Basics: Forces and the momentum principle

**Pre reqs:** [Free Body Diagrams](, [Force](

The time has come to look at things that are NOT in equilibrium. The most basic question to ask yourself is: *"What do forces do to an object"*? Aristotle would say that forces make things move. Constant forces make things move constantly. Actually, Aristotle said there were two types of motion:

  • Natural motions: These motions don't need anything to happen, they just do. Example: a rock falling. You don't need to do anything to it. Example: fire rising. It just rises. (there was more to it than that, but you get the idea).
  • Violent motions: These motions are due to some interaction that forces them from their natural state. The natural state of a cart is to be at rest. If someone pushes on it, it will move. When you stop pushing (stop the violent motion) it returns to its natural state - at rest

I am talking about Aristotle, because these basic ideas are what most people think. If you push something it moves. If you stop pushing, it stops. And these people are correct. The problem is that there is always this extra force that no one thinks about - friction. Without friction, the rules change.

**New Rules (Newtonian ideas)**

If you push something with one force, it changes velocity. If you stop pushing, it stays at a constant velocity.

If you want to test your feelings for force, [try this force game I made on Scratch]( The idea is that you need to move the box to the red circle. The arrow keys exert a **force** on the object.

Perhaps the best way to talk about force and motion is with the momentum principle. It says:

*A net force on an object changes its momentum where momentum is the product of mass times its velocity.* (velocity, and therefore momentum, is a vector quantity)

Mathematically, this is:

![Screenshot 06](…)

Maybe you don't like derivatives. In that case, you could write this as:

![Screenshot 07](…)

Where ? means "change in". Also, momentum is typically represented by the letter "*p*" - I don't know why. The expression for momentum above is true only for velocities much less than the speed of light. Just saying.

The ?p notation is actually very useful. You can write it this way if the force is constant - which is most of the cases in introductory physics. You can also cheat. If ?t is really small, then the force is almost constant. I will talk about this more when I show you how to do numerical calculations.

You are probably thinking - that is not how you remember Newton's laws. Well, you are correct. I can rewrite the derivative of the momentum as:

![Screenshot 08](…)

There, is that better? I hope you are happy now. Really, these two are essentially the same thing. Yes, I know that the mass only comes out of the derivative if it does not change with time (not a terrible assumption). So, I will typically use these two forms interchangeably. In some cases, it makes more sense to use the "acceleration" form. If the forces are changing, sometimes it is easier to use the momentum form.

**Important implications**

  • Note that it is the NET force. I don't think we need to talk about this. Hopefully, you have already learned how to add vectors.
  • Force and momentum are VECTORS. This means the force changes the vector momentum. The momentum changes if you speed up or slow down. The momentum ALSO changes if you change direction even if your speed stays the same. (I will save circular motion for a later post.)

**What about Newton's Laws?**

Everybody likes to get caught up on these, so I will list them.

  • Newton's First Law: If an object does not have a net force acting on it, its momentum will be constant.
  • Newton's Second Law: The net force is equal to the rate at which momentum changes.
  • Newton's Third Law: Forces come in pairs. Forces are an interaction between two objects. The force object 1 exerts on object 2 is the same magnitude (but opposite direction) as the force object 2 exerts on object 1.

Ok, that is the basics of the momentum principle.

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