Chapter 1
Measurement and Problem
Solving
Units of Chapter 1
Why and How We Measure
SI Units of Length, Mass, and Time
More about the Metric System
Unit Analysis
Unit Conversions
Significant Figures
Problem Solving
1.1 Why and How We Measure
Physics attempts to describe nature in an
objective way through measurement.
Measurements are expressed in units; officially
accepted units are called standard units.
Major systems of units:
1. Metric
2. British (used by the U.S., but no longer by the
British!)
1.2 SI Units of Length, Mass, and Time
Length, mass, and time are fundamental quantities;
combinations of them will form all the units we will
use through Chapter 14.
In this text, we will be using the SI system of units,
which is based on the metric system.
1.2 SI Units of Length, Mass, and Time
SI unit of length: the meter. The original definition is
on the left, the modern one is on the right.
1.2 SI Units of Length, Mass, and Time
SI unit of mass: the kilogram
Originally, the kilogram was
the mass of 0.10 m3 of water.
Now, the standard
kilogram is a platinumiridium cylinder kept at the
French Bureau of Weights
and Measures.
1.2 SI Units of Length, Mass, and Time
SI unit of time: the second
The second is defined as a certain number of
oscillations of the cesium-133 atom.
1.2 SI Units of Length, Mass, and Time
In addition to length, mass, and time, base units in
the SI system include electric current, temperature,
amount of substance, and luminous intensity.
These seven units are believed to be all that are
necessary to describe all phenomena in nature.
1.3 More about the Metric System
The British system of units is used in the U.S., with
the basic units being the foot, the pound (force,
not mass), and the second.
However, the SI system is virtually ubiquitous
outside the U.S., and it makes sense to become
familiar with it.
1.3 More about the Metric System
In the metric system, units of the same type of
quantity (length or time, for example) differ from
each other by factors of 10. Here are some
common prefixes:
1.3 More about the Metric System
The basic unit of volume in the SI system is the
cubic meter. However, this is rather large for
everyday use, so the volume of a cube 0.1 m on a
side is called a liter.
Recall the original definition of a kilogram; one
kilogram of water has a volume of one liter.
1.4 Unit Analysis
A powerful way to check your calculations is to use
unit analysis.
Not only must the numerical values on both sides
of an equation be equal, the units must be equal
as well.
1.4 Unit Analysis
Units may be manipulated algebraically just as
other quantities are.
Example:
Therefore, this equation is dimensionally correct.
1.5 Unit Conversions
A conversion factor simply lets you express a quantity
in terms of other units without changing its physical
value or size.
The fraction in blue is the conversion factor; its
numerical value is 1.
Scientific Notation
Scientific notation is a shorter method to write out very
large or small numbers.
123,000,000,000 = 1.23 x 1011
In the number above…
1.23 is called the coefficient
10 is the base (ALWAYS)
11 is the exponent
Scientific Notation
To write in scientific notation, put the decimal after
the first non-zero digit and drop the zeroes.
1.23000000000.
This is your coefficient = 1.23
To find the exponent, count the number of places
from the decimal to the end of the number.
This is your exponent: 11
And we find our number is equal to: 1.23 x 1011
Scientific Notation
Very small numbers work the same way but have
negative exponents!
0.00000000324
0.000000003.24
Count to the new decimal…
This is your coefficient = 3.24
This is your exponent = -9
Scientific Notation = 3.24 x 10-9
Scientific Notation
And you can go backwards to get to decimal form!
9.14 x 101 = 91.4
9.14 x 102 = 914
9.14 x 103 = 9140
9.14 x 104 = 91400
9.14 x 105 = 914000
9.14 x 106 = 9140000
3.54 x 10-1 = 0.354
3.54 x 10-2 = 0.0354
3.54 x 10-3 = 0.00354
3.54 x 10-4 = 0.000354
3.54 x 10-5 = 0.0000354
3.54 x 10-6 = 0.00000354
Scientific Notation Practice
1.
2.
3.
4.
5.
6.
7.
8.
429000
1.394 x 107
104
42000001
1.572 x 10-3
0.0000159
100
1 x 104
1.
2.
3.
4.
5.
6.
7.
8.
4.29 x 105
13940000
10000
4.2000001x107
0.001572
1.59 x 10-5
1 x 102
10000
Metric Prefixes
You are required to know what these prefixes are!
109 = giga = 1000000000
106 = mega = 1000000
103 = kilo = 1000
1=1
10-2 = centi = .01
10-3 = milli = .001
10-6 = micro = .000001
10-9 = nano = .000000001
Metric Prefixes
Prefixes are reversible too!
Kilogram = 103 grams (1000 g)
Gram = 10-3 kilograms (0.001 kg)
Nanometer = 10-9 meters (0.000000001 meters)
Meter = 109 nanometers (1000000000 nanometers)
Metric Prefixes
• How many meters in 93 nanometers?
