Vector Addition

1
Vector Addition Displacement
Do not worry about your difficulties in
mathematics, I assure you that mine are
greater.A. Einstein
2
Adding Vectors by Components
Ill illustrate by using this sample problem.
There are standard problem-solving rules I will
recommend you follow throughout this course, so
lets start using them now (they will be
introduced officially later in a slightly
different format).
i.e., require
3
The zeroeth step (i.e., not part of the
official regimen, but you should do it anyway)
is to understand, think about, or make sense out
of the problem.
In this example, the words on the previous slide
dont make much sense to me, so understanding
requires drawing a diagram.
Drawing a diagram is also the official first
step.
4
Ill work this example on the blackboard in
class. My lecture notes contain a detailed
step-by-step solution that I recommend you study
outside of class.
Step 1 draw a complete, fully-labeled diagram
for the problem.
Step 2 choose, draw, and label your axes, with
arrow(s) indicating the positive direction(s).
Step 3 lightly draw in components of all
vectors that are not parallel to your chosen axes.
Step 4 select an OSE (official starting
equation).
In class, I will jump to a digression on
components.
5
Step 1 draw a complete, fully-labeled diagram
for the problem.
A 66.0 ? 28? B 40.0 ? 56?
Vectors should be drawn using darker lines than
the rest of the sketch.
How do you know the diagram is complete and
fully-labeled? Anything you use in your
calculations must appear in or be justified by
the diagram. You will often have to add items to
your diagram as you proceed.
6
Step 2 choose, draw, and label your axes, with
arrow(s) indicating the positive direction(s).
Already done the axes were an integral part of
the problem statement.
In most problems, you will get to choose your
axes.
7
Step 3 lightly draw in components of all
vectors that are not parallel to your chosen axes.
When doing vector problems in chapter 3, label
the components appropriately (Ax, By etc.) In
other problems, dontit just adds too much
confusion to the diagram).
For computer-generated figures, I will often used
dashed lines for components. Dont waste your
time on exams making dashed lines. Its OK to
offset components slightly from axes or other
parts of the diagram.
8
The resultant is , so lets draw that in.
I decided to use the parallelogram method to
indicate the resultant.
If you dont show the parallelogram, may look
wrong to your eyes.
9
Remember, DO NOT do this
No! No! No!
No!
10
Step 4 select an OSE (official starting
equation).
OSE if then Cx Ax Bx, Cy Ay
By, Cz Az Bz.
All further steps must follow logically from this
OSE and the diagram.
Do not substitute any numbers until the very end!
11
Digression vectors and components.
A
12
On page 49 of your text, Giancoli clearly says
that com-ponents of a vector are themselves
vectors.
Physicists often use the words x-component to
mean the vector which is the x-component of the
vector in question or to mean the magnitude of
the x-component of the vector in question and
leave it to you to figure it out from the
context.
Physicists often use the words x-component to
mean the vector which is the x-component of the
vector in question or to mean the magnitude of
the x-component of the vector in question and
leave it to you to figure it out from the
context. Bad!
Physicists often use the words x-component to
mean the vector which is the x-component of the
vector in question or to mean the magnitude of
the x-component of the vector in question and
leave it to you to figure it out from the
context. Bad! Physicists!
In Physics 23, a component is a scalar. The
magnitude of the scalar tells how big the
component is. The sign tells which direction the
component points.
13
If you happen to be working on homework in the
Physics Learning Center
and Dr. Bieniek (professor in charge of Physics
23) hears you say that a component is a vector
he will most likely start talking in a loud
voice about how stupid your instructor is for not
properly teaching you about components.
We wouldnt want that, now, would we?
14
In this course, the symbol Ax is understood to be
a scalar which contains information about both
the magnitude and direction along the x-axis of
the x-component of the vector
Anybody confused yet?
If not, youve been sleeping!
15
I want this class to make you hear little voices
in your head.
Ax
The symbol talks to you.
It asks two questions
How big am I?
Which way do I point?
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Remember, Ax says you need to tell how big I am
and which way I point.
Ax also says I am a scalar. Never set me equal
to a vector.
Notice how vectors and scalars are never set
equal in our OSE.
17
Continuing on the blackboard...
Step 5 using your selected OSE, replace generic
quantities with specifics and solve algebraically
Must do each component separately note trig
functions.
Make sure to solve all parts of the problem!
Now jump here.
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Step 5 using your selected OSE, replace generic
quantities with specifics and solve algebraically.
This is going to take a lot of slides, because I
want to keep the diagram handy.
This is still an OSE. Do not change it! Do not
insert a sign here!
x-component Cx Ax Bx
Well have to do each component separately.
