Monday, July 30, 2007

The Horse and The Cart (IN SYLLABUS!!)

Let's discuss something that is commonly brought up in a class on Newton's 3rd Law to test the concepts of students. If you find that what follows doesn't really make sense to you, then please try to find out what's happening.

Newton's 3rd Law states that for every action, there is an equal and opposite reaction. This basically means that forces always occur in pairs. In any situation where one object is exerting a force on the other, then the other object is also exerting a force that is equal in magnitude but opposite in direction on the initial object. So for example, The Sun exerts a gravitational pull on Earth, and likewise, the Earth exerts the same gravitational pull on the Sun. But since the mass of the Earth is so much smaller than that of the Sun, the effects of the force on the Earth is much greater, and so we orbit around the Sun, while the Sun merely does a small wobble in reaction to the forces exerted by the Earth.

We come now to a very famous puzzle which features a horse and a cart.



The horse(I know that's a donkey, but the idea's the same.) is pulling on the cart to try to pull the cart forward. But by Newton's third law, doesn't the cart pull on the horse as well? So if the horse pulls on the cart, and the cart pulls back on the horse, how does the whole thing move at all? Shouldn't they be locked in a never-ending battle that is guaranteed, by Newton's 3rd Law, to always end in a tie?

Now before you read on, please pause for awhile to figure out this problem on your own. If you've already had your lesson on Newton's 3rd Law, and you thought you had no problems with the 3rd Law, then you should be able to figure this out on your own. If you can't, then perhaps your understanding is not as clear as you thought!

Figured it out yet?

Answer:

If you couldn't figure out what was wrong with the paradox presented above, then you have not understood the true spirit of Newton's 3rd Law. Let us state Newton's 3rd Law again, this time using the modern phrasing: If Body A exerts a force on Body B, then Body B exerts a force that is of equal magnitude but in the opposite direction on Body A. Now let's look at the horse and cart system again:



By Newton's 3rd Law, the horse exerts a force on the cart, and the cart exerts an equal but opposite reaction on the horse. That is true. But this does not mean that no motion is possible. Why? Because the pair of forces are acting on different bodies.

Let us look at the horse alone. What are the forces exerted on it? There are only two forces: the force exerted on it by the cart, and the force exerted by the ground. Remember also that the force exerted on the horse by the ground is equal and opposite to the force exerted on the ground by the horse. So long as the horse is able to exert a force greater than the force exerted by the cart on him, then there will be a nett force, and the horse will begin moving.

Looking at the cart alone, there are two forces acting on it: the force exerted by the horse, and the force exerted by the ground. So long as the force exerted by the horse is greater than the force exerted by the ground, then the cart will start moving as well.

So there is no paradox! If you thought there was a paradox, that was because you thought that the force acting on the horse by the cart and the force acting on the cart by the horse cancel each other out. But they do not because they are acting on different things. So once again, Newton saves the day!

Remember, you may think you understand certain concepts in physics, but a more rigorous examination will prove otherwise. Even after 6 years of studying physics, there are aspects Newton's laws that I still have not fully understood. Don't be satisfied with what you've learnt in class and believed was correct. Test them out yourselves!

155 comments:

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I used this problem as a test of understanding of Newton's laws starting back in the 60s in my introductory astronomy and physics classes. It appeared on every exam after Newton's Laws, including the final. I'm a generous grader, but on this one, I was merciless by the final. I forget where I originally got it. "World of Mathematics" which I read as a teenager?

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