Unit I
Units and Measurement
Objectives
- Express lengths, mass, time in SI units.
- Convert distances between different units.
- Describe time intervals in hours, minutes, and seconds.
- Convert time in mixed units to time in seconds.
- Describe the mass of objects in grams and kilograms.
It All Starts with a Ruler!!!
I. Two Systems of Units
1. Metric system and
International System of
Units
meter
kilogram
second
Kelvin
2. English system
inches, feet, yards, and miles.
pound
Fahrenheit
3. SI units
meter (m): unit of length
kilogram (kg): unit of mass
second (s): unit of time
meter, (SI unit symbol: m), is
the fundamental unit of length in
the International System of Units (SI).
Originally intended to be one ten-millionth
of the distance from the Earth's equator to
the North Pole (at sea level).
Since 1983, it has been defined as "the
length of the path travelled by light in
vacuum during a time interval of
1/299,792,458 of a second."
National Prototype Metre Bar ( alloy of ninety
percentplatinum and ten percent iridium)
in International Bureau of Weights and
Measures (BIPM: Bureau International des Poids
et Mesures) to be located in Sèvres, France.
kilogramme ( kg), is the base
unit of mass in the International System of
Units (SI)
Is defined as being equal to the mass of
the International Prototype of the Kilogram
(platinum–iridium alloy) in International
Bureau of Weights and Measures in
Sèvres, France
Second (sec or s)
The second is the duration of 9,192,631,770
periods of the radiation corresponding to the
transition between the two hyperfine levels of
the ground state of the caesium 133 atom.
UNITS (Systéme Internationale)
4. Examples
The units for length, mass, and time (as well as a
few others), are regarded as base SI units.
These units are used in combination to define
additional units for other important physical
quantities such as force and energy.
II THE CONVERSION OF UNITS
A) relation between different units
1 ft = 0.3048 m
1 mi = 1.609 km
1 liter = 10-3 m3
Example 1.1
Grandma traveled 27 minutes at 44 m/s.
How many miles did Grandma travel?
44𝑚
𝑠
=
44𝑚
𝑚𝑖𝑙𝑒
1609𝑚
1𝑠
𝑚𝑖𝑛
60𝑠
= 44 ×
1.64𝑚𝑖𝑙
27𝑚𝑖𝑛 ×
𝑚𝑖𝑛
= 44.3𝑚𝑖𝑙𝑒𝑠
60
mile/min=1.64mile/min
1609
44.3 miles
B) How to convert
Example 1 The World’s Highest Waterfall
The highest waterfall in the world is Angel Falls in
Venezuela,
with a total drop of 979.0 m. Express this drop in feet.
Since 3.281 feet = 1 meter, it follows that
(3.281 feet)/(1 meter) = 1 and (1 meter) / (3.281 feet)=1
For meter  feet:
 3.281 feet 
Length  979.0 meters 
  3212 feet
 1 meter 
Convert 100km to miles
• A football field is 100 yards long.
• What is this distance expressed in meters?
C) unit convert chart
D) Summary
Reasoning Strategy: Converting Between Units
1. In all calculations, write down the units explicitly.
2. Treat all units as algebraic quantities. When
identical units are divided, they are eliminated
algebraically.
3. Use the conversion factors in reference tables.
Be guided by the fact that multiplying or dividing
an equation by a factor of 1 does not alter the
equation.
time
• Two ways to think about time:
– What time is it?
• 3 P.M. Eastern Time on April 21, 2004,
– How much time has passed?
• 3 hr: 44 min: 25 sec.
• A quantity of time is often called a time interval.
Converting Mixed Units
1.
2.
You are asked for time in seconds.
You are given a time interval in mixed
units.
1 hour = 3,600 sec
1 minute = 60
sec
3.
Do the conversion:
1 hour = 3,600 sec
26 minutes = 26 × 60 = 1,560 sec
4.
Add all the seconds:
t = 3,600 + 1,560 + 31.25 = 5,191.25 sec
Time Units
E) Practice
Example 2 Interstate Speed Limit
Express the speed limit of 65 miles/hour in terms of meters/second.
Use 5280 feet = 1 mile and 3600 seconds = 1 hour and
3.281 feet = 1 meter.
m
 miles 
 miles  1609m  1 hour 
Speed   65
11   65


