Force Equilibrium Calculator
Analyse force equilibrium using Lami's theorem, concurrent forces, or general (non-concurrent) force systems
Inputs
Formula Explanation
Formula ① — Vector Equilibrium
When a body is in static equilibrium, the vector sum of all forces acting on it is zero: = 0.
Formula ② — Lami's Theorem
For three concurrent forces in equilibrium, each force divided by the sine of the angle between the other two equals the same constant. Angles satisfy = 360°.
Formula ③ — Concurrent Forces (Coordinates)
Resolve all forces into X and Y components. Equilibrium requires both = 0 and = 0.
Formula ④ — Non-Concurrent Forces (Coordinates)
For forces not sharing a single action point, add moment equilibrium: = 0, = 0, and = 0.
Key Concepts
Force Equilibrium
When an object receives several forces simultaneously and remains stationary, those forces are said to be in equilibrium. The vector sum of all forces is zero.
Moment Equilibrium
Force equilibrium alone does not guarantee rotational rest. When forces are balanced but moments are not, the body will rotate. Full equilibrium requires both conditions.
Lami's Theorem
For three concurrent coplanar forces in equilibrium, each force is proportional to the sine of the angle between the other two forces. The three angles sum to 360°.
Coordinate Method
Resolve all forces into X and Y components, then sum each direction independently. For non-concurrent forces, add a third moment equation = 0.
Worked Example
A rod pivots at A. End B carries a 250 N load. A rope at C (200 mm from A) makes 30° with the rod. Find rope tension T and the wall's normal force.
Step 1 — Moment equilibrium about A
Step 2 — Solve for T
Step 3 — Normal force on wall
Rope tension , wall normal force .
Further Reading
Structural Reactions
Beam support reactions are solved using the three equilibrium equations: ΣX=0, ΣY=0, ΣM=0, giving pin/roller reactions at each support.
Truss Analysis
Each truss joint is a concurrent force system. Applying ΣX=0 and ΣY=0 at each joint reveals member forces (tension or compression).
Cable and Pulley Systems
Rope tensions and pulley reactions are found by applying Lami's theorem or the coordinate method at each junction point.
Three-Force Member
If a rigid body in equilibrium has exactly three forces, those forces must be concurrent (meet at a point) or all parallel — this is the three-force member principle.