Chapter 1
Measurement
Fundamental Quantities in Physics
Units & Conversion
Scott Hildreth – Chabot College – Adapted from Wiley Fundamentals of Physics 10e; Pearson University Physics 13e
Three KEYS for Chapter 1
• Fundamental quantities in physics
(length, mass, time)
– Units (meters, kilograms, seconds...)
– Dimensional Analysis
• Force = kg meter/sec2
• Power = Force x Velocity
= kg m2/sec3
Three KEYS for Chapter 1
• Fundamental quantities in physics
(length, mass, time)
– Units (meters, kilograms, seconds...)
– Dimensional Analysis
• Significant figures in calculations
– 6.696 x 104 miles/hour
– 67,000 miles hour
Three KEYS for Chapter 1
• Fundamental quantities in physics
(length, mass, time)
– Units (meters, kilograms, seconds...)
– Dimensional Analysis
• Significant figures in calculations
• Estimation (order of magnitude ~10#)
Standards and units
• Length, mass, and time = three fundamental
quantities (“dimensions”) of physics.
• The SI (Système International) is the most widely
used system of units.
– Meeting ISO standards are mandatory for
some industries. Why?
• In SI units, length is measured in meters, mass in
kilograms, and time in seconds.
Converting Units
•A conversion factor is
•A ratio of units equal to 1
•Used to convert between units
• Units obey same algebraic rules as variables &
numbers
Converting Units
km
Converting Units
km
1000 m = 1 km
Multiplying by 1
doesn’t change
the overall
value, just the
units.
Converting Units
km
Converting Units
km
Converting Units
km
cm
Converting Units
km
cm
Unit consistency and conversions
• An equation must be dimensionally consistent.
Terms to be added or equated must always
have the same units. (Be sure you’re adding
“apples to apples.”)
• OK: 5 meters/sec x 10 hours =~ 2 x 102 km
(distance/time) x (time) = distance
Unit consistency and conversions
• An equation must be dimensionally consistent.
Terms to be added or equated must always
have the same units. (Be sure you’re adding
“apples to apples.”)
• OK: 5 meters/sec x 10 hours =~ 2 x 102 km
5 meters/sec x 10 hour x (3600 sec/hour)
= 180,000 meters = 180 km = ~ 2 x 102 km
Unit consistency and conversions
• An equation must be dimensionally consistent.
Terms to be added or equated must always
have the same units. (Be sure you’re adding
“apples to apples.”)
• OK: 5 meters/sec x 10 hours =~ 2 x 102 km
• NOT: 5 meters/sec x 10 kg = 50 Joules
(velocity) x (mass) = (energy)
Unit prefixes
• Larger & smaller units for fundamental quantities.
• Learn these – and prefixes like Mega, Tera, Pico, etc.!
• Skip Ahead to Slide 24 – Sig Fig Example
Measurement & Uncertainty
No measurement is exact; there is always
some uncertainty due to limited instrument
accuracy and difficulty reading results.
Measurement & Uncertainty
• The precision – and also uncertainty - of
a measured quantity is indicated by its
number of significant figures.
–Ex: 8.7 centimeters
• 2 sig figs
• Specific rules for significant figures exist
• In online homework, sig figs matter!
• In exams, sig figs matter!!
Significant Figures
Number of significant figures = number of
“reliably known digits” in a number.
Often possible to tell # of significant figures by the
way the number is written:
•
23.21 cm = four significant figures.
•
0.062 cm = two significant figures
(initial zeroes don’t count).
Significant Figures
• Significant figures are not decimal places
0.00356 has 5 decimal places, but just
3 significant figures

Generally, round to the least number of
significant figures of the given data

25 x 18 → 2 significant figures;

25 x 18975 → still 2

Round up for 5+ (13.5 → 14, but 13.4 → 13)
Significant Figures
In general, trailing zeros are NOT significant
In other words, 3000 may have 4 significant figures
but usually 3000 will have only 1 significant figure!
Numbers ending in zero are ambiguous.
