Moment of a Force Objects External Effects a particle translation a 3-D body translation and rotation The moment of a force about a point or an axis: a measure of the tendency of the force to rotate a body about that point or the axis.

Definition: Moment Common-sense definition: Vector representation: Magnitude: F o d Direction: is perpendicular to the plane defined by the force and the rotation center and can be determined by the right-hand rule. Vector representation: r is a position vector from the rotation center to any point on the line of action of the force.

Example 1: Find the moments of F , F and F about point o. 2 3 o

(a) the moment of force F about point A Example 2: Determine (a) the moment of force F about point A (b) the perpendicular distance from point A to the line of action of the force. 30o

Moment of a Force about a Line Definition: l F Example: What is the moment about x axis? z M y x

Couples Definition two equal, noncollinear, parallel forces of opposite senses are called a couple. Effects rotation only, no translation

Moment of a Couple Common sense definition F Magnitude: d Vector representation F Magnitude: d Direction: right-hand rule -F F r -F

Couples Characteristics The magnitude of the moment of the couple The direction of the moment of a couple Equivalent couples - same and F d F F d F F d F

Equivalent Force/Couple Systems External effects translation rotation Conditions for two force/couple systems to be equivalent:

The Force-and-Couple Resultant of a Force System

Equivalent Force / Couple Systems Example 1: (a) Replace F by an equivalent force/couple system applied at A. (b) Replace F by an equivalent force/couple system applied at B. 5m F=100N B 5m A 30m

Equivalent Force / Couple Systems Example 2: Reduce the force/couple system by an equivalent single force y 1000N 500N 500N 45o x 3m 2m 1m

Equivalent Force / Couple Systems Questions: Can any general force/couple system be reduced to a single force/couple system? Can any general force/couple system be reduced to a single force? If two force/couple systems have the same moments about point P and have the same net force, is valid? Q is an arbitrary point other than P.

Distributed Force Systems Force distributed over a line Force distributed over a surface Force distributed over a volume x f (x) x y f (x,y) F(x,y,z)

Force Distributed Over a Line x f (x) d L Examples: L O x f0 x f (x)=f0 L O

Centroid of an Area Definition: Example 1: Find the centroid of a rectangle. h w

Centroid of an Area Composite body: Example 2: Find the centroid of the following area. h w w 4h w