Lami’s theorem in coplanar concurrent force systems: Which precise statement correctly expresses Lami’s theorem for three coplanar forces acting at a point in equilibrium?

Introduction / Context:
Lami’s theorem is a core tool in engineering mechanics for solving problems with three coplanar, concurrent forces in equilibrium. It provides a direct proportionality between each force and the sine of the angle between the other two forces, avoiding component resolution when only three forces act at a point.

Given Data / Assumptions:

• Three forces are coplanar and concurrent at a point.
• The system is in static equilibrium (resultant equals zero).
• Angles are the included angles between the directions of the other two forces.

Concept / Approach:

Lami’s theorem states: F1 / sin(α) = F2 / sin(β) = F3 / sin(γ), where α is the angle between F2 and F3, β between F3 and F1, and γ between F1 and F2. This relation arises from the equilibrium polygon (triangle of forces) and the sine rule applied to that triangle.

Step-by-Step Solution:

Verification / Alternative check:

Resolving each force along two perpendicular axes and imposing ΣFx = 0 and ΣFy = 0 leads to the same sine relationships after elimination, confirming Lami’s theorem.

Why Other Options Are Wrong:

(a) is too broad; equilibrium is not automatic. (b) uses a triangle condition but is not the precise theorem. (d) claims inverse proportionality, which is incorrect. (e) is not applicable.

Common Pitfalls:

Forgetting the forces must be exactly three, coplanar, and concurrent; misidentifying the included angles; mixing angle definitions.

Final Answer:

Three coplanar forces in equilibrium are each proportional to the sine of the angle between the other two