 29
s
 hour 
 hour  mile  3600 s 
Speed  29
meters
second
More practice
1. Convert 789 cm2 to m2
1m=100cm, 1m2=100cm *100cm=10000cm2
789𝑐𝑚2 = 789𝑐𝑚2 ×
1𝑚2
2
=0.0789𝑚
10000𝑐𝑚2
2. Convert 75.00 km/h to m/s
75.00 km x 1000 m x 1 h___ = 20.83m/s
h
1 km
3600 s
III Limits of Measurement
A). Accuracy and Precision
• Accuracy - a measure of how
close a measurement is to the
true value of the quantity being
measured.
Example: Accuracy
• Who is more accurate when
measuring a book that has a true
length of 17.0cm?
Susan:
17.0cm, 16.0cm, 18.0cm, 15.0cm
Amy:
15.5cm, 15.0cm, 15.2cm, 15.3cm
• Precision – a measure of how
close a series of measurements
are to one another. A measure of
how exact a measurement is.
Example: Precision
Who is more precise when measuring
the same 17.0cm book?
Susan:
17.0cm, 16.0cm, 18.0cm, 15.0cm
Amy:
15.5cm, 15.0cm, 15.2cm, 15.3cm
Example: Evaluate whether the following are
precise, accurate or both.
Accurate
Not Accurate Accurate
Not Precise Precise
Precise
--Why significant figures is important?
--What does significant figures in a number consist of?
--How to record measurement with proper significant
figures? (digram)
--List the rules in counting significant figures (zero rules)
--Rules in calculation:
Multiplication rule
Division rule
Addition & subtraction rule
Rule in calculate average
B) Significant Figures
• The significant figures in a
measurement include all of the
digits that are known, plus one
last digit that is estimated.
Centimeters and Millimeters
40.16 cm
The length of this
miniature piezo electric
motor is:
8.0 mm
B.1) Finding the Number of Sig Figs:
• All non-zero digits are significant.
• Zeros between two non-zero digits are
significant.
• Leading zeros are not significant.
• Trailing zeros in a number containing a decimal
point are significant.
• trailing zeros in a number not containing a
decimal point can be ambiguous. (scientific
notation is the solution)
• One convention about trailing zero
A bar placed over ( or under) the last significant
figure; any trailing zeros following this are
insignificant
Example:
500 has 1 s.f.
500 has 2 s. f.
500 has 2 s.f.
500. has 3 s.f.
Scientific notation
Write number in form: 𝑎 × 10𝑏
1 ≤ 𝑎 < 10, b is proper exponent
Standard decimal notation
Scientific notation
2
2×100
300
3×102
4,321.768
4.321768×103
−53,000
−5.300×104
6,720,000,000
6.72000×109
0.2
2×10−1
0.000 000 007 51
7.51×10−9
How many sig figs?
100
2
10302.00
7
100.00
5
970
2
0.001
1
0.00250
3
5
1.0302x104
5
10302
B.2)Sig Figs in Addition/Subtraction
Express the result with the same
number of decimal places as the
number in the operation with the least
decimal places.
Ex: 2.33 cm
+ 3.0 cm
5.3 cm
(Result is rounded to one decimal place)
B.3) Sig Figs in Multiplication/Division
• Express the answer with the same sig
figs as the factor with the least sig
figs.
• Ex: 3.22 cm
x 2.0 cm
6.4 cm2
(Result is rounded to two sig figs)
More example
2330 cm
+ 3.0 cm
2330 cm
2330𝑚
= 780𝑚/𝑠
3.0𝑠
B.4) Constant and Counting Numbers
• Constant number have infinite sig.
figs.
• Counting numbers have infinite sig
figs.
• Ex: 3 apples
• Eg. π=3.1415926……
C) practice
1.Calculate Volume of sphere with r  0.55m,
4
4
3
3
V  r   (0.55)
3
3
 0.6969m 3  0.70m 3
2. Perimeter of the big circle
P  2r  2 (0.55m)
 3.4557 m  3.5m
2. Try the following
7.895 + 3.4=
(8.71 x 0.0301)/0.056 =
𝐷 = 2.015𝑚 + 4.5 𝑚/𝑠 × 2.35𝑠 = 13m
A= 𝜋𝑟 2 = 𝜋(1.25𝑚)2 =
4.91m2
IV Dimension Analysis – some simple rules
1.In 𝒎𝒖𝒍𝒕𝒊𝒑𝒍𝒊𝒄𝒂𝒕𝒊𝒐𝒏 𝒐𝒓 𝒅𝒊𝒗𝒊𝒔𝒊𝒐𝒏: The
product unit is the product of the individual
unit of each of those variables. (Ditto for
ratios.)
2. 𝐈𝐧 𝒂𝒅𝒅𝒊𝒕𝒊𝒐𝒏 𝒐𝒓 𝒔𝒖𝒃𝒕𝒓𝒂𝒄𝒕𝒊𝒐𝒏: Different
terms can only added together in a sum if
each term in the sum has the same unit
type. (Ditto for subtraction.)
Example 1
- impossible: 40m + 20m/s
or 12.5 s - 20m2
- Can Do:
50.0m + 20.55m=70.6m
and 40m/s +11m/s =51m/s
- Can Do, but need to convert into same
unit:
1𝑚
40m + 11cm = 40m + 11cm ×
= 40.11m
100𝑐𝑚
Example 2
1.5 m  3.0 kg  ?
The above expression yields:
a)4.5 m kg
b)4.5 g km
c)A or B
d)Impossible to evaluate (dimensionally invalid)
IV Scalars and Vectors
1. Definition
A scalar quantity is one that can be described
by a single number:
temperature, speed, mass
A vector quantity deals inherently with both
magnitude and direction:
velocity, force, displacement
2. Graph a Vector
Arrows are used to represent vectors. The
direction of the arrow gives the direction of
the vector.
By convention, the length of a vector
arrow is proportional to the magnitude
of the vector.
8 lb
4 lb
3. Vector Addition and Subtraction
Often it is necessary to add one vector to another.
A)
example 1
3m
5m
8m
B) Example 2
Example 2 continue
2.00 m
6.00 m
Example 2 continue
R  2.00 m   6.00 m 
2
R
2
2
2.00 m  6.00 m
2
2
 6.32m
R
2.00 m
6.00 m
C)
When a vector is multiplied
by -1, the magnitude of the
vector remains the same, but
the direction of the vector is
reversed.
Example 3
Given

B
 
AB

A

A

A
 
AB

B

B