AMME1802 Chap.2 Moments, Couples and Resultants
Moments, Couples and Resultants
Moments describe rotational tendency, while couples create rotation without a net force. This chapter connects perpendicular distance with the cross product, then reduces a general planar loading to an equivalent force and couple at a selected point. The central habit is maintaining separate force and moment ledgers until the final equivalent system is stated.
The perpendicular-distance method is efficient when the line of action is obvious; the cross product is safer when coordinates and components are supplied. Both require a declared centre and sign convention. A free couple can be added to any moment ledger without relocating it, but a force moved to a parallel line needs a compensating couple.
Equivalent systems are tested by matching resultant force and moment about a common point. If a single resultant line is requested, its offset must reproduce the full signed moment, not merely the moment magnitude.
What this chapter covers
- 01
Moment from perpendicular distance
- 02
Moment from r cross F
- 03
Right-hand sign convention
- 04
Free couples
- 05
Equivalent force-couple systems
- 06
Changing the reference point of a reduction
- 07
Varignon checks using moments of separate Cartesian force components
- 08
A final equivalence check using force and moment ledgers
Worked example · free
Equivalent loading at a point
- +1Add the two force vectors component by component.
- +1For each force calculate xFy-yFx about O.
- +1Add the free couple with its signed rotational sense.
- +1State the resultant vector and total couple at O.
- +1Check one force moment by perpendicular distance or a determinant.
Key terms
- Moment arm
- The perpendicular distance from a moment centre to a force line of action.
- Couple
- A zero-resultant force pair that produces a free moment.
- Equivalent system
- A loading with the same resultant force and moment effects as another loading.
- Moment centre
- The point about which rotational tendency is calculated.
- Free vector
- A vector whose external effect is unchanged by moving its point of application.
- Force ledger
- A structured list of signed force components before summation.
- Moment ledger
- A structured list of signed force moments and applied couples.
- Reference moment
- The signed moment of a loading evaluated about a specifically named point.
Moments, Couples and Resultants FAQ
When should I use a cross product?
Use r cross F when coordinates or force components are already available, especially in three dimensions. In a simple planar geometry, force times perpendicular distance may be shorter. Both must give the same signed moment when the same centre is used.
Can a force be moved freely?
A rigid-body force may slide along its own line of action without changing its external effect. Moving it to a parallel line changes the moment, so a compensating couple must be added to preserve equivalence.
Why is a couple called free?
Its two forces sum to zero, and the couple moment has the same value about every reference point. That point-independence allows the couple vector to be added directly when reducing a loading at a chosen point.
How do I verify an equivalent system?
Compute the resultant force and the resultant moment about one common point for both loadings. Matching only the force is insufficient because parallel forces at different locations rotate the body differently. A second reference-point calculation provides a useful confirmation.
Assessment move
For each problem, mark the moment centre and either draw the perpendicular distance or label the position vector. Practise translating between distance and cross-product methods. Keep a signed moment ledger and finish by imagining the rotation to verify clockwise or counterclockwise sense. Separate each page into a force ledger and a moment ledger.
The force ledger carries x and y components; the moment ledger carries xFy, minus yFx and any free couples. This layout makes it difficult to add a moment to a force or to lose a length unit. Rework simple forces by both methods until force times perpendicular distance and r cross F agree with the same sign.
Then practise transfers: slide a force along its line, move it to a parallel line with a compensating couple, and reduce several loads at two different reference points. Confirm that the resultant force stays fixed while the reported couple changes with the point. When solving for a single line of action, plot the signed intercept on the original sketch and substitute it back into the moment equation.
Finish revision by explaining why an arbitrary force-couple system in three dimensions may not collapse to one force. Prepare a set of moment sketches in which the same force is applied at different points along one line, then along parallel lines. Predict which moves preserve the moment and which require a couple before calculating.
Include one problem with millimetres and newtons so unit conversion happens before the cross product. For planar reductions, tabulate every force with its x component, y component and signed moment contribution. Recalculate the total about another point and verify the transformation between reference moments.
Practise finding a resultant intercept on both sides of the origin so a negative distance is interpreted geometrically rather than erased. End with a verbal equivalence test: name the resultant force and the moment about a common point for both systems.
