Moment of a Force – Comprehensive Definitions Which statement correctly characterizes the moment (torque) of a force about a point?

Introduction / Context:
The moment (or torque) is central to statics and dynamics. Multiple equivalent definitions exist—algebraic, geometric, and vectorial—which are useful in different problem settings.

Given Data / Assumptions:

• A force F acts on a rigid body.
• A reference point O is specified for taking moments.
• Planar setting for scalar moment; vector cross product is the general form.

Concept / Approach:
Scalar magnitude of moment is M_O = F * d, where d is the perpendicular distance from O to the line of action. The direction (sign) depends on rotation sense. Geometrically, the moment magnitude also equals twice the area of the triangle formed by O and the force vector’s line, consistent with vector cross product properties.

Step-by-Step Solution:
Turning effect: the essence of torque—tendency to rotate. Perpendicular distance form: M_O = F * d (most common in statics). Geometric area form: |r × F| equals 2 × triangle area with sides r and F. All statements express the same physical quantity.

Verification / Alternative check:
Vector definition: M_O = r × F. Magnitude |M_O| = |r| * |F| * sin(θ). This equals F times the perpendicular distance and also equals twice the triangle area (½ |r||F|sinθ × 2).

Why Other Options Are Wrong:
“none of the above” is incorrect because all listed statements are valid descriptions of moment.

Common Pitfalls:
Using non-perpendicular distances in moment calculations. Forgetting sign convention when summing moments for equilibrium.

Final Answer:
all of the above