# Free-body diagrams

#### Prerequisites

We will be considering many cases that include many objects. In order not to get confused about which forces go where, we introduce a free-body diagram. This allows us to keep careful track of both Newton's Zeroth Law ("N0") and Newton's Second Law ("N2").

• Each object only feels forces acting on itself (N0)
• Each object satisfies its own N2 response equation.

A process that helps us disentangle all the different forces in a complex situation is the free-body diagram.  In this, we isolate one object at a time and look for the forces acting on it. Explicitly, for each object in the system being considered, we

• draw a dot representing the object;
• find all the objects touching it;
• decide what kind of touching forces these objects exert on our object (N, T, f);
• identify all the non-touching forces acting on our object (W, E, M);
• draw arrows starting at the dot in the direction of the forces, with lengths indicating the approximate relative size of the forces;
• label each force with our labeling convention so we can keep track of which force is which.

Here's an example.  Suppose a hand is pushing a box that is sitting next to another box.  The hand is pushing but not hard enough so that the boxes begin to slide.  So both boxes are at rest.

First consider box A. It's being touched by the hand, box B, and the table.  "Thinking like the box", we can see that the hand is pushing into the box from the left toward the right -- a normal force.  (In physics and math, "normal" just means "perpendicular", so a "normal force" is a contact force that is perpendicular to the surface.)  Because we are being pushed into box B, box B is pushing back on us from the right toward the left.  We know that has to be the case from N2!  If there were no force back on box A from box B, there would be an unbalanced horizontal force pushing right and box A would begin to speed up.  But the table is touching us too.  So even if there were not a box B we might not start to slide.  So there might be a force from the table holding us back.  Since that force would be parallel to the surfaces touching (box A and table) it is a friction force.  Note that we don't know how big the friction force is -- it might even be 0.  We only know that the normal force from box B and the friction force from the table that push us to the left must balance the force of the hand pushing us to the right, since our velocity is not changing (from 0).

As for up-down forces, we know that the object has a weight -- it's being pulled down by the earth.  Since it isn't changing its vertical velocity, there has to be an upward force from the table to balance it.  Since it is pushing into the box perpendicular to the surface, it is a normal force.  The final free-body diagram for A is shown above.

Note the very useful and important point!

Every force that appears in the free-body diagram for A has its "cause-feels" label pair end in A.  That's because in a free-body diagram for A only forces felt by object A belong.

The net force on object A is the sum of all these forces and this is what goes into N2.

Now consider object B.   It feels object A pushing it from the left to the right but it isn't changing its velocity from 0.  So there must be another force acting on it to balance it.  Since it's touching the table there could be (must be) a friction force acting on it from the table.  It also has a weight and the table must be holding it up.  The result is the diagram shown at the right.

Note the interesting result that the force exerted by the hand does NOT appear in this diagram.  That's because the hand is only exerting a touching force -- and it is not touching block B.  So even though the hand is indirectly responsible for pushing block A into block B, all block B knows is that it's block A that's pushing it.

Again, as with block A, for the diagram for block B, every force in the diagram has its "cause-feels" label end in B. That's because in a free-body diagram for B, only forces felt by object B belong.

Although we don't actually know a lot of these forces, if we know some, we can infer others by using a variety of principles.  In particular:

• Newton 2:  If the object is not changing its velocity in a particular direction, the sum of all the forces (and components of forces) in that direction must be 0. All of these forces must be in the free-body diagram for the same object.
• Newton 3:  If two objects are exerting forces on each other, the forces they exert are equal and opposite -- and this is true for each type of force, so if there is a normal and a friction force, the normals they exert on each other are equal and opposite, and the frictions they exert on each other are equal and opposite. Each of these pairs must be in the free-body diagrams for different objects.

Joe Redish 9/22/11, revised by Ben Dreyfus 2/11/15

Article 358