Basics: Free Body Diagrams

**Pre Reqs:** [Intro to Forces](http://scienceblogs.com/dotphysics/2008/09/basics-what-is-a-force.php), [Vectors](http://scienceblogs.com/dotphysics/2008/09/basics-vectors-and-vector-ad…)

Hopefully now you have an idea of what a force is and what it isn't. What do you do with them? The useful thing to do with forces is to determine the total force acting on an object. At the beginning of the introductory physics course, you will likely look at cases where the total force is the zero vector. This is called equilibrium. Even if you are looking at cases where the forces don't add up to the zero vector (I say that instead of just "zero" to remind you that the total force is still a vector). Physicists like to represent forces on an object by drawing a Free Body Diagram. This is simply a representation of an object and a graphical representation of all the forces acting on that object.

Simply put, in a free body diagram, all the forces acting on the given object are represented as arrows. Let me start with a simple case, a box sitting on a table.

There are only two forces acting on this box (essentially). The table pushing up on the box and the gravitational force of the Earth pulling down on the box. The free body diagram for this box would look like this:


Note that I have used proper vector notation on my force vectors. The force of the table pushing up on the box is labeled as N because these types of forces are called "normal forces" - maybe I will talk about that more later. Another useful thing is to include the labels "table-box" and "Earth-box" to indicate that each force is an interaction between two objects. A final note on this first example is the length of the arrows representing the forces. They are the same length indicating that they are the same magnitude of force. Since these forces are the same magnitude, but different directions, the total force on this box is zero vector.

One final note. I put a dot in the middle of the box. That is where I started all the forces from. It doesn't *really* matter where the force is, but this might make it a little easier.

**A more complicated Example**

Now suppose that I have two blocks, block A sitting on top of block B that is sitting on a table. In this case, I can draw a free body diagram for both block A and block B:


Here you can see the advantage of extreme-labeling of the forces. I know it is a pain to keep writing "the force of block B on block A" but you can see something. All the forces on block A end in "block A" and all the forces on B end in "B". This notation can really help you keep track of which forces are on which block. A common mistake is to include the gravitational force of the Earth pulling on block A on the block B diagram. The thinking is that gravity is pulling block A down onto block B - which is true. However, the gravitational interaction is between Earth and A and Earth and B.

**Newton's 3rd Law**

Here you might notice something else. I have left the force of B on A and A on B as both red vectors and they are both the same length. This is a fundamental property of forces. If Newton were around today, he would state this property as:

*Forces come in pairs. For every force there is another force on a different object that has the same magnitude, but opposite direction*.

So, in a sense, the two forces are the same thing. They are a representation of the interaction between block A and B.

Finally, notice that the force of the table pushing on block B is much larger than the other forces. Why is this? Well, block B also has gravity pulling down (the Earth pulling on block B) and block A is pushing down. In order to make the total force zero vector, the table has to push up with a greater magnitude. Notice that when I have two forces acting on the same object in the same direction, I can just put the forces in a line. This is useful in that it looks like one force of a longer length.

**One more example**

Here is a slightly more complicated example for a block sitting at rest on an inclined plane.


In this case, there are three forces on the block. I imagine that everyone is ok with the gravitational force of the Earth pulling on block A - right? Here you see why the force of the plane pushing on the block is called the normal force. It is because that force is perpendicular to the surface (normal). There is another force between the block and the plane that is NOT normal. It is the friction force and it is parallel to the surface.

**Adding Vectors on the inclined plane**

Suppose you want to calculate the frictional force or something using the assumption that the forces all add up to zero vector. Here you can use a small trick. Since N and the friction force are perpendicular, you can put the x-y axis tilted so that those two forces are ONLY in x or y direction:


This would give the equation for the x-direction as (I am going to call the normal force N, friction F and the gravitational force G):


Note that these are not vectors, here the symbols stand for the magnitudes of the vectors. Also, I will leave it as a geometry exercise for you to show that the angle between the gravitational force and the y-axis is the same as the angle of the incline.

**Identifying Forces**

I understand it can be difficult to determine what forces are acting on an object. All forces that you will see can be in one of two groups:

• Long range forces: These are forces between two objects where the objects do not have to touch (thus long range). Really, there are only two interactions you will see that do this. The gravitational interaction (between objects with mass) and the electromagnetic interaction between things with electric charges.
• Contact forces: Secretly, there is no such thing as contact forces (see this post) but we will pretend for simplicity. Contact forces are from things that are touching that object. Examples: friction, normal force, tension from a rope, hand pushing on something, air resistance.
• When you are identifying forces, first look for long range. In the first semester physics this will probably JUST be gravity. All the other forces on that object are from things touching it.

In your journey to create free body diagrams, I encourage you to properly label your forces. This will help you find forces that really shouldn't be there.