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My question is very general, so I would like to narrow down what my confusion is about: I understand that the forces in a N3 pair act on opposite bodies, which means they cannot be added. I also understand how to solve the horse and cart problem when considering the horse and the cart to be one system, and thus ‘ignoring’ the internal forces, as they are not relevant in the overall acceleration of the system. However, when I presented this knowledge to my Physics teacher, he said that although it was true, it didn’t answer the horse’s question. This is because in the story he told us about the horse and the cart problem, the horse doesn’t refer to any type of system that encapsulates both it and the cart. Instead, it asks “How can I pull this cart”.

I have thought extensively about this problem, and I keep coming up with the same problem. I will try to describe my reasoning below, so it is easy to pick up the mistakes I have made:

A horse exerts a ‘backwards’ force on the ground (H-G), and thus the ground exerts a ‘forwards’ force on the horse (G-H). The same thing (Although opposite) happens at the cart’s wheels: The carts wheels exert a ‘forwards’ force on the ground (W-G), and thus the ground exerts a ‘backwards’ force on the wheels of the cart (G-W). Now, if the horse and the cart were seen as a closed system, one could simply say that: “if (G-H) is greater in magnitude than (G-W) then the system will accelerate.

But, I would like to examine the horse, which is where my confusion lies. The forces at play here are: (H-C), the ‘forwards’ force on the cart, exerted by the horse. And, (C-H) the ‘backwards’ force on the horse, exerted by the cart.

This is my thought process: The force (G-H) moves (or, tries to move,) the horse forward. It is my understanding that this is the ‘source’ of force in the whole problem, because without it, none of the other forces would exist. Then, due to the fact that the horse is trying to move forward, the horse exerts a force (H-C), which tries to move the cart forward. I understand that these two forces are not a N3 pair, by I can’t help but think that they are directly linked. If the force (G-H) becomes greater, then it is my understanding that the force (H-C), will increase equally. Then, because (H-C) and (C-H) are a N3 pair, the force (C-H) will be equal in magnitude to (H-C), and thus also equal in magnitude to the force (G-H). Then, if I examine the forces acting on the horse, I see (C-H) acting ‘backwards’ and (G-H) acting ‘forwards’. I understand, by my previous logic, that these two forces are equal, and thus balance to give a resultant force of 0 on the horse, which thus can’t accelerate. I understand that I have some flaw in my reasoning, but I cannot seem to figure out what it is.

Qmechanic
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UnrulyTank
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1 Answers1

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Thanks to friction with the ground. The ground exerts a force on the horse, which makes it move. This is the same mechanism that allows you to walk. Of course, because of Newton's third low, the horse exerts the opposite force on the ground. However, as you correctly point out, this has another application point.

Now coming to the confusion about the system horse-cart. Your reasoning is almost correct. There is only one flaw. You reason that:

If the force (G-H) becomes greater, then it is my understanding that the force (H-C), will increase equally.

This is correct. But this does not imply that the forces are equal in magnitude, as you say here:

Then, if I examine the forces acting on the horse, I see (C-H) acting ‘backwards’ and (G-H) acting ‘forwards’. I understand, by my previous logic, that these two forces are equal

This is the flaw in your reasoning.

In particular, you can demonstrate that

$$F_{HC} = F_{GH} \frac{m_C}{m_H+m_C}$$

As you see when you increase (G-H), also (H-C) increases, but they are not equal. They are so only in the limit of infinite $m_C$.

Then the total force on the horse is

$$F_{HC} - F_{GH} = F_{GH} \frac{m_H}{m_H+m_C}$$

Notice that this gives the acceleration

$$ a = \frac{F_{GH}}{m_H+m_C}$$

which of course coincide with the point of view of horse and cart as a single body.

fra_pero
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