Equilibrium of a Particle: Forces in Balance Explained

A particle is in equilibrium when the vector sum of all forces acting on it is zero — the forces perfectly balance, so the particle stays at rest or keeps moving with constant velocity. Lami's theorem is a shortcut for the special case of three concurrent forces in equilibrium: each force is proportional to the sine of the angle between the other two forces.

Equilibrium matters because most structures around you survive only by balancing forces: a hanging lamp, a kite held by its string, a picture frame on two nails, and a crane lifting a load. Engineers use these rules to size ropes and supports so nothing accelerates unexpectedly. Lami's theorem turns a three-force balancing problem into one quick proportion instead of two simultaneous component equations, which is why it appears in almost every mechanics exam.

Key Concepts

• Particle: an object treated as a single point, so rotation and torque are ignored — only the balance of forces matters.
• Concurrent forces: forces whose lines of action all pass through the same point.
• First condition of equilibrium: the vector sum of forces is zero, ΣF = 0, which splits into ΣFx = 0 and ΣFy = 0.
• Equilibrium types: static (at rest) or dynamic (moving with constant velocity); both satisfy ΣF = 0.
• Lami's theorem: for three concurrent forces P, Q, R in equilibrium, P / sin α = Q / sin β = R / sin γ, where α is the angle between Q and R, β between P and R, and γ between P and Q.
• Free-body diagram: a sketch showing every force on the particle — the essential first step before any calculation.

Formulas

• ΣFx = 0 and ΣFy = 0 — component equations for equilibrium of a particle.
• Lami's theorem: P / sin α = Q / sin β = R / sin γ — each force divided by the sine of the angle between the other two is a constant.

Assumptions: forces act at a single point (particle model), they are coplanar, and exactly three forces are present for Lami's theorem. Limits: Lami's theorem cannot be used with four or more forces, with non-concurrent forces, or when the body can rotate — then torque balance is also required.

Worked Example

A 50 N weight hangs from two strings: the left string makes 60° with the horizontal, the right string 30° with the horizontal. Find both tensions.

1. Draw the free-body diagram: T₁ (left, 60° above horizontal), T₂ (right, 30° above horizontal), and weight W = 50 N downward.
2. Angles between forces: between the strings = 90°; between T₁ and W = 150°; between T₂ and W = 120°. (They sum to 360°.)
3. Apply Lami: T₁ / sin 120° = T₂ / sin 150° = 50 / sin 90°.
4. T₁ = 50 × sin 120° = 50 × 0.866 = 43.3 N; T₂ = 50 × sin 150° = 50 × 0.5 = 25 N.
5. Check with components: horizontal — 43.3 × cos 60° = 21.65; 25 × cos 30° = 21.65 ✓. Vertical — 43.3 × sin 60° + 25 × sin 30° = 37.5 + 12.5 = 50 N ✓.

Common Misconceptions

• "In Lami's theorem, each force is divided by the sine of its own angle." Wrong — each force is divided by the sine of the angle between the other two forces, which is the angle opposite to it in the force triangle.
• "Equilibrium means the object is at rest." A particle in equilibrium may move with constant velocity (dynamic equilibrium). Rest is only one special case.
• "Lami's theorem works for any number of forces." It applies to exactly three concurrent forces. With more forces, resolve into components and use ΣFx = 0, ΣFy = 0.

Real-World Applications

• Hanging signs and lamps: designers compute cable tensions so that no cable is overloaded.
• Kites and tethered balloons: the string tension, wind force, and weight balance in a three-force system.
• Cranes: the load's weight balances against the tensions in the supporting cables.
• Surveying tripods: the three legs share the instrument's weight through concurrent forces at the head.

A particle is in equilibrium when all its forces add to zero, giving ΣFx = 0 and ΣFy = 0. When exactly three concurrent forces balance, Lami's theorem — each force over the sine of the angle between the other two — solves for the unknowns in a single step. Always start with a free-body diagram and verify with components.