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Physical Quantities and Measurement
What is Physics?
Natural Philosophy
science of matter and energy
fundamental principles of engineering and technology
an experimental science: theoryexperiment
simplified models
Quantum
Field Theory
Relativistic
Mechanics
Quantum
Mechanics
Classical
Mechanics
speed
range of validity
size
Phys211C1 p1
Quantifying predictions and observations
physical quantities: numbers used to describe physical phenomena
height, weight e.g.
may be defined operationally
standard units: International System (SI aka Metric)
defined units established in terms of a physical quantity
derived units established as algebraic combinations of other units
Quantity
Length
Time
Mass
Temperature
Electric Current
Unit
meter (m)
second (s)
kilogram (kg)
kelvin (K)
ampere (A)
Phys211C1 p2
Scientific Notation: powers of 10
5,820
= 5.82x103
= 5.82E3
.000527 = 5.28x10-4 = 5.28E-4
note: 103 = 1x103 =1E3 not 10E3!
Common prefixes
Prefix
femto
pico
nano
micro
milli
centi
kilo
mega
giga
Abbreviation
f
p
n
µ
m
c
k
M
G
Power
of Ten
10-15
10-12
10-9
10-6
10-3
10-2
103
106
109
1/1,000,000,000,000,000
1/1,000,000,000,000
1/1,000,000,000
1/1,000,000
1/1,000
1/100
1,000
1,000,000
1,000,000,000
How big (in terms of everyday life/other things) is a
meter
nanometer
centimeter
kilometer
gram
kilogram
Phys211C1 p3
Dimensional Analysis: consistency of units
Algebraic equations must always be dimensionally consistent.
You can’t add apples and oranges!
d  vt distance  speed  time
 m
10 m   2 5 s 
 s 
converting units
treat units as algebraic quantities
multiplying or dividing a quantity by 1 does not affect its value
cm
1inch  2.540cm  2.540
1
inch
cm
 12inches 
1 ft  1 ft 
 30.48cm
2.540
ft 
inch

Phys211C1 p4
Some Useful Conversion factors:
1 inch
= 2.54 cm
1m
= 3.28 ft
1 mile
= 5280 ft
1 mile
= 1.61 km
Units Conversion Examples
Example 1-1 The world speed record, set in 1983 is 1019.5 km/hr. Express this
speed in m/s
Example how man cubic inches are there in a 2 liter engine?
Phys211C1 p5
Significant Figures and Uncertainty
Every measurement of a physical quantity involves some error
random error
averages out
small random error  accurate measurement
systematic error
does not average out
small systematic error  precise measurement
less precise
20
20
15
15
10
10
5
5
less accurate
15
10
5
0
0
7
number
0
10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55
0
0
0
0
0
0
0
0
4
8
15 13
5
3
0
0
0
7
number
0
10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55
0
0
0
0
0
2
6
10 18 11
1
0
0
0
0
0
7
number
0
10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55
0
0
0
0
0
1
0
4
4
12 10
6
4
3
2
1
Phys211C1 p6
Indicating the accuracy of a number: x ± Dx or x± dx
nominal value: the indicated result of the measurement
numerical uncertainty: how much the “actual value” might be
expected to differ from the nominal value
sometimes called the numerical error
1 standard deviation
A measured length of 20.3 cm ± .5 cm means that the actual length is expected to lie between
19.8 cm and 20.8 cm. It has a nominal value of x = 20.3 cm with an uncertainty of Dx .5 cm.
fractional uncertainty: the fraction of the nominal value
corresponding to the numerical uncertainty
Dx
.5 cm

 .025
x 20.3 cm
percentage uncertainty: the percentage of the nominal value
corresponding to the numerical uncertainty
Dx
.5 cm
100% 
100%  2.5%
x
20.3 cm
20.3 cm  2.5%
Phys211C1 p7
Uncertainties in calculations
Adding and subtracting: add numerical uncertainty
c  ab
or
Dc  Da  Db
c  a b
Multiplying or Dividing: add fractional/percentage uncertainty
Dc Da Db
c  ab


c
a
b
or
Dc
Da
Db
100% 
100%  100%
ca b
c
a
b
Powers are “multiple multiplications”
Dc
Da
c  aN
N
c
a
Phys211C1 p8
More complex algebraic expressions must be broken down operation
by operation
a = 3.13±.05 b = 7.14 ±.01 c = 14.44 ±.2%
x  a  b  c  ( a  b)  c
D(a  b) Da Db



b
a
( a  b)
( a  b)  a  b 
D ( a  b)
 ( a  b) 
D ( a  b) 
( a  b)
 Dc 100%   1  c 
Dc  

 100
 c
Dx  D(a  b)  Dc 
x
Phys211C1 p9
Significant Figures: common way of implicitly indicating uncertainty
number is only expressed using meaningful digits (sig. figs.)
last digit (the least significant digit = lsd) is uncertain
3
3.0
3.00
one digit
two digits (two significant figures = 2 sig. figs.)
three digits,etc.
(300 how many digits?)
Combining numbers with significant digits
Addition and Subtraction: least significant digit determined by
decimal places (result is rounded)
.57 + .3 = .87 =.9
11.2 - 17.63 = 6.43 = 6.4
Multiplication and Division: number of significant figures is
the number of sig. figs. of the factor with the fewest sig. figs.
1.3x7.24 = 9.412 = 9.4
17.5/.3794 = 46.12546 = 46.1
Integer factors and geometric factors (such as p) have infinite
precision
p x 3.762 = 44.4145803 = 44.4
Phys211C1 p10
Estimates and Order of magnitude calculations
an order of magnitude is a (rounded) 1 sig fig calculation, whose
answer is expressed as the nearest power of 10.
Estimates should be done “in your head”
check against calculator mistakes!
Additional Homework: with
a = 3.13±.05
b = 7.14 ±.01 c = 14.44 ±.2%
evaluate expressions (nominal value and uncertainty expressed
as numerical uncertainty and percentage uncertainty)
x  abc
x  ab / c
x  a3/ 2
x  a  bc
x  a2  c / b
x  a 2  b2
Phys211C1 p11