Equilibrium

The Principle of Moments states that for a body in rotational equilibrium, the sum of clockwise moments about any pivot point must equal the sum of anticlockwise moments about that same point.

A Moment (or torque) is defined as the product of the force and the perpendicular distance from the pivot to the line of action of the force: $M = F \cdot d_{\perp}$.

For three coplanar forces to be in equilibrium, their vectors must form a closed triangle when placed head-to-tail, indicating a resultant of zero.

The choice of pivot is arbitrary in an equilibrium system, but selecting a point where an unknown force acts simplifies calculations by making its moment zero.

Free-Body Diagrams (FBD): Always start by isolating the object and drawing all external forces acting on it as vectors originating from their points of application.

Resolving Forces: Break diagonal forces into horizontal ($F_x = F \cos \theta$) and vertical ($F_y = F \sin \theta$) components to solve $\sum F_x = 0$ and $\sum F_y = 0$ independently.

Vector Triangle Method: For systems with exactly three forces, rearrange the force vectors head-to-tail to form a closed triangle and use trigonometry (Sine or Cosine Rule) to find unknowns.

Moment Equations: Set up an equation where $\sum M_{\text{clockwise}} = \sum M_{\text{anticlockwise}}$ about a strategic pivot point to solve for unknown distances or magnitudes.

Static vs. Dynamic: Static equilibrium implies the object is at rest ($v=0, \omega=0$), while dynamic equilibrium implies constant motion ($v \neq 0$ or $\omega \neq 0$) without acceleration.

Moment vs. Force: A force causes linear acceleration, whereas a moment (force at a distance) causes angular acceleration.

Strategic Pivot Selection: Always try to place the pivot at the point of application of an unknown or unwanted force. This eliminates that force from the moment equation because its distance $d$ is zero.

Check Perpendicularity: Ensure the distance used in $M = F \cdot d$ is the perpendicular distance from the pivot to the line of action of the force, not just the distance along a beam.

Sign Convention: Consistently assign one direction (e.g., clockwise) as positive and the other as negative to avoid arithmetic errors when summing moments.

Sanity Check: If an object is in equilibrium, the upward forces must equal the downward forces. Use this as a quick final check after solving moment equations.