Physical quantities units and standard
What do we measure? - Physical quantities
Units - a unit is a measure of the quantity that is defined to be exactly 1 (one)
Examples: meter, mile, gram, kilogram
Standard - a reference to which all the other examples of the quantity are compared
Base quantities, their units and standards
The International System of Units
Quantity Unit name Unit symbol Standard
Length Meter
m
Distance traveled by light in
1/299,792,458 second
Time
Second
s
Time required for 9,192,631,770 periods
of radiation emitted by cesium atoms
Mass
Kilogram kg
Platinum-iridium cylinder in
International Bureau of Weights and
Measures at Sevres, near Paris
1
Other systems of units
•CGSE: centimeter, gram, second
1m = 100cm
1kg = 1000g
•British engineering system
This system has force instead of mass as one of its basic quantities, which are
feet, pounds, and seconds.
1 m = 3.281 ft; 1 inch = 2.54 cm
1 kg = 0.06585 slug (Not the same as weight!)
on Earth 1 kg weighs 2.205 lb, on the Moon 1 kg weighs 0.368 lb
2
Multiples of Units
10-24
yocto-
y
10-21
zepto-
z
10-18
atto-
a
10-15
femto-
f
10-12
pico-
p
10-9
nano-
n
10-6
micro-

10-3
milli-
m
10-2
centi-
c
103
kilo-
k
106
mega-
M
109
giga-
G
1012
tera-
T
1015
peta-
P
1018
exa-
E
1021
zetta-
Z
1024
yotta-
Y
Conversion of units:
Multiply by the appropriate representation of 1
to cancel the unwanted units away
Converting between metric units, is easy, as all it
involves is powers of 10.
Example 1: Convert 3kg into gram
1000 g
3kg  3kg
 3000 g
1kg
Example 2: Convert 10 mph into m/s
10
mile
mile
1h
1609 m
 10



h
h
3600 s 1 mile
 4.47 m/s
3
VECTORS
•A vector has magnitude as well as direction
•Some vector quantities: displacement, velocity, force, momentum
•A scalar has only a magnitude
•Some scalar quantities: mass, time, temperature

a
Geometric presentation:
Notations:

a-
letter with arrow;
Magnitude (length of the vector):
Some properties:
 

A  B  C

A
a – bold font

a a

B

C
4
Vector addition (geometric)
Two vectors:

b

c
  
a b  c

b

a
Several vectors

d
   
a b c  d

a
Subtraction
  
a b  c

b

a

c

a

c

b

b

b

c

b

a

c
5
Question 1: Which of the following arrangements will produce the largest
resultant when the two vectors of the same magnitude are added?
A
B
C
Question 2: A person walks 3.0 mi north and then 4.0 mi west.
The length and direction of the net displacement of the person are:
1) 25 mi and 45˚ north of east
2) 5 mi and 37˚ north of west
3) 5 mi and 37˚ west of north
4) 7 mi and 77˚ south of west
Question 3: Consider the following three vectors:
What is the correct relationship between the three vectors?
  
1. C  A  B
  
2. C  A  B



3. C  

4. C  
 
A B
 
A B
 
A B


6
2 - Dimensional Vectors (2D)
y
ay
ĵ

a

ax

a x  a x iˆ
iˆ
x

a a

a  a x , a y 

a  a x iˆ  a y ˆj

a  a,  
 

a  ax  a y
a x  a cos 
a  a x2  a y2
a y  a sin 

a x ,a y – components ( scalars) of vector a
 

a x ,a y – component vectors of vector a
iˆ, ˆj - unit vect ors in x, and y directions


ˆa  a a - unit vecto r in the direction of vector a
tan  
ay
ax
7
Example 1:
ax = 3m, ay = - 4m. Find a and .
a  a x2  a y2 
3m 2  4m2
tan   a y a x  4 / 3

 5m
  arctan( 4 / 3)  53
Question: What is the unit vector in the direction of vector a ?



1) a  0.60m i  0.80m  j



2) a  0.75i  1.00 j



3) a  0.30i  0.40 j



4) a  0.60i  0.80 j
Example 2: a = 4,  = 30◦. Find ax , ay .
a x  a cos   4 cos 30   4 3 2  2 3  2  1.73  3.46
a y  a sin   4 sin 30   4  1 2  2



a  3.46i  2  j
Example 3:  = 30◦, ay = 3. Find ax ,and a.
a y  a sin  
a  a y sin   3 sin 30   6
a x  a cos   6 cos 30   6  3 2  3  1.73  5.19
8
z
k̂

b
3 - Dimensional Vectors (3D)


x
iˆ
ĵ
y
bx  b sin  cos 
b y  b sin  sin 

b  bx , b y , bz 




b  bx i  b y j  bz k

b  b, ,  

2
2
2
b  b  bx  b y  bz
bz  b cos 
9
Right-hand rule
Z
Z
Y
Y
X
X
10
Vector addition
Geometric:
Algebraic:

a

b
  
c  a b
Triangle inequality:
c  ab
Multiplication by
number
q x  cp x
c x  a x  bx


q  cp
c y  a y  by
q y  cp y
q z  cp z


q c p
c z  a z  bz
Properties:
   
a b b a
     
a b c  a  b c

 

c a  b  ca  cb






11

B


A
Angle between vectors:
Scalar product (dot product)
   
