Equilibrium of a Particle

What is a Particle?

In statics, a particle is a body whose size and shape are neglected — all forces act at a single point.

Conditions for Equilibrium

A particle is in static equilibrium when the net force acting on it is zero:

ΣF = 0

In 2D:

• ΣFx = 0
• ΣFy = 0

In 3D:

• ΣFx = 0
• ΣFy = 0
• ΣFz = 0

Free Body Diagram (FBD)

A FBD is an isolated diagram showing ALL forces acting on the particle:

1. Isolate the particle
2. Draw all applied forces
3. Draw all reaction forces (tension in cables, normal forces)
4. Label magnitudes and angles

Resolving Forces into Components

For a force F at angle θ from horizontal:

• Fx = F cos θ
• Fy = F sin θ

Example Problem

A 500 lb traffic light hangs from two cables. Cable A is at 30° from horizontal; Cable B is at 45°.

FBD at the junction point:

• W = 500 lb ↓
• T_A along cable A
• T_B along cable B

Equilibrium equations:

ΣFx = 0:  T_A cos(30°) - T_B cos(45°) = 0
ΣFy = 0:  T_A sin(30°) + T_B sin(45°) - 500 = 0

Solving simultaneously:

• T_A cos(30°) = T_B cos(45°) → T_A = T_B × (cos45/cos30) = 0.816 T_B
• 0.816 T_B × sin(30°) + T_B × sin(45°) = 500
• T_B (0.408 + 0.707) = 500
• T_B = 448 lb, T_A = 366 lb

Tips

• Always draw FBD first
• Assume unknown forces in tension (positive)
• Negative result = assumed direction was wrong

Summary

Equilibrium means ΣF = 0. The FBD is the most important tool. Resolve all forces into x and y components, then solve simultaneously.