Physics: Problem Solving Chapter 4 Vectors. Physics: Problem Solving Chapter 4 Vectors.
Vectors (Physics)
-
Upload
jonathan-mullen -
Category
Documents
-
view
222 -
download
0
Transcript of Vectors (Physics)
-
8/7/2019 Vectors (Physics)
1/66
VectorsVectorsVectorsVectors
-
8/7/2019 Vectors (Physics)
2/66
Sign Conventions
Its importantto haveaconsistentsetofalgebraicsignsto indicate direction forvectors
Why do we need them?A (+) ora(-) willtell usthe directionthe vector is going.
Sign conventions:(+): vector pointsright orup.
(-): vector pointsleft ordown.
-
8/7/2019 Vectors (Physics)
3/66
Sign Conventions
Sign conventions:(+): vector pointsright orup.
(-): vector pointsleft ordown.
-
8/7/2019 Vectors (Physics)
4/66
Topic:Vector Addition
In red isa displacement
vector.
Displacement distance& direction from theorigin/reference.
Vectorsare usually represented by alinewith an arrowhead atoneend whichindicatesthe direction.
The displacementabove is 8 blocks
North (magnitudeand direction)
-
8/7/2019 Vectors (Physics)
5/66
Topic:Vector Addition
Ifthe person now walks 4 blocks South,what isthe new displacement?
-
8/7/2019 Vectors (Physics)
6/66
Topic:Vector Addition
Ifthe person now walks 4 blocks South,what isthe new displacement?
Tofigurethisout, wemust use vector
addition:+ = ?
-
8/7/2019 Vectors (Physics)
7/66
Topic:Vector Addition
Ifthe person now walks 4 blocks South,what isthe new displacement?
Tofigurethisout, wemust use vector
addition:+ = ?
-
8/7/2019 Vectors (Physics)
8/66
Topic:Vector Addition
Ifthe person now walks 4 blocks South,what isthe new displacement?
Tofigurethisout, wemust use vector
addition:+ =
This method iscalled the tip totailmethod. The new displacement is4 blocksNorth.
-
8/7/2019 Vectors (Physics)
9/66
Topic:Vector Addition
Note how the displacement vector isdrawn:Itstartsatthetailofthefirst vector and endsatthetip ofthelast
vector.
-
8/7/2019 Vectors (Physics)
10/66
Topic:Vector AdditionLets gobacktoouroriginalexample: personstanding 8 blocks North. What ifshe walks 5blocks East. What isthe new displacement?What istheresultant?
-
8/7/2019 Vectors (Physics)
11/66
Topic:Vector AdditionLets gobacktoouroriginalexample: personstanding 8 blocks North. What ifshe walks 5blocks East. What isthe new displacement?What istheresultant?
-
8/7/2019 Vectors (Physics)
12/66
-
8/7/2019 Vectors (Physics)
13/66
Topic:Vector AdditionLets gobacktoouroriginalexample: personstanding 8 blocks North. What ifshe walks 5blocks East. What isthe new displacement?What istheresultant?
The direction is indicatedby the purple vector.
How aboutthe
magnitude?
-
8/7/2019 Vectors (Physics)
14/66
Topic:Vector Addition
A2 + B2 = C2
-
8/7/2019 Vectors (Physics)
15/66
Topic:Vector AdditionTo summarize:
1. If we haveparallel vectors, then wecan addthem algebraically, using thetip totail method.
2. a. If we havevectors thatareatrightangles,
we draw the vectorstip totail.b. We draw theresultant to getthe direction.
c. We usethePythagorean theorem to getthe
magnitudeoftheresultant.These methods workbecause wecan moveavectorsaround aslong asthelength and directionstay thesame!
-
8/7/2019 Vectors (Physics)
16/66
Subtracting VectorsSubtracting VectorsSubtracting VectorsSubtracting Vectors
A B = ?
-
8/7/2019 Vectors (Physics)
17/66
Example Problem
What is 30 m/s north minus 15 m/seast?
-
8/7/2019 Vectors (Physics)
18/66
Example Problem
What is 30 m/s north minus 15 m/seast?
Question:
How many vectorsarethere in this problem?
-
8/7/2019 Vectors (Physics)
19/66
Example Problem
What is 30 m/s north minus 15 m/seast?
Question:
How many vectorsarethere in this problem?
-
8/7/2019 Vectors (Physics)
20/66
Words Math
What is30 m/s north minus15 m/seast?
north30As
m!X east15Bs
m!X
-
8/7/2019 Vectors (Physics)
21/66
What is30 m/s north minus15 m/seast?
north30As
m!X
Mathematically
east15Bs
m!X
A B = ?
Words Math
-
8/7/2019 Vectors (Physics)
22/66
What is30 m/s north minus15 m/seast?
north30As
m!X
Mathematically
east15Bs
m!X
A B = ?
Words Math
The Problem: Weonly know how toadd vectors.
-
8/7/2019 Vectors (Physics)
23/66
north30 sm
!
Xeast15B s
m
!
