Position Vectors The position vector r is defined as a fixed vector which locates a point in space relative to another point. r = xi + yj + zk x y z O P(x,y,z) r yj xi zk r AB A = from point B = to point r OA =r A

Position Vectors r A + r = r B r= r B - r A =(x B i+y B j+z B k)-(x A i+y A j+z A k) r= (x B -x A )i+(y B -y A )j+(z B -z A )k x y z O B(x B,y B,z B ) rBrB rArA r A(x A,y A,z A )

Moment of a force about a point A measure of the tendency of the force to cause a body to rotate about the point or axis. The moment of a force about a point O; M O = r x F A O F r MOMO

Moment of a force about a point M O = rFsinθ = F (rsinθ) = Fd A O F r MOMO r θ d MxMx MyMy MzMz

Moment of a force about a line In order to find the projected component of the moment about an axis. r a a’ F MoMo MaMa

Moment of a force about a line as a Cartesian vector;

Example The magnitude of the force shown in the figure is 140 N. Determine; a. The moment of F about point D. b. The moment of F about a line joining D and A. c. The angle btw. BG & DG. z x y C D(4,0,2) G(2,0,3) B(4,6,0) A(0,6,0) E(2,0,2) F

Moment of a Couple A couple consists of two forces of equal magnitude having parallel lines of action but opposite directions. d F F Moment of couple

Example Add couples whose forces act along diagonal of the rectangular prism. x y z 5 N 10 N 10 5

Equivalent Force Systems Translation of a force to a parallel position Resultant of a force system Distributed force systems

Translation of a force to a parallel position A force can be moved to any parallel position provided that a couple moment of the correct orientation and size is simultaneously provided. z x y F=6i+3j+6k A(2,1,10) P(6,10,12) Replace this force by an equivalent force system at point P?

Resultant of a force system Any force system can be replaced by any point by a single force and a single couple moment. M C :couple moments

Example a. If θ=60°, F 1 =15N, F 2 =80N, F 3 =50N compute the equivalent resultant force. b. Determine the intercepts of this resultant force with x and y axes. c. Find the magnitude and direction of force F 3 such that the simplest resultant is only A COUPLE MOMENT at point O F 1 =15N, F 2 =80N. d. If F 1 =15N, F 2 =80N, F 3 =50N find the direction of force F 3 such that the simplest resultant is only a force at point O. e. Determine the magnitude of force F1 and the magnitude and direction of F 3 such that the equivalent force and the couple moment at point O are ZERO. F1F1 F3F3 F2F2 x O 4m 3m 2m θ

Distributed Force Systems Volume Distribution If the force is distributed over the volume of a body. (e.g. Force of gravitational attraction. N/m 3 ) Area Distribution If the force is distributed over an area. (e.g. Hydraulic pressure of fluid. N/m 2 ) Line Distribution If the force is distributed along a line. (e.g. Vertical load supported by a suspended cable. N/m)

Center of a Geometry Center of Mass Center of Gravity Centroid

Line Force Distribution w: load intensity (N/m) y x w(x) x FRFR

Example y x w(x) F R =? 9m 200 N/m