can think of the triceps in some sense "pushing" the elbow in.

  3. In https://www.strongerbyscience.com/how-to-bench/#Elbow_Extension the author of this article goes into detail of how the triceps/lateral force helps. I don't really understand the reasoning very well, but maybe it will make sense to someone else. The idea is that the lateral/triceps force essentially lowers the torque on the hand with respect to the shoulder. That is true, but I don't see how that matters - the upper arm and forearm are not a rigid body. If for example the only force acting on $C$ is $(-1,-1)$, there might be no torque with respect to the shoulder, but clearly the hand will still move down (this is ignoring the force on $D$ by the triceps, but I'm not sure how that would make a difference).

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