phy 113 c general physics i 11 am-12:15 pm tr olin 101 plan for lecture 3: chapter 3 – vectors
DESCRIPTION
PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors Abstract notion of vectors Displacement vectors Other examples. iclicker question Have you attended a tutoring session yet? Have you attended a lab session yet? - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/1.jpg)
PHY 113 C Fall 2013 -- Lecture 3 19/3/2013
PHY 113 C General Physics I11 AM-12:15 PM TR Olin 101
Plan for Lecture 3:
Chapter 3 – Vectors
1. Abstract notion of vectors
2. Displacement vectors
3. Other examples
![Page 2: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/2.jpg)
PHY 113 C Fall 2013 -- Lecture 3 29/3/2013
![Page 3: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/3.jpg)
PHY 113 C Fall 2013 -- Lecture 3 39/3/2013
![Page 4: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/4.jpg)
PHY 113 C Fall 2013 -- Lecture 3 49/3/2013
iclicker question
A. Have you attended a tutoring session yet?B. Have you attended a lab session yet?C. Have you attended both tutoring and lab sessions?
![Page 5: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/5.jpg)
PHY 113 C Fall 2013 -- Lecture 3 59/3/2013
Mathematics Review -- Appendix B Serwey & Jewett
iclicker question
A. Have you used this appendix?B. Have you used the appendix, and find it helpful?C. Have you used the appendix, but find it unhelpful?
iclicker question
Have you changed your webassign password yet?A. yesB. no
![Page 6: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/6.jpg)
PHY 113 C Fall 2013 -- Lecture 3 69/3/2013
Question from Webassign #2
1 2 3 4 5 6 7 8 9 10
8
46
2
-8-6 -4-20
![Page 7: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/7.jpg)
PHY 113 C Fall 2013 -- Lecture 3 79/3/2013
Mathematics Review -- Appendix B Serwey & Jewett
aacbbx
cbxax
24
0
:equation Quadratic
2
2
)cos()sin(
:calculus alDifferenti
1
ttdtd
eedtd
antatdtd
tt
nn
)cos(1)sin(
11
:calculus Integral1
tdtt
edte
natdtat
tt
nn
a
b
c
q
bacacb
q
q
q
tan
sin
cos
:ryTrigonomet
![Page 8: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/8.jpg)
PHY 113 C Fall 2013 -- Lecture 3 89/3/2013
Definition of a vector
1. A vector is defined by its length and direction.
2. Addition, subtraction, and two forms of multiplication can be defined
3. In practice, we can use trigonometry or component analysis for quantitative work involving vectors.
4. Abstract vectors are useful in physics and mathematics.
![Page 9: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/9.jpg)
PHY 113 C Fall 2013 -- Lecture 3 99/3/2013
Vector addition:
ab
a – b
Vector subtraction:a
-b
a + b
![Page 10: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/10.jpg)
PHY 113 C Fall 2013 -- Lecture 3 109/3/2013
Some useful trigonometric relations
(see Appendix B of your text)
g
c
b
a
Law of cosines:
a2 = b2 + c2 - 2bc cos
b2 = c2 + a2 - 2ca cos
c2 = a2 + b2 - 2ab cos g
Law of sines:
sin c
sin b
sin a
g
![Page 11: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/11.jpg)
PHY 113 C Fall 2013 -- Lecture 3 119/3/2013
Some useful trigonometric relations -- continued
(from Appendix B of your text)
g
c
b
a
Law of cosines:
a2 = b2 + c2 - 2bc cos
b2 = c2 + a2 - 2ca cos
c2 = a2 + b2 - 2ab cos g
![Page 12: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/12.jpg)
PHY 113 C Fall 2013 -- Lecture 3 129/3/2013
Some useful trigonometric relations -- continued
Example:
20o
c=?
15
10
Law of cosines:
a2 = b2 + c2 - 2bc cos
b2 = c2 + a2 - 2ca cos
c2 = a2 + b2 - 2ab cos g
654.60922.43
0922.4320cos151021510 222
c
c o
![Page 13: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/13.jpg)
PHY 113 C Fall 2013 -- Lecture 3 139/3/2013
Possible realization of previous example:
20o
c=?
15m
10m
Start
South
East
A pirate map gives directions to buried treasure following the indicated arrows. A wily physics students decides to take the easterly direct route after computing the distance c.
treasure
![Page 14: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/14.jpg)
PHY 113 C Fall 2013 -- Lecture 3 149/3/2013
Quantitative representation of a vector
Cartesian coordinates
i
j
A
Ax
Ay
jiA ˆˆyx AA
![Page 15: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/15.jpg)
PHY 113 C Fall 2013 -- Lecture 3 159/3/2013
Quantitative representation of a vector
reference direction
AA
q
Note: q can be specified in degrees or radians; make sure that your calculator knows your intentions!
