Free-body diagrams
A free-body diagram isolates a body and shows every external force on it. Drawing one correctly is the most important step in solving any statics problem.
Why free-body diagrams
Equilibrium equations are only as good as the forces put into them. A free-body diagram (FBD) makes those forces explicit: a sketch of the body cut free from its surroundings, with every force that the surroundings exert on it. Experienced engineers draw an FBD for every problem, however simple.
Steps to draw one
First decide which body or point to isolate. Sketch its outline free of supports and contacts. Then add, at the correct points, its weight, every contact force where it touches something, and every applied load, each with a label, direction, and known magnitude or a symbol for an unknown. Finally add axes and key dimensions or angles.
Common forces
Weight acts at the centre of gravity, straight down. A cable or rope pulls along its own length, away from the body, and can only pull, never push. A smooth surface pushes perpendicular to itself. A spring pushes or pulls along its axis with force proportional to its stretch or compression.
Unknown directions
When you do not know whether an unknown force pushes or pulls, assume a direction. If the answer comes out negative, the force acts the other way. This is perfectly valid and avoids guesswork.
Internal versus external forces
Only external forces, those applied by things outside the isolated body, belong on an FBD. Forces between parts inside the body cancel in pairs by Newton's third law and are omitted, unless you cut the body there to expose them.
Particles in equilibrium
For a particle, all forces pass through one point, so the only equilibrium condition is \(\sum \mathbf{F} = \mathbf{0}\): in 2D, \(\sum F_x = 0\) and \(\sum F_y = 0\). A ring where three cables meet, or a small hook carrying a load, are typical particles.
Mistakes to avoid
Common errors include forgetting the weight, drawing forces that the body exerts on others rather than forces acting on it, letting cables push, mislabelling angles, and including internal forces. Checking each arrow against 'what touches the body here?' catches most of them.
Worked example
A 20 kg traffic light hangs from a ring A, which is held by two cables, AB at 30 degrees and AC at 45 degrees above the horizontal, and the light's cable AD hanging straight down. The FBD of ring A shows three forces: T_AD = 20 x 9.81 = 196.2 N downward, T_AB along AB away from A, and T_AC along AC away from A. Two equations will give the two unknown tensions.
Key takeaways
A free-body diagram isolates a body and shows every external force on it, with labels, directions, axes, and dimensions. Weight acts downward at the centre of gravity, cables pull along their length, smooth surfaces push perpendicular to themselves, and springs act along their axis. Assume directions for unknowns; a negative answer means the opposite direction. Omit internal forces. For a particle, equilibrium means the forces sum to zero, giving two equations in 2D.
Flashcard
Body isolated with all external forces
Check your understanding
On a free-body diagram, a cable force is always drawn: