Statics – Full Equilibrium Conditions for Coplanar Force Systems For any system of coplanar forces acting on a rigid body, which conditions must be satisfied simultaneously for equilibrium?

Introduction / Context:
Equilibrium analysis is the cornerstone of statics. For planar (coplanar) problems, the correct set of independent equations is needed to solve reactions, internal forces, and support conditions for structures and mechanisms.

Given Data / Assumptions:

• Rigid body assumption.
• All forces lie in one plane (2D problem).
• Moments are taken about any convenient point in the plane.

Concept / Approach:
For a rigid body in the plane, there are three independent equilibrium conditions: the sums of force components in two orthogonal directions must be zero, and the sum of moments about any point must be zero. Using all three ensures no net translation or rotation.

Step-by-Step Solution:
Write ΣFx = 0 to prevent acceleration along x. Write ΣFy = 0 to prevent acceleration along y. Write ΣM_O = 0 (about any point O) to prevent angular acceleration. These three equations fully describe planar equilibrium.

Verification / Alternative check:
Taking moments about different points yields equivalent conditions when ΣFx = 0 and ΣFy = 0 hold. If any condition is omitted, rotational or translational equilibrium may be violated.

Why Other Options Are Wrong:
Using only ΣM = 0 ignores possible net forces; the body could translate without rotation. Selecting only ΣFx = 0 or ΣFy = 0 is incomplete.

Common Pitfalls:
Forgetting to include couple moments or distributed loads as equivalent forces/moments. Choosing an inconvenient moment center leading to algebraic complexity.

Final Answer:
all of the above