Free body diagrams

A free body diagram (FBD) is a stripped-down picture: one object, every force on it drawn as an arrow. Two rules drive it.

The object becomes a dot. No wheels, no shape, no detail. The dot stands for the object's center of mass, and that single point is enough.

Only forces acting on that object appear. Forces the object pushes on other things (the floor, the rope, the next cart) belong on someone else's diagram. This is where most FBD mistakes start: a busy scene tempts you to draw every arrow you see, even arrows that don't belong.

The point of all this stripping-down is to make Newton's second law, $\vec{F}_{net} = m\vec{a}$, easy to use. You can't add forces until you know what they are and where they point. The FBD tells you that.

Use this procedure for every FBD. Go slowly the first dozen times; speed comes later. Most wrong answers come from skipping Step 1 or Step 3.

1. Pick the object. Choose one object (or one group of things that move together). If two boxes are tied together by a rope, you usually need two FBDs, one for each box.
2. Replace the object with a dot. Drop the picture of the block, the cart, the planet. Put a dot in its place. The dot stands for the object's center of mass. That's all the geometry you need.
3. List every force on the object. Two questions, in order. First: does anything touch it? Each contact gives a normal force, maybe friction, maybe tension. Second: is it near a planet? If yes, gravity acts. If you can't name what's causing a force, that force isn't real and doesn't go on the diagram.
4. Draw each force as an arrow from the dot, and label it. Tail at the dot, head pointing the way the force pushes or pulls. Label each arrow with a symbol ($F_g$, $F_N$, $F_T$, etc.). Direction matters; getting the lengths to match is nice but not required yet.

Notice what's not on the list. There's no step for "draw an arrow for where the object is moving." Velocity is not a force and never goes on an FBD. There's no step for "split into components," either. That comes later, after the FBD is built, when you set up the algebra.

Six forces cover almost every problem in AP Physics 1. Each one has a symbol, a direction rule, and a color used consistently in this lesson (and in the Skill Check below).

Quick test for whether a force belongs: name what's causing it. Gravity? Earth. Normal force? A specific surface. Tension? A specific rope. If you can't name a real, external cause, the force isn't real. Imaginary forces wreck more FBDs than anything else. (More on that in §6.)

Setup. A 3 kg block sits at rest on a smooth flat table. Build its FBD.

Step 1. The object is the block. Not the table, not the block plus table. Just the block.

Step 2. Replace the block with a dot.

Step 3. Anything touching the block? Yes, the table on the bottom. The table pushes up perpendicular to its surface (normal force). Friction? Only if the block tends to slip. Nothing is pushing it sideways, so no tendency to slip, so no friction. Near a planet? Yes, so gravity acts down. Two forces total.

Step 4. Two arrows from the dot: $F_g$ down, $F_N$ up.

Notice how little the FBD says about the table. From the block's point of view, the table is just the source of one upward push. Whatever else the table is doing (pressing on the floor, holding up a lamp, sitting in a kitchen) doesn't show up here, because none of it acts on the block.

Also notice what's not drawn. There's no arrow for the block pushing down on the table. That force is real, but it acts on the table, not on the block. The FBD shows only forces on the block. Mixing the two up is the most common mistake on this kind of problem (more on it in §6).

Setup. A 5 kg block sits on a ramp tilted 30° above horizontal. The ramp is just rough enough to keep the block from slipping. Build the block's FBD.

Step 1. The object is the block.

Step 2. Replace it with a dot.

Step 3. Anything touching the block? Yes, the ramp surface underneath. The ramp pushes perpendicular to its surface (normal force), pointing out of the ramp. Friction? Yes. The block tends to slide down (gravity pulls along the slope, nothing else holds the block), so static friction points up the slope to oppose that tendency. Near a planet? Yes, so gravity acts down. Three forces total.

Step 4. Three arrows from the dot. $F_g$ straight down (toward Earth, not toward the ramp). $F_N$ perpendicular to the ramp, pointing away. $F_f$ along the ramp, pointing up the slope.

Two things to flag.

Gravity doesn't point into the ramp. It points toward Earth, which is straight down on the page, no matter how the ramp is tilted. A common slip is to draw gravity perpendicular to the ramp because the block is "on" the ramp. The ramp doesn't change the direction of gravity.

The normal force isn't equal to $mg$ on an incline. On a flat surface, $F_N = mg$ falls out of $\sum F_y = 0$. On a ramp, only the perpendicular part of gravity ($mg\cos\theta$) presses into the surface, so $F_N = mg\cos\theta$, which is less than $mg$. The takeaway: the normal force is whatever the surface gives to keep the object from sinking through. Its size depends on what else is acting, and you find it from the algebra.