Does the last zero mean uncertainty to
a factor of 10, or just 1?
Significant Figures
Numbers ending in zero are ambiguous
Is 20 cm precise to 10 cm, or 1? We need rules!
•
20 cm = one significant figure
(trailing zeroes don’t count w/o decimal point)
•
20. cm = two significant figures
(trailing zeroes DO count w/ decimal point)
•
20.0 cm = three significant figures
Rules for Significant Figures
•When multiplying or dividing numbers, or
using functions, result has as many sig figs as
term with fewest (the least precise).
•ex: 11.3 cm x 6.8 cm = 77 cm.
•When adding or subtracting, answer is no
more precise than least precise number used.
•
ex: 1.213 + 2 = 3, not 3.213!
Significant Figures
•Calculators will not give right # of sig
figs; usually give too many but
sometimes give too few (especially if
there are trailing zeroes after a
decimal point).
•top image: result of 2.0/3.0
•bottom image: result of 2.5 x 3.2
Scientific Notation
•Scientific notation commonly used
•Uses powers of 10 to write large & small numbers
Scientific Notation
•Scientific notation allows the number of
significant figures to be clearly shown.
•Ex: cannot easily tell how many significant
figures in “36,900”.
•Clearly
and
3.69 x 104
3.690 x 104
has three
has four!
Remember trailing zeroes DO count with a decimal point
(always in Scientific Notation!)
Measurement & Uncertainty
No measurement is exact; there is always
some uncertainty due to limited instrument
accuracy and difficulty reading results.
Photo illustrates this –
it would be difficult to
measure the width of
this board more
accurately than ± 1 mm.
Uncertainty and significant figures
• Every measurement has uncertainty
–Ex: 8.7 cm (2 sig figs)
• “8” is (fairly) certain
• 8.6? 8.8?
• 8.71? 8.69?
• Good practice – include uncertainty
with every measurement!
–8.7  0.1 meters
Uncertainty and significant figures
• Uncertainty should match
measurement in the least precise
digit:
–8.7  0.1 centimeters
–8.70  0.10 centimeters
–8.709  0.034 centimeters
–8  1 centimeters
• Not…
–8.7 +/- 0.034 cm
Relative Uncertainty
•Relative uncertainty: a percentage, the ratio of
uncertainty to measured value, multiplied by 100.
•ex. Measure a phone to be 8.8 ± 0.1 cm
What is the relative uncertainty in this
measurement?
Uncertainty and significant figures
• Physics involves
approximations; these can
affect the precision of a
measurement.
Uncertainty and significant figures
• As this train mishap
illustrates, even a small
percent error can have
spectacular results!
Conceptual Example: Significant figures
Using a protractor, you measure an angle to be 30°.
(a) How many significant figures should you quote in this
measurement?
Conceptual Example: Significant figures
Using a protractor, you measure an angle to be 30°.
(a) How many significant figures should you quote in this
measurement? What uncertainty?
2 sig figs! (30. +/- 1 degrees or 3.0 x 101 +/- 1 degrees)
Conceptual Example: Significant figures
Using a protractor, you measure an angle to be 30°.
(b) What result would a calculator give for the cosine of this
result? What should you report?
Conceptual Example: Significant figures
Using a protractor, you measure an angle to be 30°.
(b) What result would a calculator give for the cosine of this
result? What should you report?
0.866025403, but to two sig figs, 0.87!
1-3 Accuracy vs. Precision
Accuracy is how close a measurement comes
to the true value.
ex. Acceleration of Earth’s gravity = 9.81 m/sec2
Your experiment produces 10 ± 1 m/sec2
•
You were accurate! How accurate? Measured
by ERROR.
•
|Actual – Measured|/Actual x 100%
•
| 9.81 – 10 | / 9.81 x 100% = 1.9% Error
Accuracy vs. Precision
•Accuracy is how close a measurement comes
to the true value
• established by % error
•Precision is a measure of repeatability of the
measurement using the same instrument.
