Equilibrium of a point under concurrent forces A particle is subjected to several forces acting through a point. Which condition ensures equilibrium of the point in a plane?

Introduction / Context:
For concurrent forces in a plane, equilibrium requires the vector sum of all forces to be zero. In practical computation we ensure this by resolving the forces along two perpendicular axes and setting both component sums to zero.

Given Data / Assumptions:

• Forces are concurrent at a point (no couple-only system).
• Planar problem (two dimensions).
• Rigid-body effects of moments are irrelevant for concurrency at the same point.

Concept / Approach:
Vector equilibrium of concurrent forces demands ΣF = 0. In components this is equivalent to ΣF_x = 0 and ΣF_y = 0 for any orthogonal axes. These two scalar equations completely specify equilibrium in the plane for concurrent systems.

Step-by-Step Solution:

Verification / Alternative check:
Graphical polygon of forces: if the head-to-tail polygon closes, the component sums are zero; the two methods are equivalent.

Why Other Options Are Wrong:

• “Algebraic sum of forces is zero” is ambiguous (forces are vectors, not scalars).
• “Resolved parts are equal” does not enforce zero; equilibrium needs zero resultant, not equality.
• “Only moments zero” is insufficient; forces could still have a net vector.

Common Pitfalls:
Mixing vector and scalar language; forgetting that any two perpendicular directions suffice, not specifically horizontal/vertical.

Final Answer:
The sums of the resolved components in any two mutually perpendicular directions are both zero