Conservation of Linear Momentum

The conservation law for linear momentum follows is very close to being just a restatement of Newtons first law that the velocity vector is constant for a body on which no forces act. So you have in a sense already been using this conservation law in various problems involving Newtons laws.

Until now though we have typically been applying Newtons laws to individual objects, typically a brick on an incline or hanging from a string. We did consider bricks tied to each other in the Atwood machine but you will notice that until now there was no consideration of ``internal'' motion in the object. In the Atwood machine for example the masses always moved together.

One of the beauties of Newtons laws is that there are no restrictions on what type of macroscopic objects we can apply them to. We now embark on the use of Newtons laws to examine systems of particles which in general can have internal motion as well. In this context we shall learn about the conservation laws for Linear and Angular momentum and about the center of mass for an object.

First it turns out to be handy to introduce the quantity called linear momentum. For a single particle the linear momentum is a vector quantity defined as

Using this quantity we can re-write Newtons second law as

Interestingly this was actually the form which Sir Isaac Newton himself chose to work with. A special case that we have oft considered before is when

If our object under considerations consists of many individual ``sub'' objects then we can still define a momentum for that system of particles

To distinguish it from the linear momentum of individual particles I use a large for the momentum of a many particle system. Taking the time derivative of this equation I get

Reading from the beginning to the end of this euation we see that as expected we can equally well apply Newtons second law to a collection of objects as to a single object. Specifically what we shall be interested in here is to consider the case where there is no net force acting on the system of particles. Again we find

Or in words

The linear momentum of a system of particles is conserved when no net force acts on that system

It is important to note that no net force does not mean that no forces act. Internal forces are allowed to occur because Newtons third law guarantees that these forces come in action-reaction pairs and hence they cancel out exactly when we add them up to get the net force, . Thus the presence of large internal forces does not invalidate the conservation of linear momentum.