19
Trig that youll be using for the rest of this
course
20
x-component Cx Ax Bx
(copied from a previous slide)
Replace generic component quantities with the
information given in the problem.
Cx
Cx (A cos ?)
Cx (A cos ?)
Cx (A cos ?)
Cx (A cos ?)
Cx (A cos ?) (-B
cos ?)
Cx (A cos ?) (-B
cos ?)
Cx (A cos ?) (-B
cos ?)
You decide the signs to put in. Use the finger
trick that Ill show you to prevent embarassing
sign errors.
21
y-component Cy Ay By
Cy (A sin ?) (B
sin ?)
If this were an algebra-only problem, you would
be done
Cx A cos ? - B cos ?
Cy A sin ? B sin ?
22
However, the problem said calculate and
give the resultant in terms of its (a) components
and (b) magnitude and angle with the x axis, so
we are not done.
The algebra is done, so we can plug in numbers
Cx A cos ? - B cos ? Cy A sin ? B sin ?
Cx 66.0 cos 28.0? - 40.0 cos 56? Cy 66.0
sin 28.0? 40.0 sin 56?
keep at least one extra digit during intermediate
calculations and round at the end
how to make mistakes skip steps
Cx 58.27 - 22.37 Cy 30.99 33.16
Cx 35.9 Cy 64.1
dimensionless, because the problem didnt give
any units for the original vectors
23
Cx 35.9 Cy 64.1
Before I finish, lets think a bit. Is the above
answer reasonable?
24
Cx 35.9 Cy 64.1
Cy
Cx
My figure suggests that Cy is bigger than Cx, and
both point in the positive direction. The math
agrees.
25
Calculate the magnitude and direction of the
resultant C (Cx2 Cy2)½ C (35.902
64.152)½ C (35.902 64.152)½ C 73.5
This is listed as an OSE, but I will not require
that you designate algebra and trig facts as
OSEs on your exams unless I make it clear that
you need to do so.
rememberput boxes around your answers
26
Use the finger trick again to determine the
angle that makes with the x-axis.
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Dang, that was a lot of work for just one little
piece of one simple problem!
No, it only seemed that way because I did it in
detail.
Example 3-2 on page 52 of your text is done
equivalently to the above example. Our solutions
typically involve more steps than Giancolis
because leaving out steps is dangerous to your
grade.
28
Unit Vectors
Most physicsts use unit vectors, and you may
encounter them in your visits to the PLC. Unit
vectors are a shortcut for expressing vectors.
The unit vector î (thats a hat above the i) is
a vector of length 1 in the x direction. The
unit vector j (thats a hat above the j) is a
vector of length 1 in the y direction. In
solving our example above, we could have
written OSE . (Ax Bx) î
(Ay By) j 35.9 î 64.1 j The line
above completely specifies the resultant.
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Remember!
Vectors and scalars are different. A vector can
never equal a scalar. Some of you will try to
make a vector equal a scalar, either
on paper
BAD!!
BAD!! EMBARASSING!
on the board in front of the class
or on a test.
BAD! PAINFUL!
One word
DON'T
These stupid animations are intended to help you
remember the point!
30
Reminder
We will use this convention Qx is the
x-component of . Qx has a magnitude Qx and
is pointed along either the x or -x direction.
The sign buried in the symbol Qx tells the
direction. The scalar Qx includes both the
magnitude of Qx and the sign.
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Components, one more time
Qxî
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Handout Dr. Bienieks Homework and Boardwork
Operating Procedures handout.
Handout Dr. Bienieks Litany for Kinematics
Problems handout.
I highly recommend you look at Dr. Bienieks
Physics 23 lecture notes on this subject. They
will be available on the web at
http//www.umr.edu/phys23 under the lectures
and announcements link. They will probably be
under lec-02 (they are not there yet because he
hasnt given that lecture yet). In fact, I highly
recommend you spend some time looking at all of
Dr. Bienieks lecture notes as they become
available.
33
Chapter 2 Describing Motion Kinematics in One
Dimension
Now that we are experts in vectors, we backtrack
to Chapter 2.
Kinematics is the study of how objects move,
without asking about the causes of the motion.
Chapter 2 deals with the kinematics of motion in
one dimension.
34
2.1 Reference Frames and Displacement
Virtually all problems require a set of
coordinate axes which provide a reference frame
for position and motion.
In this class, unless you are specifically told
to use a particular set of axes, you always have
the freedom of choice of coordinate axes.
Like these axes better? Use them.
Like these axes? Use them.
But you really want to keep x and y
perpendicular.
35
To study motion, we need to measure position and
how it changes.
I suppose I had better start at the beginning and
define position
The î and j are just shortcuts for writing both
components in one equation.
