How does Newton's 3rd Law apply to arm wrestling?

Question

I understand that when we refer to Newton's third law, forces do not cancel out because action-reaction pair forces are acting on different objects. I tried to apply this to arm wrestling, but I am a little confused.

Let's say you have box A on the left and box B on the right (representing the arms). Box A exerts 50 N towards the right. Box B exerts 40 N to the left. I.e. $\text{Force}_{AonB} = 50N$ and $\text{Force}_{BonA} = 40N$. Applying Newton's third law now, $\text{Force}_{BonA} = 50N$ and $\text{Force}_{AonB} = 40N$. Thus there is a net force of 90N in each direction.

What is my misunderstanding here?

@AaronStevens The reason why I am confused is that since the net force on each box is 90N, how is one side winning (in an arm wrestling match)? — Jay, Jun 21 '19 at 01:21
Possible duplicate of With Newton's third law, why are things capable of moving? — BioPhysicist, Jun 21 '19 at 01:35
The third law of Newton describes the relationship that exists between forces that already exist. It doesn't describe some new forces. So, if hand A applies a force of 50 N on B then B will also apply a force of 50 N on A--it can't just apply 40 N. — , Jun 21 '19 at 02:40

score 0 · Answer 1 · answered Jun 21 '19 at 01:40

Because they are in contact, the two forearms exert a normal force on one another. The normal forces on each forearm are a Newton's 3rd law pair, and so the normal force on each forearm will have the same magnitude. However, the normal force between forearms is not the only force on the forearms. There is an additional force on each forearm due to the rest of the arm/body. These body forces are not related to the normal forces by Newton's 3rd law. The Newton's 3rd law pair for these body forces on the forearm are forces exerted by each forearm back on its respective body. A person who wins the arm wrestling contest has to exert a body force that overcomes the normal force pushing their forearm.

score 0 · Answer 2 · answered Jun 21 '19 at 01:58

Wining at arm wrestling is not only about forces, or about the application of Newton's laws.

For one thing, you need to maximize torque about the pivot point of your opponents arm in such a way and direction that minimizes your opponents ability (muscular strength) to resist the torque, and maximizes your ability (muscular strength) to overcome your opponents. Both will depend on the direction in which you attempt to pull or push you opponents arm. It can also depend on how you plant your feet on the ground to maximize not just the force of your arms and but the leverage of your entire body that you can bring to bear.

I'm certainly no expert in arm wrestling, but I suspect the physics of winning at arm wrestling goes beyond the simple application of Newton's third (or second, for that matter) law.

This certainly doesn't answer your question, but hopefully encourages you to do more research on the subject.

Hope this helps.

score 0 · Answer 3 · answered Jun 21 '19 at 05:43

I think the issue here is that humans aren't rigid bodies. That means the analysis is more difficult.

Let's leave aside arm wrestling for now and imagine a single box being pushed on by two people Alice and Bob, from opposite sides. Alice exerts a $50 N$ force, Bob exerts a $40 N$ force. What happens?

Alice experiences a $90N$ force. So does Bob. This is the sum of the force exerted by the other party as well as the reaction force exerted by the box.
The box experiences a net $10N$ force and moves towards Bob's side.
Alice is "winning" because she's capable of withstanding the $90N$ force while Bob isn't. It doesn't necessarily mean Alice is stronger than Bob, but it does mean that Bob (for whatever reason) is unable to resist being pushed.

The last bullet point is crucial. To illustrate it further, let's say you're standing still and I push you. Do you fall over? Not necessarily. If I push with a very small amount of force (e.g. I blow on you) then your muscles are more than capable of maintaining position. If I push harder, then you might have to brace yourself - e.g. move your legs such that one leg is behind you, and then using that leg to push back against the force I'm exerting. This position lets your muscles more effectively fight back against me. However if I'm much stronger than you, then even this might not work, and you'll fall over.

We can make this situation more artificial as well. To go back to Alice and Bob, suppose Bob is wearing roller skates. This means the friction he has with the ground is much weaker. Alice can easily push the box towards Bob's side because Bob can't get enough traction to push back. You can see that a large part of our ability to resist being toppled while standing comes from friction, and just because the box moves towards Bob's side doesn't mean Bob is physically weaker.

The same thing is happening in arm wrestling, except this time the ability to resist being pushed mainly comes from the muscles of the arm.

score 0 · Answer 4 · answered Jun 21 '19 at 06:35

Let's pretend you were armwrestling a robot 10x stronger than a human (capable of 500N instead of 50). Even if "it" goes full force on you, your arm does not get crushed, because the robot is able to exert pressure on your hand only as much as you are able to push back. Luckily your arm has plenty room to move out of harm's way. You lose the bet, but not the arm :)

Back to your example:

Inside the moving system (hands), each hand applies a force of 40N onto the other. The 3rd law is not violated. The external forces applied to the system by the two arm wrestlers do cancel each other out this way (partially) and the 10N difference is the resultant force that pushes the system forward.

How does Newton's 3rd Law apply to arm wrestling?

4 Answers4