Overview
This lecture explains the concept of a couple in mechanics, how force-couple systems work, and the properties and notation of couples and moments.
Definition and Properties of Couples
- A couple consists of two equal, opposite, and parallel (non-colinear) forces acting on a rigid body.
- The moment produced by a couple is a vector called the couple moment.
- The magnitude of the couple’s moment is ( M = F \times d ), where ( F ) is the force and ( d ) is the distance between forces.
- The couple moment is independent of the reference point and is thus a free vector.
Free Vectors vs. Sliding Vectors
- Forces are sliding vectors; they can move along their line of action but not off it.
- Moments (couples) are free vectors; they can move anywhere and still have the same rotational effect.
- The rotational direction of a moment is determined using the right-hand rule.
Sign Convention for Couples
- Counterclockwise couples produce a positive moment (out of the page, along the positive Z-axis).
- Clockwise couples produce a negative moment (into the page, along the negative Z-axis).
Couples in Parallel Planes and Force-Couple Systems
- The effect of a couple remains the same even if the forces act in separate parallel planes.
- The couple’s moment vector remains unchanged regardless of the location of the parallel planes.
Replacing Forces and Equivalence
- To move a force away from its line of action, it must be compensated by adding a couple (moment).
- Two systems are equivalent if they have the same resultant force and same moment about any reference point.
- Replacing a force at one point with the same force at another point plus an appropriate couple creates a force-couple system.
Key Terms & Definitions
- Couple — Two equal, opposite, and parallel forces separated by a distance, producing a pure rotational moment.
- Moment (of a couple) — The turning effect produced by a couple, given by force times the perpendicular distance between them.
- Free Vector — A vector whose effect does not depend on its point of application (e.g., moments).
- Sliding Vector — A vector that can be moved along its line of action but not away from it (e.g., forces).
- Force-Couple System — A combination of a force and a couple that is equivalent to the original force at a different location.
Action Items / Next Steps
- Review the definition and properties of couples and force-couple systems.
- Practice calculating moments of couples and identifying free vs. sliding vectors.