93 nanometers = 93 x 10-9 meters
• How many kilometers in 93 x 10-9 meters?
1 km = 103 meters OR 10-3 km = 1 meter
93 x 10-9 x 10-3 = 93 x 10-12 kilometers
Metric Prefixes
109 = giga
106 = mega
103 = kilo
1=1
10-2 = centi
10-3 = milli
10-6 = micro
10-9 = nano
1 nanometer = 10-9 meter
1 nanometer = 10-12 kilometer
1 nanometer = 10-15 megameter
…
1 megameter = 10? centimeters
1 kilogram = 10? nanograms
1 milliliter = 10? gigaliters
1.6 Significant Figures
Calculations may contain two types of numbers:
exact numbers and measured numbers.
Exact numbers are part of a definition, such as the 2
in d = 2r.
Measured numbers are just that—for example, we
might measure the radius of a circle to be 10.3 cm,
but that measurement is not exact.
1.6 Significant Figures
When dealing with measured numbers, it is useful
to consider the number of significant figures.
The significant figures in any measurement are the digits
that are known with certainty, plus one digit that is
uncertain.
It is easy to create answers that have many digits
that are not significant using a calculator. For
example, 1/3 on a calculator shows as
0.33333333333. But if we’ve just cut a pie in three
pieces, how well do we really know that each one is
1/3 of the whole?
1.6 Significant Figures
Significant figures in calculations:
1. When multiplying and dividing quantities, leave as
many significant figures in the answer as there are in the
quantity with the least number of significant figures.
2. When adding or subtracting quantities, leave the same
number of decimal places in the answer as there are in
the quantity with the least number of decimal places.
Rounding: If the first digit dropped is 5 or greater,
increase the preceding digit by one.
Significant Figures Practice
Determine the number of significant figures
1. 871
1. 3
2. 0.00983
2. 3
3. 1473829.3
3. 8
4. 8.700 x 103
4. 4
5. 129300 +/- 100
5. 4
Significant Figures Practice
1. 1.11 x 3.458 =
1. 3.84
2. 22.27 – 14.4=
2. 7.9
3. 9 / 4.3 =
3. 2
4. 14.98 + 438.54 =
4. 453.52
5. 54.3 + 18.9 / 74.13 =
5. 54.6
Trigonometry
• Trigonometry is math that deals with the
relationships between sides and angles of
triangles
Equilateral
Isosceles
Scalene
Trigonometry
• We define triangles by the three interior
angles.
Acute
Right
Obtuse
Trigonometry
• The three interior angles of a triangle add up to
180 degrees.
60°
60°
23°
?°
?°
32°
18°
60°
Trigonometry
• The interior and exterior angles when against a
plane are equal to 180 degrees.
60°
150°
135°
120°
23°
120°
?°
?°
?°
?°
76°
60°
60°
Trigonometry Practice
• Using the two given angles, determine the 10
unknown angles. Yes, it’s possible!
?°
?°
?°
?°
123°
?°
?°
?°
94°
?°
?°
?°
Trigonometry
• Pythagoras’ Theorem:
a2 + b2 = c2
a=3
b=4
c=?
a = 13
b = 11
c=?
Trigonometry
x = adjacent
y = opposite
r = hypotenuse
Hypotenuse > Opposite
Hypotenuse > Adjacent
Hypotenuse < Opposite + Adjacent
Trigonometry
x = adjacent
y = opposite
r = hypotenuse
S o h C a h To a !
Trigonometry Practice
x = adjacent
y = opposite
r = hypotenuse
S o h C a h To a !
?
?
40°
15 m
Trigonometry Practice
x = adjacent
y = opposite
r = hypotenuse
S o h C a h To a !
60°
37 m
?
?
Trigonometry Practice
x = adjacent
y = opposite
r = hypotenuse
S o h C a h To a !
?
104 m
20°
?
1.7 Problem Solving
The flowchart at left
outlines a useful problemsolving strategy. It can be
used for most types of
physics problems.
Problem Solving Practice
• A closed cylindrical container used to store
material from a manufacturing process has
an outside radius of 50.0 cm and a height
of 1.30 m. What is the total outside surface
area of the container?
Problem Solving Practice
• Aend = πr2
• Aend x 2 = 2 x πr2
• Abody = 2πr x h
• r = 50.0 cm
• h = 1.30 m
• A = 2 x πr2 + 2πr x h
• A = 5.65 m2
Review of Chapter 1
SI units of length, mass, and time: meter,
kilogram, second
Liter: 1000 cm3; one liter of water has a mass of
1 kg
Unit analysis may be used to verify answers to
problems
Significant figures – digits known with certainty,
plus one
Review of Chapter 1
Problem-solving procedure:
1. Read the problem carefully and analyze it.
2. Where appropriate, draw a diagram.
3. Write down the given data and what is to be
found. (Make unit conversions if necessary.)
4. Determine which principle(s) are applicable.
5. Perform calculations with given data.
6. Consider if the results are reasonable.