Measurement
Why are precise measurements
and calculations essential to a
study of physics?
Measurement & Precision


The precision of a measurement depends on the
instrument used to measure it.
For example, how long is this block?
Measurement & Precision



Imagine you have a piece of string that is
exactly 1 foot long.
Now imagine you were to use that string to
measure the length of your pencil. How
precise could you be about the length of the
pencil?
Since the pencil is less than 1 foot, we must
be dealing with a fraction of a foot. But what
fraction can we reliably estimate as the length
of the pencil?
Measurement & Precision

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Suppose the pencil is slightly over half the 1 foot
string. You guess, “Well it must be about 7 inches,
so I’ll say 7/12 of a foot.”
Here’s the problem: If you convert 7/12 to a
decimal, you get 0.583.
Can you reliably say, without a doubt, that the pencil
is 0.583 and not 0.584 or 0.582?
You can’t. The string didn’t allow you to distinguish
between those lengths… you didn’t have enough
precision.
So, what can you estimate, reliably?
Measurement & Precision

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Basically, you have one degree of freedom… one
decimal place of freedom.
So, the only fractions you can use are tenths!
You can only reliably estimate that the pencil is 0.6
ft long. It’s definitely more than 0.5 ft long and
definitely less than 0.7 ft long.
Thus, precision determines the number of significant
figures we use to report measurements.
In order to increase the precision of their
measurements, physicists develop more-advanced
instruments.
How big is the beetle?
Measure between the head
and the tail!
Between 1.5 and 1.6 in
Measured length: 1.54 in
The 1 and 5 are known with
certainty
The last digit (4) is estimated
between the two nearest fine
division marks.
Copyright © 1997-2005 by Fred Senese
How big is the penny?
Measure the diameter.
Between 1.9 and 2.0 cm
Estimate the last digit.
What diameter do you
measure?
How does that compare to
your classmates?
Is any measurement EXACT?
Copyright © 1997-2005 by Fred Senese
What Length is Indicated by
the Arrow?
Significant Figures

Indicate precision of a measured value


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1100 vs. 1100.0
Which is more precise? How can you tell?
How precise is each number?
Determining significant figures can be tricky.
There are some very basic rules you need to
know. Most importantly, you need to practice!
Counting Significant Figures
The Digits
Digits That Count
Example
# of Sig Figs
Non-zero digits
ALL
4.337
4
Leading zeros
(zeros at the BEGINNING)
NONE
0.00065
2
Captive zeros
(zeros BETWEEN non-zero digits)
ALL
1.000023
7
Trailing zeros
(zeros at the END)
ONLY IF they follow a
significant figure AND
there is a decimal
point in the number
Leading, Captive AND Trailing
Zeros
Combine the
rules above
Scientific Notation
ALL
89.00
but
8900
4
0.003020
but
3020
4
7.78 x 103
2
3
3
Calculating With Sig Figs
Type of Problem
MULTIPLICATION OR DIVISION:
Find the number that has the fewest sig
figs. That's how many sig figs should
be in your answer.
ADDITION OR SUBTRACTION:
Example
3.35 x 4.669 mL = 15.571115 mL
rounded to 15.6 mL
3.35 has only 3 significant figures, so
that's how many should be in the
answer. Round it off to 15.6 mL
64.25 cm + 5.333 cm = 69.583 cm
rounded to 69.58 cm
Find the number that has the fewest
64.25 has only two digits to the right of
digits to the right of the decimal point.
the decimal, so that's how many
The answer must contain no more
should be to the right of the decimal
digits to the RIGHT of the decimal
in the answer. Drop the last digit so
point than the number in the problem.
the answer is 69.58 cm.
Scientific Notation

Number expressed as:

Product of a number between 1 and 10 AND a power of 10

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5.63 x 104, meaning
5.63 x 10 x 10 x 10 x 10
or 5.63 x 10,000
ALWAYS has only ONE nonzero digit to the left of the
decimal point
ONLY significant numbers are used in the first number
First number can be positive or negative
Power of 10 can be positive or negative
When to Use Scientific
Notation

Astronomically Large Numbers


Infinitesimally Small Numbers


mass of planets, distance between stars
size of atoms, protons, electrons
A number with “ambiguous” zeros

59,000
 HOW PRECISE IS IT?
Powers of 10

Positive Exponents
10  10
1
10  10  10  100
2
10  10  10  10  1000
3
10  10  10  10  10  10,000
4
Exponent of Zero Means “1”
100 = 1
Powers of 10

Negative Exponents
1
10  101  0.1
10
2
1
 101  101  100
 0.01
3
1
10  101  101  101  1000
 0.001
10
4
 101  101  101  101  10,1000  0.0001
Exponent of Zero Means “1”
100 = 1
Converting From Standard to
Scientific Notation

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
Move decimal until it is behind the first sig fig
Power of 10 is the # of spaces the decimal moved
Decimal moves to the left, the exponent is positive
Decimal moves to the right, the exponent is negative

428.5

4.285 x 102
(decimal moves 2 spots left)

0.0004285

4.285 x 10-4
(decimal moves 4 spots right)
Converting From Scientific to
Standard Notation




Move decimal point
# of spaces the decimal moves is the power of 10
If exponent is positive, move decimal to the right
If exponent is negative, move decimal to the left

4.285 x 102

428.5
(move decimal 2 spots right)

4.285 x 10-4

0.0004285
(decimal moves 4 spots left)
Systems of Measurement

Why do we need a standardized system of
measurement?

