In engineering mechanics, Lami theorem is applicable only to which type of system of forces in equilibrium?

Introduction / Context:
This question belongs to the topic of engineering mechanics and deals with Lami theorem, a very useful relation for solving problems involving three forces in equilibrium. Understanding the exact conditions under which Lami theorem can be applied is important for correctly choosing it as a tool in statics problems rather than misusing it in more general force systems.

Given Data / Assumptions:
- The question asks about the type of force system for which Lami theorem is applicable.- We assume the forces are acting on a single rigid body in equilibrium.- The options mention different combinations of concurrent and coplanar forces.

Concept / Approach:
Lami theorem states that if a body is in equilibrium under the action of three forces, then each force is proportional to the sine of the angle between the other two forces. Symbolically, if the three forces are F1, F2, and F3, making angles opposite to them α, β, and γ respectively, then F1 / sin α = F2 / sin β = F3 / sin γ. This theorem is valid only under specific conditions: the three forces must be concurrent, meaning they meet at a single point, and coplanar, meaning they lie in the same plane. Also, the body must be in equilibrium so that the vector sum of the forces is zero.

Step-by-Step Solution:
- Lami theorem deals explicitly with three forces acting on a point in equilibrium.- For three forces to be in equilibrium, their vector sum must be zero, and they must act at the same point or be concurrent.- The theorem uses angles between forces, so all three forces must be in a single plane, that is, be coplanar.- Therefore, the correct condition is that the three forces are coplanar and concurrent.- From the options, this corresponds to coplanar and concurrent forces.

Verification / Alternative check:
Consider a standard example: three strings meeting at a point supporting a mass, each string making a different angle with the vertical. The tensions in the three strings are the three concurrent forces. All forces lie in a single vertical plane. Lami theorem is used to relate these three tensions and solve for magnitudes. If the forces were not concurrent or not coplanar, the standard Lami relation would not hold and we would need more general vector methods.

Why Other Options Are Wrong:
- Non concurrent forces: Lami theorem cannot be used when forces do not meet at one point because the equilibrium condition then involves moments and distributed effects.- Coplanar forces only: Coplanarity alone is not enough; the forces must also be concurrent.- Any type of forces: This is too broad and incorrect because Lami theorem is not a universal equilibrium condition.- Non coplanar concurrent forces: For such three dimensional systems, Lami theorem in its standard planar form does not apply.

Common Pitfalls:
A frequent error is to try to apply Lami theorem to more than three forces or to forces that are not concurrent. Some students also forget about the requirement of coplanarity and attempt to use it in three dimensional configurations. When solving statics problems, always check that you have exactly three forces, that they intersect at a point, and that they lie in a plane before using this theorem.

Final Answer:
Lami theorem is applicable only for coplanar and concurrent forces in equilibrium.