3.2 Free-body diagrams and their importance

Free-body diagrams (FBDs) are the single most important skill you'll use in statics. They take a real-world object and strip it down to just the forces acting on it, giving you a clean picture you can translate directly into equilibrium equations. Get the FBD wrong, and every calculation that follows will be wrong too.

This section covers how to draw FBDs for both particles and rigid bodies, how to identify and classify the forces that belong on them, and how to connect them to the equilibrium equations you'll solve.

Free-body diagrams for particles and rigid bodies

Graphical representations of isolated objects

A free-body diagram isolates a single object (either a particle or a rigid body) from its surroundings and shows every external force acting on it. The word "free" means you've mentally cut the object free from all supports, cables, and contact surfaces. What's left is just the object and arrows representing forces.

FBDs serve two purposes:

• They let you set up equilibrium equations by making every force visible and organized.
• They act as a visual check to confirm you haven't missed a force or double-counted one before you start solving.

Replacing supports and connections with reaction forces

The key step in drawing an FBD is replacing every support, connection, or contact with the reaction force(s) it provides. You remove the physical constraint and substitute the force it would exert.

• For particles, all forces are drawn as vectors originating from a single point, since a particle has no physical size.
• For rigid bodies, forces act at specific points on the body. Each vector originates from its actual point of application, because where a force acts affects the moments it creates.
• Moments on rigid bodies are shown as curved arrows (clockwise or counterclockwise) at the point where they act.

All forces are drawn as vectors with arrows showing direction. Label each one with a symbol (known magnitude or an unknown variable) so you can reference it in your equations.

Identifying forces on a body

Classifying forces as applied or reaction forces

Every force on an FBD falls into one of two categories:

• Applied forces are forces imposed on the body by the environment or other objects. Examples: the body's own weight ($W = mg$), wind loads, tension from an attached cable, or a person pushing on the object.
• Reaction forces are forces that supports and constraints exert in response to the applied forces. They prevent the body from moving. Examples: the normal force from a surface, the tension in a rope that holds something in place, or the horizontal and vertical forces at a pin joint.

The distinction matters because applied forces are usually known (or given), while reaction forces are usually the unknowns you're solving for.

Common types of supports and their reaction forces

Different supports constrain motion in different ways, so they produce different reaction forces:

A few additional points:

• Friction forces should appear on the FBD whenever the object contacts a rough surface and sliding (or the tendency to slide) is relevant. For example, a block on an incline has both a normal force perpendicular to the surface and a friction force parallel to it.
• Internal forces (forces between parts within the isolated system) are never shown on an FBD. By Newton's Third Law, they come in equal-and-opposite pairs that cancel out. You only show them if you cut the system apart and isolate one piece.

Free-body diagrams for equilibrium problems

Equilibrium equations derived from Newton's First Law

Newton's First Law tells you that a body in equilibrium has zero net force and zero net moment. Translated into equations:

For a particle (no size, so moments don't apply):

• $\sum F_x = 0$
• $\sum F_y = 0$
• $\sum F_z = 0$

That gives you up to 3 independent equations (or 2 in a 2D problem).

For a rigid body (size matters, so moments come into play):

• $\sum F_x = 0$
• $\sum F_y = 0$
• $\sum F_z = 0$
• $\sum M_A = 0$ (moments about any chosen point A)

In 3D, you can write up to 6 independent equations (3 force, 3 moment). In 2D, you get 3 ($\sum F_x = 0$, $\sum F_y = 0$, and $\sum M_A = 0$).

Solving for unknown forces using equilibrium equations

Here's the general process for using an FBD to solve an equilibrium problem:

1. Isolate the body and draw it separated from all supports and connections.
2. Draw all external forces and moments, replacing each support with its appropriate reaction forces (use the table above). Include the body's weight acting at its center of gravity.
3. Label everything. Known forces get numerical values; unknown forces get variable names ($A_x$, $B_y$, etc.) with assumed directions.
4. Choose a coordinate system and, for rigid bodies, choose a convenient point for summing moments (picking a point where an unknown force acts eliminates that unknown from the moment equation).
5. Write the equilibrium equations and solve the system for the unknowns.
6. Check your results. If an unknown comes out negative, the actual direction is opposite to what you assumed. That's fine; just note the correct direction.

One critical constraint: you can only solve for as many unknowns as you have independent equilibrium equations. If you have more unknowns than equations, the problem is statically indeterminate, meaning statics alone can't solve it. You'd need additional relationships, such as material deformation equations from strength of materials.