Date post: | 13-Jan-2016 |
Category: |
Documents |
Upload: | moris-wood |
View: | 226 times |
Download: | 1 times |
CE 201 - Statics
Lecture 5
Contents
Position Vectors Force Vector Directed along a Line
POSITION VECTORS
If a force is acting between two points, then the use of position vector will help in representing the force in the form of Cartesian vector.
As discussed earlier, the right-handed coordinate system will be used throughout the course
x
y
z
A
B
Coordinates of a Point (x, y, and z)
A coordinates are (2, 2, 6)
B coordinates are (4, -4, -10)
x
y
z
B
A
Position Vectors
Position vector is a fixed vector that locates a point relative to another point.
If the position vector ( r ) is extending from the point of origin ( O ) to point ( A ) with x, y, and z coordinates, then it can be expressed in Cartesian vector form as:
r = x i + y j + z k
x
y
z
r
A (x,y,z)
x i
y j
z k
O
If a position vector extends from point B (xB, yB, zB) to point A(xA, yA, zA), then it can be expressed as rBA.
By head – to – tail vector addition, we have:
rB + rBA = rA
then,
rBA = rA - rBx
y
z
rBAA (xA,yA,zA)
rArB
B(xB, yB, zB)
Substituting the values of rA and rB, we obtain
rBA = (xA i + yA j + zA k) – (xB i + yB j + zB k)
= (xA – xB) i + (yA – yB) j + (zA – zB) k
So, position vector can be formed by subtracting the coordinates of the tail from those of the head.
FORCE VECTOR DIRECTED ALONG A LINE
If force F is directed along the AB, then it can be expressed as a Cartesian vector, knowing that it has the same direction as the position vector ( r ) which is directed from A to B.
The direction can be expressed using the unit vector (u)u = (r / r)where, ( r ) is the vector and ( r ) is its magnitude.We know that:F = fu = f ( r / r)
x
y
z
rB
AF
u
Procedure for Analysis
When F is directed along the line AB (from A to B), then F can be expressed as a Cartesian vector in the following way:
Determine the position vector ( r ) directed from A to BDetermine the unit vector ( u = r / r ) which has the
direction of both r and FDetermine F by combining its magnitude ( f ) and direction
( u )
F = f u
Examples
Examples 2.12 – 2.15 Problem 86 Problem 98