Polar coordinates
![Page 16: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/16.jpg)
PHY 113 C Fall 2013 -- Lecture 3 169/3/2013
Quantitative representation of a vector
AA
q
Polar & cartesian coordinates
iqcosAAx
qsinAAy
x
y
AA
AA
qqq tan
cossin:note Also
![Page 17: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/17.jpg)
PHY 113 C Fall 2013 -- Lecture 3 179/3/2013
Vector components:
ax
ay
22yx aa a
jiyxa ˆˆˆˆ yxyx aaaa
yxba
yxbyxaˆˆ
ˆˆ and ˆˆFor
yyxx
yxyx
baba
bbaa
![Page 18: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/18.jpg)
PHY 113 C Fall 2013 -- Lecture 3 189/3/2013
ax
ay
q
y
a = 1 m
Suppose you are given the length of the vector a as shown. How can you find the components?
A. ax=a cos q, ay=a sin qB. ax=a sin y, ay=a cos yC. Neither of theseD. Both of these
iclicker question
![Page 19: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/19.jpg)
PHY 113 C Fall 2013 -- Lecture 3 199/3/2013
Vector components; using trigonometry
An orthogonal coordinate system
A
kjizyxA ˆˆˆˆˆˆ zyxzyx AAAAAA
![Page 20: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/20.jpg)
PHY 113 C Fall 2013 -- Lecture 3 209/3/2013
Vector components:
ax
ay
yxba
yxbyxaˆˆ
ˆˆ and ˆˆFor
yyxx
yxyx
baba
bbaa
yxa ˆˆ yx aa
by
bx
yxb ˆˆ yx bb ba
![Page 21: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/21.jpg)
PHY 113 C Fall 2013 -- Lecture 3 219/3/2013
Examples
Vectors ScalarsPosition r Time tVelocity v Mass mAcceleration a Volume VForce F Density m/VMomentum p Vector components
![Page 22: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/22.jpg)
PHY 113 C Fall 2013 -- Lecture 3 229/3/2013
Vector componentszyxR ˆˆˆ 1111 zyx
zyxR ˆˆˆ 2222 zyx
zyxRR ˆ(ˆ)(ˆ)( )21212121 zzyyxx
Vector multiplication “Dot” product 1ˆˆ;cos AB xxBA qAB
“Cross” product zyxBA ˆˆˆ;sin|| AB qAB
![Page 23: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/23.jpg)
PHY 113 C Fall 2013 -- Lecture 3 239/3/2013
Example of vector addition:
a
ba + b
![Page 24: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/24.jpg)
PHY 113 C Fall 2013 -- Lecture 3 249/3/2013
a
ba + b
gcos2
:Dallas and Chicagobetween Distance22 bababa
o
oooo
74
18021590
g
g
mi 788bag
![Page 25: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/25.jpg)
PHY 113 C Fall 2013 -- Lecture 3 259/3/2013
Webassign version:
f
Note: In this case the angle f is actually measured as north of east.
![Page 26: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/26.jpg)
PHY 113 C Fall 2013 -- Lecture 3 269/3/2013
Another example:
ji
BARjiB
jiA
ˆ9.16ˆ7.37
ˆ6.34ˆ0.20
ˆ7.17ˆ7.17
:units kmin ectorsPosition v
![Page 27: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/27.jpg)
PHY 113 C Fall 2013 -- Lecture 3 279/3/2013
iclicker question
A. Because physics professors like to confuse studentsB. Because physics professors like to use beautiful
mathematical concepts if at all possibleC. Because all physical phenomena can be described by
vectors.D. Because there are some examples in physics that
can be described by vectors
Why are we spending 75 minutes discussing vectors
![Page 28: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/28.jpg)
PHY 113 C Fall 2013 -- Lecture 3 289/3/2013
Example: Vector addition of velocities
Vb
Vw
Vtotal
![Page 29: PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors](https://reader035.vdocuments.net/reader035/viewer/2022062812/56816303550346895dd37b5e/html5/thumbnails/29.jpg)
PHY 113 C Fall 2013 -- Lecture 3 299/3/2013
Example: Displacement in two dimensions
(0,0)
(8,5)
43.9)5()8( 22 d