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Scientific community is global.
An international “language” of measurement allows
scientists to share, interpret, and compare experimental
findings with other scientists, regardless of nationality or
language barriers.
By the 1700s, every country used its own system of
weights and measures. England had three different
systems just within its own borders!
Metric System & SI

The first standardized system of measurement: the
“Metric” system

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Developed in France in 1791
Named based on French word for “measure”
based on the decimal (powers of 10)
Systeme International d'Unites
(International System of Units)

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
Modernized version of the Metric System
Abbreviated by the letters SI.
Established in 1960, at the 11th General Conference on
Weights and Measures.
Units, definitions, and symbols were revised and simplified.
Components of the SI System


In this course we will primarily use SI units.
The SI system of measurement has 3 parts:

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base units
derived units
prefixes
Unit: measure of the quantity that is defined to be
exactly 1
Prefix: modifier that allows us to express multiples
or fractions of a base unit
As we progress through the course, we will
introduce different base units and derived units.
SI: Base Units
Physical Quantity
Unit Name
Symbol
length
meter
m
mass
kilogram
kg
time
second
s
electric current
ampere
A
temperature
Kelvin
K
amount of substance
mole
mol
luminous intensity
candela
cd
SI: Derived Units
Physical Quantity
Unit Name
Symbol
area
square meter
m2
volume
cubic meter
m3
speed
meter per
second
m/s
acceleration
meter per
second squared
m/s2
weight, force
newton
N
pressure
pascal
Pa
energy, work
joule
J
Prefixes
Prefix Symbol
Numerical Multiplier
Exponential
Multiplier
yotta
Y
1,000,000,000,000,000,000,000,000
1024
zetta
Z
1,000,000,000,000,000,000,000
1021
exa
E
1,000,000,000,000,000,000
1018
peta
P
1,000,000,000,000,000
1015
tera
T
1,000,000,000,000
1012
giga
G
1,000,000,000
109
mega
M
1,000,000
106
kilo
k
1,000
103
hecto
h
100
102
deca
da
10
101
1
100
no prefix means:
Prefixes
Prefix Symbol
no prefix means:
Numerical Multiplier
Exponential
Multiplier
1
100
deci
d
0.1
10¯1
centi
c
0.01
10¯2
milli
m
0.001
10¯3
micro
m
0.000001
10¯6
nano
n
0.000000001
10¯9
pico
p
0.000000000001
10¯12
femto
f
0.000000000000001
10¯15
atto
a
0.000000000000000001
10¯18
zepto
z
0.000000000000000000001
10¯21
yocto
y
0.000000000000000000000001
10¯24
Unit Conversions
Method
“Staircase”
Factor-Label
Type
Visual
Mathematical
What to
do…
Move decimal point the
same number of places as
steps between unit
prefixes
Multiply measurement by
conversion factor, a fraction that
relates the original unit and the
desired unit
When to
use…
Converting between
different prefixes between
kilo and milli
Converting between SI and nonSI units
Converting between different
prefixes beyond kilo and milli
“Staircase” Method
Draw and label this staircase every time you need
to use this method, or until you can do the
conversions from memory
“Staircase” Method: Example

Problem: convert 6.5 kilometers to
meters




Start out on the “kilo” step.
To get to the meter (basic unit) step, we need
to move three steps to the right.
Move the decimal in 6.5 three steps to the
right
Answer: 6500 m
“Staircase” Method: Example

Problem: convert 114.55 cm to km




Start out on the “centi” step
To get to the “kilo” step, move five steps to the
left
Move the decimal in 114.55 five steps the left
Answer: 0.0011455 km
Factor-Label Method

Multiply original measurement by conversion
factor, a fraction that relates the original unit
and the desired unit.

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
Conversion factor is always equal to 1.
Numerator and denominator should be equivalent
measurements.
When measurement is multiplied by
conversion factor, original units should cancel
Factor-Label Method: Example


Convert 6.5 km to m
First, we need to find a conversion factor that
relates km and m.


We should know that 1 km and 1000 m are
equivalent (there are 1000 m in 1 km)
We start with km, so km needs to cancel when we
multiply. So, km needs to be in the denominator
1000 m
1 km
Factor-Label Method: Example

Multiply original measurement by conversion
factor and cancel units.
1000 m
6.5 km 
 6500 m
1 km
Factor-Label Method: Example


Convert 3.5 hours to seconds
If we don’t know how many seconds are in an
hour, we’ll need more than one conversion
factor in this problem
60 minutes 60 seconds
3.5 hours 

 12600 seconds
1 hour
1 minute
round to appropriat e number of sig figs (2)
Answer :13000 seconds