X
TheTrick:
Make it AdditionA B = A +( B)
-
8/7/2019 Vectors (Physics)
24/66
north30 sm
!
Xeast15B s
m
!
X
TheTrick:
Make it AdditionA B = A +( B)
Guess: what is B?
-
8/7/2019 Vectors (Physics)
25/66
Time Out!
C
This is C
-
8/7/2019 Vectors (Physics)
26/66
Time Out!
C
Point in the direction of C!
-
8/7/2019 Vectors (Physics)
27/66
Time Out!
C
Point in the direction of C!
-
8/7/2019 Vectors (Physics)
28/66
Time Out!
C
Point in the direction of C!
- C
-
8/7/2019 Vectors (Physics)
29/66
Continue
-
8/7/2019 Vectors (Physics)
30/66
north30 sm
!
X
east15B sm
!
X
TheTrick:
Make it AdditionA B = A +( B)
- B isB in theopposite direction
-
8/7/2019 Vectors (Physics)
31/66
north30 sm
!
X
east15B sm
!
X
TheTrick:
Make it AdditionA B = A +( B)
A B = A +( B)
- B isB in theopposite direction
-
8/7/2019 Vectors (Physics)
32/66
-
8/7/2019 Vectors (Physics)
33/66
Graphical Addition of
Vectors
1. Makeascale drawing.2. Movethe vectorstip-
to-tail.3. Draw theresultant
vector (freetailtofreetip).
4. Measurethelength
oftheresultant witharuler.5. Measurethedirection oftheresultant with a
protractor.
A +( B)
1)
Which aretip-to-tail?
3)
2)
4)
-
8/7/2019 Vectors (Physics)
34/66
Graphical Addition of
Vectors
1. Makeascale drawing.2. Movethe vectorstip-
to-tail.3. Draw theresultant
vector (freetailtofreetip).
4. Measurethelength
oftheresultant witharuler.5. Measurethedirection oftheresultant with a
protractor.
A +( B)
1)
Which aretip-to-tail?
3)
2)
4)
-
8/7/2019 Vectors (Physics)
35/66
Graphical Addition of
Vectors
1. Makeascale drawing.2. Movethe vectorstip-
to-tail.3. Draw theresultant
vector (freetailtofreetip).
4. Measurethelength
oftheresultant witharuler.5. Measurethedirection oftheresultant with a
protractor.
A +( B)
1)
Draw theresultantfor 1):
from point ___to point ___?
2)
c
ba
-
8/7/2019 Vectors (Physics)
36/66
Graphical Addition of
Vectors
1. Makeascale drawing.2. Movethe vectorstip-
to-tail.3. Draw theresultant
vector (freetailtofreetip).
4. Measurethelength
oftheresultant witharuler.5. Measurethedirection oftheresultant with a
protractor.
A +( B)
1)
Draw theresultantfor 1):
from point ___to point ___?
2)
c
ba
ac
-
8/7/2019 Vectors (Physics)
37/66
Graphical Addition of
Vectors
1. Makeascale drawing.2. Movethe vectorstip-
to-tail.3. Draw theresultant
vector (freetailtofreetip).
4. Measurethelength
oftheresultant witharuler.5. Measurethedirection oftheresultant with a
protractor.
A +( B)
1)
Draw theresultantfor 2):
from point ___to point ___?
2)
f
d
e
-
8/7/2019 Vectors (Physics)
38/66
Graphical Addition of
Vectors
1. Makeascale drawing.2. Movethe vectorstip-
to-tail.3. Draw theresultant
vector (freetailtofreetip).
4. Measurethelength
oftheresultant witharuler.5. Measurethedirection oftheresultant with a
protractor.
A +( B)
1)
Draw theresultantfor 2):
from point ___to point ___?
2)
f
d
e
f d
-
8/7/2019 Vectors (Physics)
39/66
-
8/7/2019 Vectors (Physics)
40/66
-
8/7/2019 Vectors (Physics)
41/66
Graphical Addition of
Vectors
1. Makeascale drawing.2. Movethe vectorstip-
to-tail.3. Draw theresultant
vector (freetailtofreetip).
4. Measurethelength
oftheresultant witharuler.5. Measurethedirection oftheresultant with a
protractor.
A +( B)
63
-
8/7/2019 Vectors (Physics)
42/66
Graphical Addition of
Vectors
1. Makeascale drawing.2. Movethe vectorstip-
to-tail.3. Draw theresultant
vector (freetailtofreetip).
4. Measurethelength
oftheresultant witharuler.5. Measurethedirection oftheresultant with a
protractor.
A +( B)
33.5 m/s @ 63 north ofwest
1 cm = 5 m/s
63
Yourfinalanswer!
-
8/7/2019 Vectors (Physics)
43/66
Motion Notes
(Continued):10.Subtracting Vector a.The negative sign makes one vectorpoint in
the opposite direction.
i.Ex: BA XX X has the same magnitudeasX
and the opposite direction
b.Deal with the directions, then add the
vectors.i. )B(ABA
XXXX!
