From the interaction diagram to the free body

The interaction diagram tells us who interacts with whom, but to apply Newton’s second law we need to isolate a single body and draw all the forces on it as vectors. This second diagram is called a free body diagram (FBD). The transition from one to the other is mechanical and unfolds in four moves.

Principle — From the interaction diagram to the free body

1. Circle the object of interest in the interaction diagram.
2. Count the squiggles crossing the circle: each corresponds to a force on the object.
3. Draw the free-body diagram: the isolated object with all forces as arrow vectors.
4. Choose the axes, resolve the forces, and write $\sum \vv{F} = m\vv{a}$.

The heart of the method lies in the second step: counting the outgoing squiggles. If four squiggles leave the circle, then exactly four forces act on the body — neither one too many (no forces are invented) nor one too few (none are forgotten). Each squiggle “translates” into an arrow on the FBD, oriented in the direction that specific interaction pushes or pulls the isolated body.

From the interaction diagram to the free-body diagram: the object is circled, the squiggles crossing the boundary are counted (4), and the corresponding forces are drawn as arrow vectors.

In the example, body 1 is connected to the Earth (weight $\vv{P}$), to the wall (constraint reaction $\vv{R}_{\text{par}}$), to the spring (elastic force $\vv{F}_m$) and to body 2 via a rope (tension $\vv{t}$). Four squiggles cross the circle, so the FBD of 1 carries four arrows. Once the forces are drawn, we choose the Cartesian axes and project the vector equation $\sum \vv{F} = m\vv{a}$ onto the components, obtaining the scalar equations to be solved.

Mistakes to avoid

• Do not invent forces: if there is no squiggle, there is no force.
• Do not confuse force with acceleration: $\vv{a}$ is not a force, it is the result of the forces.
• Each squiggle generates two twin forces (third law), but each acts on a different body. In the FBD of a single body, only one of the two appears.

The last point is the subtlest trap. Newton’s third law associates an action–reaction pair with each squiggle: the two forces are equal in magnitude and opposite in direction, but act on different bodies. When we isolate body 1, we draw only the force that the others exert on it; the twin reaction (which 1 exerts on the others) will appear, if at all, in the FBD of those other bodies, never in its own. Confusing the two leads to adding the same interaction twice, giving wrong equations.

Links

Topics: Dinamica Concepts: Seconda legge di Newton Skills: Diagramma di corpo libero · Diagramma di interazione Methods: Metodo del diagramma di corpo libero

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