VISUALIZE The interaction diagram of FIGURE 7.5 starts by representing every object as a circle in the correct position but separated from all other objects. The person and the crate are obvious objects. The earth is also an object that both exerts and experiences forces, but it’s necessary to distinguish between the surface, which exerts contact forces, and the entire earth, which exerts the long-range gravitational force.

FIGURE 7.5 The interaction diagram.

Figure 7.5 also identifies the various interactions. Some, like the pushing interaction between the person and the crate, are fairly obvious. The interactions with the earth are a little trickier. Gravity, a long-range force, is an interaction between each object and the earth as a whole. Friction forces and normal forces are contact interactions between each object and the earth’s surface. These are two different interactions, so two interaction lines connect the crate to the surface and the person to the surface. Finally, we’ve enclosed the person and crate in a box labeled System. These are the objects whose motion we wish to analyze.

We can now draw free-body diagrams for the objects in the system, the crate and the person. FIGURE 7.6 correctly locates the crate’s free-body diagram to the right of the person’s free-body diagram. Each object has three interaction lines that cross the system boundary and thus represent external forces. These are the gravitational force from the entire earth, the upward normal force from the surface, and a friction force from the surface. We can use familiar labels such as $\vec{n}{}_{\text{P}}$ and $\vec{f}{}_{\text{C}}$, but it’s very important to distinguish different forces with subscripts. There’s now more than one normal force. If you call both simply $\vec{n}$, you’re almost certain to make mistakes when you start writing out the second-law equations.

FIGURE 7.6 Free-body diagrams of the person and the crate.

The directions of the normal forces and the gravitational forces are clear, but we have to be careful with friction. Friction force $\vec{f}{}_{\text{C}}$ is kinetic friction of the crate sliding across the surface, so it points left, opposite the motion. But what about friction between the person and the surface? It is tempting to draw force $\vec{f}{}_{\text{P}}$ pointing to the left. After all, friction forces are supposed to be in the direction opposite the motion. But if we did so, the person would have two forces to the left, $\vec{F}{}_{\text{C on P}}$ and $\vec{f}{}_{\text{P}}$, and none to the right, causing the person to accelerate backward! That is clearly not what happens, so what is wrong?

Imagine pushing a crate to the right across loose sand. Each time you take a step, you tend to kick the sand to the left, behind you. Thus friction force $\vec{f}{}_{\text{P on S}}$, the force of the person pushing against the earth’s surface, is to the left. In reaction, the force of the earth’s surface against the person is a friction force to the right. It is force $\vec{f}{}_{\text{S on P}}$, which we’ve shortened to $\vec{f}{}_{\text{P}}$, that causes the person to accelerate in the forward direction. Further, as we’ll discuss more below, this is a static friction force; your foot is planted on the ground, not sliding across the surface.

Finally, we have one internal interaction. The crate is pushed with force $\vec{F}{}_{\text{P on C}}$. If A pushes or pulls on B, then B pushes or pulls back on A, so the reaction to force $\vec{F}{}_{\text{P on C}}$ is $\vec{F}{}_{\text{C on P}}$, the crate pushing back against the person’s hands. Force $\vec{F}{}_{\text{P on C}}$ is a force exerted on the crate, so it’s shown on the crate’s free-body diagram. Force $\vec{F}{}_{\text{C on P}}$ is exerted on the person, so it is drawn on the person’s free-body diagram. The two forces of an action/reaction pair never occur on the same free-body diagram. We’ve connected forces $\vec{F}{}_{\text{P on C}}$ and $\vec{F}{}_{\text{C on P}}$ with a dashed line to show that they are an action/reaction pair.