-
8/7/2019 Vectors (Physics)
44/66
Topic:Vector Addition IIAnother way of determining theresultof
adding 2 vectors (i.e. resultant), istheparallelogram method.
This method will workeven ifthe vectors
are notatrightanglesRightangle Otherangle
-
8/7/2019 Vectors (Physics)
45/66
Topic:Vector Addition IIAnother way of determining theresultof
adding 2 vectors (i.e. resultant), istheparallelogram method.
This method will workeven ifthe vectors
are notatrightanglesRightangle Otherangle
-
8/7/2019 Vectors (Physics)
46/66
Topic:Vector Addition IINotice how I have drawn these vectorsto do
theaddition, tailtotail Toadd the vectors, draw a parallelogram (ashape in which oppositesidesare parallel) by
drawing parallel vectorsshown as dashed linesbelowRightangle Otherangle
Thetailsare drawntogether
Thetailsare drawntogether
-
8/7/2019 Vectors (Physics)
47/66
Topic:Vector Addition II Theresultant isthe diagonalofthe
parallelogram, shown in purplebelowForthedirection oftheresultant, measurewith a protractor. Forthemagnitude ofthe
resultant, measure with aruler.Rightangle Otherangle
Thetailsare drawntogether
Thetailsare drawntogether
-
8/7/2019 Vectors (Physics)
48/66
Topic: ComponentsofVectors Sometimes we need tochangeasingle
vector intotwoequivalent vectorsatrightanglestoeach other. Theequivalent vectorsarecalled components.
Any vector
ca
nbe
resol
ved
in
tocomponents. This process iscalled vector
resolution. Every vector will haveacomponentalong the
x-axisand the y-axis.
-
8/7/2019 Vectors (Physics)
49/66
Topic: ComponentsofVectors Suppose we havea vectorV in acertain
directionDraw horizontalandverticallinesfrom theta
ilof
th
ev
ectorto
makean xand y axis.
V
y
x
-
8/7/2019 Vectors (Physics)
50/66
Topic: ComponentsofVectors Imagine you areshining aflashlight downward
on the vectorfrom atop ofthe y-axisYou can imaginethatthiscreatesashadow on thex-ax
is
Thelength oftheshadow onthex-axis isthex
component (horizontalcomponent) ofthe vector.
V
Vx
-
8/7/2019 Vectors (Physics)
51/66
Topic: ComponentsofVectors Now shineaflashlighttowardsthe vectorfrom thex-axis
Similarly you can imaginethatthiscreatesashadow on the y-axis
V
Thelength oftheshadow on they-axis isthey-component
(verticalcomponent)ofthe vector
Vy
-
8/7/2019 Vectors (Physics)
52/66
Topic: ComponentsofVectorsBelow arethex & y componentsofthe
vectorV,
Vx = horizontalcomponentVy = verticalcomponent
V
Vy
Vx
-
8/7/2019 Vectors (Physics)
53/66
Topic: ComponentsofVectorsTry these:
A
B
-
8/7/2019 Vectors (Physics)
54/66
Topic: ComponentsofVectorsTry these:
A
Ax
Ay
B
Bx
By
-
8/7/2019 Vectors (Physics)
55/66
Polar Coordinate System
(Direction and Magnitude)
-
8/7/2019 Vectors (Physics)
56/66
-
8/7/2019 Vectors (Physics)
57/66
Polar Math
(Cartesian)2 2
1
os
sin
tan
r x y
x r
y r
y
x
UU
U
!
!!
! x
yr
-
8/7/2019 Vectors (Physics)
58/66
Vectoraddition TwoWays:
1. Graphically: Draw vectorsto
scale, Tip toTail, and theresultant isthestraightlinefromstarttofinish
2. Mathematically: Employ vectormath analysistosolvefortheresultant vector
-
8/7/2019 Vectors (Physics)
59/66
Graphically 2-D Right A = 5.0 m @ 0 B = 5.0 m @ 90 SolveA+B
R
Start
R=7.1 m @
45
-
8/7/2019 Vectors (Physics)
60/66
Remember You can add vectors in any orderand yield
thesameresultant.
-
8/7/2019 Vectors (Physics)
61/66
ComponentsofVectors A=Ax +Ay Ax=A cos
Ay = A sin
Aslong as you
draw thexcomponentfirst
A
Ax
Ay
-
8/7/2019 Vectors (Physics)
62/66
-
8/7/2019 Vectors (Physics)
63/66
-
8/7/2019 Vectors (Physics)
64/66
Table Method Equation Add all X componentstogether FinalRx Add all Y componentstogether FinalRy
-
8/7/2019 Vectors (Physics)
65/66
Subtracting Vectors Simply add orsubtract 180 (keep
between 0 and 360) tothe
direction ofthe vectorbeingsubtracted (flip) You just ADD the OPPOSITE vector
(th
ereis
nosu
btraction in v
ectormath)
-
8/7/2019 Vectors (Physics)
66/66
Subtracting Vectors