Transcript Unit 4 PowerPoint Slides

EGR 1101 Unit 4
Two-Dimensional Vectors in
Engineering
(Chapter 4 of Rattan/Klingbeil text)
Scalars versus Vectors

A scalar is a quantity that has magnitude
only. Examples:



Mass
Temperature
A vector is a quantity that has magnitude
and direction, and that obeys the triangle
law of addition. Examples:


Velocity
Force
Component Form & Polar Form



Vectors are commonly written in two
different forms.
In component form, a two-dimensional (2D) vector is expressed as the sum of an xcomponent and a y-component.
In polar form, a 2-D vector is expressed
as having a certain magnitude in a certain
direction.
Other Names for Component Form

Component form is sometimes called
rectangular form or Cartesian form.
Component Form



v
Suppose a vector has x-component
v x and y-component v y .
Then we can write the vector in
component form as

v  v x iˆ  v y ˆj
where iˆ is the unit vector in the
positive-x direction and ˆj is the unit
vector in the positive-y direction.
Polar Form



v
Suppose a vector has magnitude v
and angle .
Then we can write the vector in polar
form as

v
= v .
Converting Between Component &
Polar Forms



Many problems involve converting
from one form to the other. This is
easy if you remember your basic trig.
From polar form to component form:
vx = v cos()
vy = v sin()
From component form to polar form:
v = vx2 + vy2
 = tan-1(vy / vx)
This Week’s Examples
1.
2.
3.
4.
5.
Force on a vacuum cleaner
Impedance of inductor & resistor in series
Position of a ship
Forces in static equilibrium: Hanging weight
Forces in static equilibrium: TV on a ramp
A New Electrical Component: The
Inductor

Recall that a resistor has a resistance
(R), which is measured in ohms (Ω).
In diagrams, the symbol for a resistor
is

An inductor has an inductive
reactance (XL), also measured in
ohms. In diagrams, the symbol for an
inductor is
Impedance

Resistance (R) and inductive reactance
(XL) are special cases of a quantity called
impedance (Z), also measured in ohms.
Impedance (Z)
Resistance (R)
Reactance (X)
Inductive Reactance (XL)
Capacitive Reactance (XC)
Review: Total Resistance of
Resistors in Series

Recall that if two resistors are connected in
series (end-to-end), total resistance is the
sum of the two resistances:

Things aren’t quite this simple when a
resistor and an inductor are connected in
series…
Total Impedance


To find total impedance of a resistance
and an inductive reactance in series,
add them as vectors, not as scalars.
When treated as vectors, resistance
always has an angle of 0, and
inductive reactance always has an
angle of 90.
Adding Vectors


Many problems involve the addition of
two or more vectors.
Vectors can be added graphically or
algebraically.
Adding Vectors Graphically


To add two vectors P1 and P2

graphically:



Draw
 the two vectors with P2 ‘s tail placed
at P1 ‘s tip.
Then draw
a third vector that extends

from P1 ‘s tail to P2 ‘s tip. This third
 vector

is the vector sum, which we call P1  P2 .
Adding Vectors Algebraically


To add P1 and P2 algebraically:


Write
the vectors in component
form:


P2  Px 2 iˆ  Py 2 ˆj
and
P1  Px1iˆ  Py 1 ˆj

Add their x-components to get the xcomponent of the sum, and add their ycomponents to get the y-component of
the sum:
 
P  P  ( P  P ) iˆ  ( P  P ) ˆj
1
2
x1
x2
y1
y2
Matrices, Vectors, and Scalars in
MATLAB
•
•
•
•
In MATLAB, all quantities are treated
as arrays of numbers.
A matrix has several rows and several
columns.
A vector has one row and several
columns, or one column and several
rows.
A scalar has just one row and one
column.
Matrices in MATLAB
•
Example of a 2x3 matrix (one with two
rows and three columns):
 7 .6 1 .2 1 .5 


 4 .9 3 .3 2 .5 
•
To enter this in MATLAB, type:
A = [7.6, 1.2, 1.5;
4.9, 3.3, 2.5]
Vectors in MATLAB
•
A row vector is an array with just one
row.
• Example: v1=[7.6, 1.2, 1.5]
•
A column vector is an array with just
one column.
• Example: v2=[7.6; 1.2; 1.5]
Scalars in MATLAB
•
A scalar is treated as an array with just
one row and one column.
• Example: s=23.5
• Could also write this as s=[23.5]
Typical Use of Vectors in MATLAB
•
•
Suppose we want to plot some
temperature-versus-time data.
Time (hour AM)
Temperature (F)
5
59
6
58
7
60
8
63
9
70
10
78
Define vectors for time and temp, and
then use plot command.
Matrix Multiplication versus Elementby-Element Multiplication
•
•
•
In MATLAB, the * operator performs
matrix multiplication.
For A*B to be defined, the number of
columns in A must equal the number of
rows in B.
The .* operator performs element-byelement multiplication.
MATLAB Multiplication Example
•
Define two 1x3 vectors:
v1 = [1, 2, 3]
v2 = [4, 5, 6]
• v1*v2 tries to perform matrix
multiplication. An error results, since
the number of columns in v1 is not
equal to the number of rows in v2.
• v1.*v2 performs element-by-element
multiplication, giving [4, 10, 18].
Other Operations in MATLAB
•
Similar comments apply to division and
exponentiation:
•
•
•
•
•
/ performs matrix division
./ performs element-by-element division
^ performs matrix exponentiation
.^ performs element-by-element expntn.
Addition and subtraction are always
performed element-by-element. So we
don’t need special .+ and .– operators.
Just use the + and – operators.
Static Equilibrium

The field called “statics” deals with
objects in static equilibrium. For such
objects, the external forces acting on
the object add to zero:
F 0


Therefore (for 2 dimensions):
F
F
x
0
y
0
Common Types of Force
The following types of forces often arise
in statics problems:





Weight
Tension
Frictional force
Normal force
Weight and Mass



Near the earth’s surface, an object’s
weight (W ) is a vector pointing straight
down.
Its magnitude (W) is equal to the object’s
mass (m) times the acceleration due to
gravity (g):
W = mg
In metric units, g  9.81 m/s2.
Free-Body Diagram


For statics problem, your first step should
be to draw a free-body diagram.
A free-body diagram shows the object of
interest and clearly indicates all of the
forces acting on that object.