• established by uncertainty in a measurement
• reflected by the # of significant figures
Accuracy vs. Precision Example
•Example:
You measure the acceleration of Earth’s gravitational
force in the lab, which is accepted to be 9.81 m/sec2
• Your experiment produces 8.334 m/sec2
•Were you accurate? Were you precise?
Accuracy vs. Precision
Accuracy is how close a measurement comes
to the true value. (established by % error)
ex. Your experiment produces 8.334 m/sec2
for the acceleration of gravity (9.81 m/sec2)
Accuracy: (9.81 – 8.334)/9.81 x 100% = 15% error
Is this good enough? Only you (or your
boss/customer) know for sure! 
Accuracy vs. Precision
Precision is the repeatability of the
measurement using the same instrument.
ex. Your experiment produces 8.334 m/sec2
for the acceleration of gravity (9.81 m/sec2)
Precision indicated by 4 sig figs
Seems (subjectively) very precise – and precisely
wrong!
Accuracy vs. Precision
Better Technique: Include uncertainty
Your experiment produces
8.334 m/sec2 +/- 0.077 m/sec2
Your relative uncertainty is
.077/8.334 x 100% = ~1%
But your error was ~ 15%
NOT a good result!
Accuracy vs. Precision
Better Technique: Include uncertainty
Your experiment produces
8.3 m/sec2 +/- 1.2 m/sec2
Your relative uncertainty is
1.2 / 8.3 x 100% = ~15%
Your error was still ~ 15%
Much more reasonable a result!
Accuracy vs. Precision
•Precision is a measure of repeatability of the
measurement using the same instrument.
• established by uncertainty in a measurement
• reflected by the # of significant figures
• improved by repeated measurements!
•Statistically, if each measurement is independent
• make n measurements (and n> 10)
•Improve precision by √(n-1)
• Make 10 measurements, % uncertainty ~ 1/3
1-6 Order of Magnitude: Rapid Estimating
Quick way to estimate calculated quantity:
• round off all numbers in a calculation to
one significant figure and then calculate.
• result should be right order of magnitude
• expressed by rounding off to nearest
power of 10
• 104 meters
• 108 light years
Order of Magnitude: Rapid Estimating
Example: Volume of a lake
Estimate how much
water there is in a
particular lake, which is
roughly circular, about 1
km across, and you
guess it has an average
depth of about 10 m.
Order of Magnitude: Rapid Estimating
Example: Volume of a lake
Volume = Area x depth
= (p x r2) x depth
= ~ 3 x 500 x 500 x 10
= ~75 x 105
= ~ 100 x 105
= ~ 107 cubic meters
Order of Magnitude: Rapid Estimating
Example: Volume of a lake
Volume = (p x r2) x depth
= 7,853,981.634 cu. m
But…. Round to power of 10 for
Order of Mag:
So ~ 107 cubic meters
1-6 Order of Magnitude: Rapid Estimating
Example: Thickness of a page.
Estimate the thickness
of a page of your
textbook.
(Hint: you don’t need
one of these!)
Solving problems in physics
• The online system offers a HUGE array of additional
resources to help you visualize how to solve problems
Solving problems in physics
• The online system offers a HUGE array of additional
resources to help you visualize how to solve problems
Solving problems in physics
• The textbook sample problems are IMPORTANT
Solving problems in physics – Step by Step!
• Step 1: Identify KEY IDEAS, relevant concepts, variables,
what is known, what is needed, what is missing.
Solving problems in physics
• Step 2: Set up the Problem – MAKE a SKETCH, label it,
act it out, model it, decide what equations might apply.
What units should the answer have? What value?
Solving problems in physics
• Step 3: Execute the Solution, and EVALUATE your
answer! Are the units right? Is it the right order of
magnitude? Does it make SENSE?
Solving problems in physics
• Good problems to gauge your learning
– “Test your Understanding” Questions throughout
the book
– Conceptual “Clicker” questions linked online
– “Two dot” problems in the chapter
• Good problems to review before exams
– Checkpoints along the way
– ODD problems with answers in the back
– Exam reviews published online