Yes, I know thats a vector equation and chapter
2 is supposed to be one dimensional. But this
OSE is better because it works for both one and
two dimensions. (How could we extend it to
three dimensions?)
It is conventional in one dimensional problems to
call the axis along which the object moves the
x-axis. Then the position is just the value of
x (including sign).
Remember, if I dont require otherwise, you can
call it anything you want.
36
Like position, displacement is a vector quantity,
defined as the change of position of an object.
If an object moves in one dimension from initial
position x0 to final position x1, then the
displacement is Dxx1-x0.
We frequently use the symbol ? for changes, so
lets use ?x for displacement
According to the Official Rules, component
versions of vector Official Starting Equations
are also OSE, so you can begin any problem with
this, if you wish
That last equation is contained in the OSE that
appears on the OSE sheet.
37
Wait a minuteon your last slide, you said
displacement is a vector quantity, but I dont
see vectors anywhere in Chapter 2.
Good point. A vector has both a magnitude and a
direction. A displacement in one dimension has a
magnitude and a direction (towards or towards
-).
A displacement in one dimension must be a vector!
However, because the sign of the displacement
specifies the direction, we dont need to carry
the vector baggage along with us in Chapter 2 (as
long as we keep signs straight).
By convention 1-dimensional kinematics is usually
taught as if there were no vectors involved.
38
Motion involves time, so that x x(t). We use
subscripts to denote corresponding positions and
times x0 is the position at time t0, xf is the
position at time tf, etc.
Caution t0 is often chosen to be zero, but it
doesnt have to be! x0 is the position at time
t0, and is not necessarily zero!
Caution t0 is often chosen to be zero, but it
doesnt have to be! x0 is the position at time
t0, and is not necessarily zero! In fact, you
get to set problems up so that t0 and x0 have
whatever values you want them to have.
39
Sample calculation I start at the 100-m finish
line of a track and walk halfway towards the
starting line. What is my displacement? Use the
Litany for Kinematics Problems.
Ill work this example on the blackboard in
class. My lecture notes contain a detailed
step-by-step solution that I recommend you study
outside of class.
1. Draw a basic representative sketch of the
physical situation.
2. Draw and label vectors for the relevant
dynamical quantities that you are given in both
initial and final states.
3. Draw an axis with an arrow at one end
indicating its positive direction. Indicate the
origin (zero position) of the axis.
4. Indicate and label with appropriate subscripts
the initial and final positions along the axis.
(Recommend show units.)
40
5. You MUST begin with an appropriate OSE.
Subsequent steps must follow logically from this
OSE and reference the diagram.
6. Replace the generic component quantities with
information given in the problem.
7. Solve for the desired quantity
algebraicallyalready done above in this simple
example. Dont do steps in your head!
8. Yes, now you can plug in numbers (if you have
them). Put a box around the final answer.
The remaining slides show a detailed solution of
this simple problem. They are intended for your
study outside of class.
41
Sample calculation I start at the 100-m finish
line of a track and walk halfway towards the
starting line. What is my displacement? Use the
Litany for Kinematics Problems.
1. Draw a basic representative sketch of the
physical situation.

On exams or quizzes, use stick figures, blobs,
filled-in circles, squares, etc. to represent
objects. Dont waste your exam time creating art.
42
2. Draw and label vectors for the relevant
dynamical quantities that you are given in both
initial and final states.
43
3. Draw an axis with an arrow at one end
indicating its positive direction. Indicate the
origin (zero position) of the axis.
44
4. Indicate and label with appropriate subscripts
the initial and final positions along the axis.
(Recommend show units.)
45
5. You MUST begin with an appropriate OSE.
Subsequent steps must follow logically from this
OSE and reference the diagram.
I dont see an OSE for displacement!
Sure you do.
46
Well use the component version of the OSE.
47
6. Replace the generic component quantities with
information given in the problem.
48
No! No! No! Dont plug in numbers yet!
49
NO!
50
7. Solve for the desired quantity
algebraicallyalready done above in this simple
example. Dont do steps in your head!
Already done. That was easy!
51
8. Yes, now you can plug in numbers (if you have
them). Put a box around the final answer.
The answer in the box, plus the diagram, are the
solution to the problem.
52
(No Transcript)
53
6. Replace the generic component quantities with
information given in the problem.
No! No! No! Dont plug in numbers yet!
54
7. Solve for the desired quantity
algebraicallyalready done above in this simple
example. Dont do steps in your head!
Already done. That was easy!
55
8. Yes, now you can plug in numbers (if you have
them). Put a box around the final answer.
The sign is correct. It means we went from
larger x to smaller x, using the axis system I
chose.