Engineering Mechanics – Moment of a Force about a Point For the force P acting on a rigid body, what is the correct expression for the moment of P about point O (i.e., torque at O), considering OB is the perpendicular distance from O to the line of action of P?

Introduction / Context:
In engineering mechanics, the moment (torque) of a force about a point measures the tendency of that force to rotate the body about the point. This concept is fundamental to solving support reactions, designing levers, and checking equilibrium.

Given Data / Assumptions:

• A force P acts on a rigid body along a known line of action.
• Point O is the reference about which the moment is taken.
• OB denotes the shortest (perpendicular) distance from O to the line of action of P.
• Sign of the moment depends on rotation sense, but the magnitude uses perpendicular distance.

Concept / Approach:
The scalar magnitude of the moment of a force about a point is given by M = P * d, where d is the perpendicular distance from the point to the force’s line of action. Any non-perpendicular segment (for example OA or OC) is not appropriate unless its perpendicular component is taken.

Step-by-Step Solution:
Identify the line of action of P. Drop a perpendicular from O to that line; its length is OB. Compute moment magnitude: M_O = P * OB. Assign sign based on clockwise or counterclockwise tendency (not asked here).

Verification / Alternative check:
Using vector formulation: M_O = r × P. The magnitude is |r| * |P| * sin(θ) which equals P times the perpendicular distance. Hence the shortest distance OB is correct.

Why Other Options Are Wrong:
P × OA / P × OC / P × AC: OA, OC, AC are not guaranteed to be perpendicular distances; using them blindly can overstate or understate the torque. P × (OA + OC): adds unrelated lengths and has no basis in the moment definition.

Common Pitfalls:
Using any convenient segment instead of the true perpendicular distance. Forgetting the sign (direction) of the moment when doing equilibrium.

Final Answer:
P × OB