In engineering mechanics, a body is said to be in mechanical equilibrium when no net effect of forces or moments tends to change its state. Which statement best describes this condition for a rigid body (e.g., a machine part or structural element)?

Introduction / Context:
Equilibrium of a rigid body is a foundational concept in engineering mechanics. Designers and analysts check equilibrium before proceeding to deformation, stress, or stability calculations. This question tests your understanding of the complete mathematical condition required for a body to be in mechanical equilibrium.

Given Data / Assumptions:

• Rigid body assumption (deformations are negligible for equilibrium checks).
• Forces and moments are external resultants acting on the body.
• Static conditions (no acceleration).

Concept / Approach:
The equilibrium of a rigid body requires two independent conditions: force equilibrium and moment equilibrium. In vector form, these are written as:
ΣF = 0ΣM = 0These must hold simultaneously. If either condition is violated, the body translates or rotates with non-zero acceleration.

Step-by-Step Solution:
1) Check translational equilibrium: enforce ΣF_x = 0, ΣF_y = 0, and (in 3D) ΣF_z = 0.2) Check rotational equilibrium: enforce ΣM about any convenient point or axis = 0. For 3D, moments about x, y, z axes must each sum to zero.3) If both are satisfied, no net linear or angular acceleration occurs. The body remains at rest or moves with constant velocity (Newton’s first law).

Verification / Alternative check:
Using Newton’s second law: ΣF = m * a and ΣM = I * α. For equilibrium, a = 0 and α = 0, which directly yields ΣF = 0 and ΣM = 0.

Why Other Options Are Wrong:
Moves horizontally at a constant speed only: ignores vertical forces and moments.Moves vertically at a constant speed only: ignores horizontal forces and moments.Only rotates uniformly about its centre of gravity: allows nonzero translation or mismatched force balance.None of these: incorrect because the force-and-moment zero condition is the correct full statement.

Common Pitfalls:

• Forgetting to check moments after balancing forces.
• Taking moments about a point that hides unknowns without ensuring independence of equations.
• Assuming rest implies equilibrium without verifying external resultant is zero.

Final Answer:
The vector sum of all external forces is zero and the algebraic sum of all external moments about any point is zero.