For a rigid body in static equilibrium under a system of forces, which complete set of equilibrium conditions must be satisfied?

Introduction / Context:
Statics establishes the conditions under which a rigid body remains at rest. For planar force systems, both translational and rotational equilibria must hold. This question checks recognition of the full set of equilibrium equations needed in two dimensions.

Given Data / Assumptions:

• Planar (2D) rigid body system.
• Resultant acceleration is zero.
• Moments are considered about any arbitrary point in the plane.

Concept / Approach:
Equilibrium requires zero net force and zero net moment. In 2D, the vector equations expand into ΣH = 0 for horizontal components, ΣV = 0 for vertical components, and ΣM = 0 for the sum of moments about any point. These three scalar equations are necessary and sufficient for planar rigid-body equilibrium.

Step-by-Step Solution:
Write force equilibrium: ΣF = 0 ⇒ ΣH = 0 and ΣV = 0.Write moment equilibrium: ΣM = 0 about any point (choice does not change truth).Recognize that all three must simultaneously hold for static equilibrium.

Verification / Alternative check:
Choosing different moment centers yields consistent results when ΣH and ΣV are also zero; otherwise contradictions arise. Problems with three unknown reactions often rely on these three equations.

Why Other Options Are Wrong:

• Any single or pair (ΣH = 0 or ΣV = 0 or ΣM = 0 alone) is insufficient to ensure full equilibrium.
• none of these: Incorrect because all listed conditions are required.

Common Pitfalls:
Forgetting to enforce moment equilibrium; taking moments about a point that eliminates all unknowns of interest; sign mistakes in component resolution.

Final Answer:
all the above.