A  B | A || B | cos   Ax Bx  Ay B y  Az Bz
 
A B
cos     
| A || B |
Properties of the dot product
   
A B  B  A
  
   
A  (B  C)  A  B  A  C
2  
2
A  A  A A
Ax Bx  Ay B y  Az Bz
Ax2  Ay2  Az2 Bx2  B y2  Bz2
Projections

Ax  A  iˆ

Ay  A  ˆj

Az  A  kˆ
 
A  B  A||B B  AB|| A
12
Projections
B projected onto A:
Component of B
perpendicular to A:
A projected onto B:
Component of A
perpendicular to B:
B|| A  B cos 

B


B
B A  B sin 
A||B  A cos 
A B  A sin 


B


B


A

A

A

A
 
A  B  A||B B  AB|| A
13

Question 1: A  A
Question 2: What is angle (in degrees) between
the following two vectors?
1)
0
2) 45
A  1.00 iˆ  1.00 ˆj
3) 90
B  1.00 iˆ  1.00 ˆj
4) 135
 
What is A  A equal to?
1) 0
2) A2
3) A1/2
4) A
Question 3: Which pair of vectors will have the largest value for A·B?
A
A
A
1)
2)
B
3)
60°
Aprojected onto B
B
B
30°
Aprojected onto B
Aprojected onto B

AB  AB cos 90  0

AB  AB cos 60

AB  AB cos 30
More visual: A·B = BAprojected onto B
14
Vector product (cross product)
 
  
  
A  B  AB sin 
A B  A
A B  B
 
A  B  A B B  AB A
Right-hand rule
A×B
B
A
15
Properties of vector product
A  B is a vector! (A  B is a scalar)

z k
A B  B  A

j
A A  0
iˆ  iˆ  0, ˆj  ˆj  0,
iˆ  ˆj  kˆ, ˆj  kˆ  iˆ,
kˆ  kˆ  0
kˆ  iˆ  ˆj
y
x

i
Right-handed
coordinate
systems
A  B  ( Ay Bz  Az By )iˆ  ( Az Bx  Ax Bz ) ˆj  ( Ax By  Ay Bx )kˆ
iˆ
 Ax
Bx
ˆj
Ay
By
kˆ
Az
Bz
 
A B
sin    
| A || B |
16
Question 1: Vectors A, B and C are on the plane of the screen. They are
drawn to scale. Compare the magnitude of these two cross products:
Bsinθ
A) |A×B| > |A×C|
C
B) |A×B| = |A×C|
C) |A×B| < |A×C|
 
 
| A  B | AB sin   AC | A  C |

Bcosθ
B
A
And they both point out of the screen.
The cross product selects the part of B that is
perpendicular to the direction of A.
Question 2:
 

  
If the vectors P , Q and R satisfy th e relations P  Q  R and P 2  Q 2  R 2 ,


what is the angle between P and Q ?
17
Questions 3 & 4: Right-handed Cartesian coordinate system.
What is the direction
of the +x axis?
What is the direction
of the +z axis?
1. Into page
2. Out of page
Question 3: Consider two nonzero
vectors A and B with an angle Φ
between them.
Question 4: What is angle (in degrees)
between vectors A and B?
A  ( A  B)  ____?
| A  B | 1.2 m 2
A  B  1.2 m 2
A) A2 B sin  cos 
B) A2 B
C) AB sin  cos 
D) zero
A) 45
B) 90
C) 135
D) 180
18
Question 5:
Two vectors are:
A  1.00ˆi  2.00ˆj
B  3.00ˆi  4.00ˆj
Question 6:
1. 0
 
A B - ?
Then A  B  ____ ?
10kˆ
2.  10kˆ
3. 3ˆi  8ˆj
1.
4. 2ˆi  2ˆj
2. 4.5
3. 2.25
4. 2.25m 2
Question 7: What is angle (in degrees)
between vectors A and B?
A  3.00 ˆi  4.00 ˆj
B  3.00 ˆi  4.00 ˆj
1.
2.
3.
4.
106
111
116
123
19