Conditions of equilibrium for a rigid body in a plane State the necessary conditions that must be satisfied simultaneously for a coplanar force system to be in static equilibrium.
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AAlgebraic sum of horizontal components is zero only.
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BAlgebraic sum of vertical components is zero only.
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CAlgebraic sum of moments about any point is zero only.
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DAll three: ΣH = 0, ΣV = 0, and ΣM = 0.
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EOnly ΣM = 0 is sufficient for equilibrium.
Answer
Correct Answer: All three: ΣH = 0, ΣV = 0, and ΣM = 0.
Explanation
Introduction / Context:To ensure a rigid body in a plane does not translate or rotate, three independent equilibrium equations are required. These are the foundation of structural analysis, machine design, and statics problem solving.
Given Data / Assumptions:
- Coplanar forces acting on a rigid body.
- Classical statics (no acceleration).
- Any reference point may be chosen for taking moments.
Concept / Approach:
Equilibrium demands zero net translation and zero net rotation. Translational equilibrium in 2D requires ΣH = 0 and ΣV = 0. Rotational equilibrium requires ΣM = 0 about any point (if it is zero about one point and ΣH = ΣV = 0, it is zero about all points).
Step-by-Step Solution:
Impose ΣFx = 0 → no acceleration along x.Impose ΣFy = 0 → no acceleration along y.Impose ΣM_O = 0 about an arbitrary point O → no angular acceleration.Verification / Alternative check:
Choosing different moment centers yields consistent results when ΣH = ΣV = 0; otherwise, contradictions appear, revealing imbalance.
Why Other Options Are Wrong:
(a), (b), or (c) alone do not prevent the other type of motion; (e) ignores translational equilibrium.
Common Pitfalls:
Forgetting to include couple moments; assuming ΣM = 0 about one special point is sufficient without verifying forces sum to zero.
Final Answer:
All three: ΣH = 0, ΣV = 0, and ΣM = 0.