Lecture #3 Additional Methods of Vector Analysis (Ref: Section 2.7 and 2.8, position vectors and force vector directed along a line) R. Michael PE 8/14/2012

Forces in Reality Previous problems, new resultant force and direction (resolve into components) or had multiple forces and their directions (find resultant. But…………………… Very seldom can you directly measure the directions of a force vector The majority of the time you can measure the force in the member and know the physical characteristics of the member. KEY: position vectors and force vector directed along a line Basically all this means is we have a force and we know start and endpoint of force defined by coordinates (i.e. it’s line of action), then we can find the force components.

Force Magnitude and Slope Ratio Given the slope ratio of the line (3/4) (meaning 3 inches of rise for every 4 inches vertical covered) Determine the resultant of the triangle (3/4/5). Fx = F * 4/5 Fy = F * 3/5 Do Example! Start out with simple 2D:

Additional Methods of Force Resolution Additional Methods can be used in the following situations: Given force magnitude where exerted and rectangular dimensions of point on its line of action Given force magnitude and two points on its line of action. Given force magnitude and slope ratio

Rectangular Coordinates of Point on the Line of Action of a Force Given the magnitude of a force (F) acting on a point at the origin of the coordinate system and the rectangular coordinates of a point, P(X p, Y p, Z p ), in its line of action

Rectangular Coordinates of Point on the Line of Action of a Force The distance from the origin to Point (P) is P total = √(P x 2 +P y 2 +P z 2 ) Rectangular Components of the Force vector can then be found by: F x = F(P x /P total ) F y = F(P y /P total ) F z = F(P z /P total )

Rectangular Coordinates of Point on the Line of Action of a Force The Direction angles of the Force Vector can be found by using the Rectangular Components of the Force or Rectangular Coordinates of the Point. cos(θ x ) = F x /F = P x /P cos(θ y ) = F y /F = P y /P cos(θ z ) = F z /F = P z /P

Force Magnitude and Two Points on its Line of Action Given two points in Space (A) with coordinates (X a, Y a, and Z a ) (B) with coordinates (X b, Y b, and Z b ) With a Force Vector (F) acting at point (A) in the direction of (B)

Force Magnitude and Two Points on its Line of Action Calculate the total displacement in rectangular components of Point (B) with respect to Point (A) d x = X b -X a d y = Y b -Y a d z = Z b -Z a Total Displacement (d) = √(d x 2 +d y 2 +d z 2 )

Force Magnitude and Two Points on its Line of Action Rectangular Components of the Force vector can then be found by: F x = F(d x /d) F y = F(d y /d) F z = F(d z /d) Note: Direction Angles can be found using Rectangular Components of Force or Rectangular Displacements.

Example 1: Find Resultant of 3D Force System (FA = 500 kN, FB = 1,000 kN:

Example 2: Tough: given resultant force – where does point A need to be located? 20 m 15 m F DA F DB F DC

Basically the same except uses position vector and unit vector Better book keeping (uses vectors) It’s the same as using distance and scalers defined in previous slides! It’s simple! THE BOOKS APPROACH: Optional Way for 3D forces when a FORCE VECTOR DIRECTED ALONG A LINE (Section 2.8) using position and unit vectors:

POSITION VECTOR Consider two points, A and B, in 3-D space. Let their coordinates be (X A, Y A, Z A ) and (X B, Y B, Z B ), respectively. A position vector is defined as a fixed vector that locates a point in space relative to another point.

POSITION VECTOR The position vector directed from A to B, r AB, is defined as r AB = {( X B – X A ) i + ( Y B – Y A ) j + ( Z B – Z A ) k }m Please note that B is the ending point and A is the starting point. ALWAYS subtract the “tail” coordinates from the “tip” coordinates!

Optional Way for 3D forces when a FORCE VECTOR DIRECTED ALONG A LINE (Section 2.8) using position and unit vectors: a) Find the position vector, r AB, along two points on that line. b) Find the unit vector describing the line’s direction, u AB = (r AB /r AB ). c) Multiply the unit vector by the magnitude of the force, F = F u AB. If a force is directed along a line, then we can represent the force vector in Cartesian coordinates by using a unit vector and the force’s magnitude. So we need to:

EXAMPLE Plan: 1. Find the position vector r AC and the unit vector u AC. 2. Obtain the force vector as F AC = 420 N u AC. Given: The 420 N force along the cable AC. Find: The force F AC in the Cartesian vector form.

EXAMPLE (continued) (We can also find r AC by subtracting the coordinates of A from the coordinates of C.) r AC = ( ) 1/2 = 7 m Now u AC = r AC /r AC and F AC = 420 u AC N = 420 (r AC /r AC ) So F AC = 420{ (2 i + 3 j  6 k) / 7 } N = {120 i j k } N As per the figure, when relating A to C, we will have to go 2 m in the x-direction, 3 m in the y- direction, and -6 m in the z-direction. Hence, r AC = {2 i + 3 j  6 k} m. Do Example!

READING QUIZ 1. A position vector, r PQ, is obtained by A) Coordinates of Q minus coordinates of P B) Coordinates of P minus coordinates of Q C) Coordinates of Q minus coordinates of the origin D) Coordinates of the origin minus coordinates of P 2. A force of magnitude F, directed along a unit vector U, is given by F = ______. A) F (U) B) U / F C) F / U D) F + U E) F – U

GROUP PROBLEM SOLVING Plan: 1) Find the forces along CA and CB in the Cartesian vector form. 2) Add the two forces to get the resultant force, F R. 3) Determine the magnitude and the coordinate angles of F R. Given: Two forces are acting on a pipe as shown in the figure. Find: The magnitude and the coordinate direction angles of the resultant force.

GROUP PROBLEM SOLVING (continued) F CB = 81 lb (r CB /r CB ) F CB = 81 lb (4 i – 7 j – 4 k)/9 F CB = {36 i – 63 j – 36 k} lb F CA = 100 lb (r CA /r CA ) F CA = 100 lb (–3 sin 40° i + 3 cos 40° j – 4 k)/5 F CA = (– i j – 80 k) lb

GROUP PROBLEM SOLVING (continued) F R = F CA + F CB = {– 2.57 i – j – 116 k} lb F R = ( ) = lb = 117 lb  = cos -1 (–2.57/117.3) = 91.3°  = cos -1 (–17.04/117.3) = 98.4°  = cos -1 (–116/117.3) = 172°

ATTENTION QUIZ 1. Two points in 3 – D space have coordinates of P (1, 2, 3) and Q (4, 5, 6) meters. The position vector r QP is given by A) {3 i + 3 j + 3 k} m B) {– 3 i – 3 j – 3 k} m C) {5 i + 7 j + 9 k} m D) {– 3 i + 3 j + 3 k} m E) {4 i + 5 j + 6 k} m 2. Force vector, F, directed along a line PQ is given by A) (F/ F) r PQ B) r PQ /r PQ C) F(r PQ /r PQ )D) F(r PQ /r PQ )

Then F = F u