From Free Body Diagram to Equations

A finished FBD is only half the job. This guide covers the translation step — turning arrows into equations you can actually solve — plus the small tactical choices (axes, moment points) that make the algebra 3× shorter.

The Governing Equations

For a body in equilibrium (at rest or constant velocity):

• ΣFx = 0
• ΣFy = 0
• ΣM (about any point) = 0

For an accelerating body (dynamics):

• ΣFx = m·ax
• ΣFy = m·ay

That's it. Every mechanics problem is these equations applied to a correct FBD.

Step 1 — Choose Axes That Match the Motion

Align one axis with the direction of (possible) motion:

• Flat ground → standard horizontal/vertical axes.
• Incline → tilt axes along/perpendicular to the slope. Then only weight needs resolving: mg sin θ along the slope, mg cos θ into it. Every other force (N, f, T) lies on an axis.
• Circular motion → point one axis toward the center (centripetal direction).

Step 2 — Resolve Angled Forces into Components

Any force F at angle θ from the x-axis becomes Fx = F cos θ and Fy = F sin θ. Write the components directly next to each arrow on your diagram — this prevents sign errors later.

Sign convention: pick positive directions once (e.g., right = +x, up = +y, counterclockwise = +M) and never change them mid-problem.

Step 3 — Write the Force Equations

March through the FBD arrow by arrow. Every arrow contributes to ΣFx and ΣFy exactly once — with a sign given by your convention. If an arrow doesn't appear in your equations, or appears twice, the diagram and math have diverged.

Step 4 — Write the Moment Equation (Statics)

Moment of a force about point O = force × perpendicular distance from O to the force's line of action, with CCW positive.

The pro move — pick the moment point that kills unknowns: take moments about the point where the most unknown forces intersect. For a simply supported beam, summing moments about the pin eliminates both pin reactions, leaving one equation with one unknown (the roller reaction). Solve it first, then use ΣFy for the rest.

Step 5 — Count Equations vs. Unknowns

• 2D particle: 2 equations → solve up to 2 unknowns.
• 2D rigid body: 3 equations → up to 3 unknowns.
• More unknowns than equations? Isolate another body (each new FBD brings 2–3 new equations) or the problem is statically indeterminate.

Worked Example: Block Sliding Down a Rough 30° Incline

FBD: weight mg down, normal N perpendicular to slope, kinetic friction f = μkN up the slope. Axes tilted along the incline.

• Along slope (+x down-slope): mg sin 30° − μkN = m·a
• Perpendicular: N − mg cos 30° = 0 → N = mg cos 30°
• Substitute: a = g(sin 30° − μk cos 30°)

Two equations, two unknowns (N, a) — solved in three lines because the axes were tilted and the FBD was complete.

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