Moments and Equilibrium

The moment of a force about a point (the pivot) is defined as the product of the force and the perpendicular distance from the pivot to the line of action of the force. This perpendicular distance is called the moment arm or lever arm. If the force F acts at a perpendicular distance d from the pivot, then Moment M = F × d. The SI unit of moment is the newton-metre (N·m). Note that the moment arm must be perpendicular to the force — if the force is applied at an angle θ to a rod of length l, the effective perpendicular distance is l sin θ, giving M = F l sin θ.

Moments have a sense of direction: they can be clockwise (CW) or anticlockwise (ACW). By convention, anticlockwise moments are taken as positive and clockwise moments as negative (though you must be consistent within a problem). When solving problems, label each moment with its direction. For example, a downward force of 10 N acting 3 m to the right of a pivot produces a clockwise moment of 30 N·m, written as −30 N·m in the sign convention where ACW is positive.

The Principle of Moments states that for an object in rotational equilibrium, the sum of all clockwise moments about any point equals the sum of all anticlockwise moments about that same point. Mathematically: Σ(ACW moments) = Σ(CW moments), or equivalently Σ M = 0 about any chosen pivot. This principle holds regardless of which point you choose as the pivot — a powerful tool that lets you strategically eliminate unknown forces from your equation by choosing the pivot at the point where those forces act.

A rigid body is in complete static equilibrium when both of the following conditions hold simultaneously. Condition 1 (Force equilibrium): The vector sum of all external forces is zero: ΣF = 0. In 2D problems this means ΣFx = 0 and ΣFy = 0 separately. Condition 2 (Moment equilibrium): The sum of all moments about any point is zero: ΣM = 0. Together, these three scalar equations (ΣFx = 0, ΣFy = 0, ΣM = 0) allow you to solve for up to three unknowns in a planar statics problem.

A couple is a pair of equal, parallel, opposite forces whose lines of action do not coincide. A couple produces pure rotation — no net force — and its moment (called the torque of the couple) is the same about every point. If the two forces each have magnitude F and are separated by a perpendicular distance d, the couple's moment is M = F × d. Couples are important in engineering: for example, tightening a bolt with a spanner involves applying a couple to produce rotation without translation.

• M — Moment (torque) (N·m): The turning effect of the force about the chosen pivot point.
• F — Force (N): The magnitude of the applied force.
• d_\perp — Perpendicular distance (moment arm) (m): The shortest distance from the pivot to the line of action of the force.
• l — Distance to point of application (m): The distance along the rod or lever from the pivot to where the force is applied.
• \theta — Angle (degrees or radians): The angle between the force vector and the rod or lever arm.

Problem. A uniform beam AB of length 4 m and weight 60 N is supported at its midpoint (the fulcrum at C, 2 m from each end). A child of weight 400 N sits at end A. Where must an adult of weight 700 N sit from C on the other side (end B side) so that the beam is in equilibrium? Ignore the thickness of the beam.

Answer. The adult must sit approximately 1.14 m from the fulcrum C on the B side.

• Resolution of Forces — To find the perpendicular component of a force for calculating its moment arm, you need to resolve forces into components using trigonometry.
• Beams and Reactions — Applying the conditions of equilibrium (ΣF=0 and ΣM=0) to beams with multiple supports is a direct extension of moments and equilibrium.
• Centre of Mass and Gravity — The weight of a uniform body acts through its centre of mass, which determines its moment arm — essential for beam and lamina equilibrium problems.
• Friction — Many equilibrium problems include frictional forces that must be included in both ΣF=0 and ΣM=0 to correctly solve for unknown reactions.