# Forces applied to horse and sled stuck in the snow

I have an assignment question that reads:

A horse pulls on a sled that is stuck in the snow and not moving. Your friend says "this happens because the horse exerts on the sled the same magnitude of force that the sled exerts on the horse. Since the sum of the forces is zero, there is no acceleration" What is wrong with your friend's reasoning?

My understanding is that the sled does not exert force on the horse, it is the force created from friction of the sled on the snow that exerts the same force on the horse and sled as the force from the horse and sled on the friction.

Is this correct?

This is a classic question testing your knowledge of Newton's third law. Consider the horse and and sled as one object. If that object is not moving, $\sum\vec{F}=0$ in all spatial dimensions. This does not mean that no forces are acting on it.

Now consider the horse as one object and the sled as a second object that is attached to it. Both objects are still not moving, so $\sum\vec{F}=0$ for both the horse and the sled. However, the horse is presumably trying to drag the sled along, and so it has a frictional interaction with the ground that exerts a force in the "forward" direction on the horse. Since $\sum\vec{F}=0$ on the horse, the force of the sled on the horse in the "backward" direction is exactly canceling out the forward force of the ground on the horse. So the horse doesn't move.

The sled doesn't move because it also has a frictional interaction with the ground that exactly cancels the force of the horse on it: $\sum\vec{F}=0$.

The friend is wrong because, while the force on the sled on the horse and the force of the horse on the sled are equal in magnitude and opposite in direction (they cancel each other out), when you are analyzing the system as a whole (the horse and the sled as one object), this pair of "3rd law" forces is considered an internal reaction. This means the two forces are not external forces and are not considered when performing the summation $\sum\vec{F}=0$. Since the problem statement is asking why the horse and sled system is not moving, you must treat both the horse and the sled as one object, then perform the summation $\sum\vec{F}=0$ on the resulting composite object.

Edit to address your understanding: The sled does exert a force on the horse. There is an equal and opposite force exerted on the sled by the horse and well according to Newton's third law. The rest of your explanation is hard to interpret exactly. For example, the phrase "the horse and sled on the friction" is confusing because a force is not created from "the horse and sled on the friction," it is created from the horse and sled on the snow (or ground).

• Sorry I really should proof read better. What I was trying to say was that the force applied to the horse and cart (as a single unit) from the friction is equal and opposite to the force being applied to the to the horse and cart from the power of the horse thus giving ∑F→=0 – Nectar-Bomb Dec 18 '16 at 23:51
• That is phrased in a much clearer manner. I only have one recommendation to clean it up further. "Power" has a very specific definition in the context of physics. It is the change in energy of a system per unit time (i.e. $P\equiv \frac{dE}{dt}$). Thus, the forward pointing force on the horse and cart is created from the frictional interaction between the horse and the ground, not the power of the horse. – UniqueWorldline Dec 19 '16 at 0:07

In much simpler terms, the error is confusing equilibrium with conservation. Both compel a certain sum of things (in this case, forces!) to be 0. But they do it for different reasons and they apply in different circumstances.

In fact these two ideas have a very strong relationship because in general equilibrium is all about taking some conserved quantity and saying that as much of it is flowing in as is flowing out, which is why the amount is not changing. The simplest example is a sink filling with water: as the water level rises the pressure in the drain increases and moves more water out of the sink, until the amount of water flowing out of the sink is equal to the water flowing into the sink and the water level remains constant. The two ideas are "this water in the sink had to come from somewhere" (conservation) and "the water flowing out of the sink is equal to the water flowing in through the faucet, because the level isn't changing" (equilibrium). The first applies to water in general and the second is an important tool which can really help a lot to understand particular systems.

Newton's third law is about conservation of momentum; in the language of forces it says that forces always come in equal-and-opposite pairs acting on different objects. On the other hand if an object is in a state of momentum equilibrium then it happens to be moving at a constant speed in a fixed direction, which can only happen if there is a force-balance on that single object, all of them adding up to the zero vector $\vec 0$.

The proposed explanation tries to explain a state of equilibrium by appealing to conservation, which is necessary but not sufficient to have equilibrium.