Chapter 1
Units, Physical
Quantities, and Vectors
PowerPoint® Lectures for
University Physics, Thirteenth Edition
– Hugh D. Young and Roger A. Freedman
Lectures by Wayne Anderson
Copyright © 2012 Pearson Education Inc.
Goals for Chapter 1
•  To learn three fundamental quantities of physics and the
units to measure them
•  To keep track of significant figures in calculations
•  To understand vectors and scalars and how to add vectors
graphically
•  To determine vector components and how to use them in
calculations
•  To understand unit vectors and how to use them with
components to describe vectors
•  To learn two ways of multiplying vectors
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The nature of physics
•  Physics is an experimental science in which
physicists seek patterns that relate the phenomena of
nature.
•  The patterns are called physical theories.
•  A very well established or widely used theory is
called a physical law or principle.
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Solving problems in physics
•  A problem-solving strategy offers techniques for setting up
and solving problems efficiently and accurately.
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Why use units
•  This table is 1000 long.
•  This table is 1 long.
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Why have different units
•  The speed of a car is 26 m/s.
•  The speed of a car is 26,000 mm/s.
•  A pencil is 15 cm long.
•  A pencil is 0.00015 km long.
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Standards and units
•  Length, time, and mass are three fundamental
quantities of physics.
•  The International System (SI for Système
International) is the most widely used system of
units.
•  In SI units, length is measured in meters, time in
seconds, and mass in kilograms.
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What is one meter
•  The meter is the basic unit of length in the SI
system of units. The meter is defined to be the
distance light travels through a vacuum in exactly
1/299792458 seconds.
•  U.S. PROTOTYPE METER BAR
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What is one kilogram
•  It is defined as the mass of a particular international
prototype made of platinum-iridium and kept at the
International Bureau of Weights and Measures. It was
originally defined as the mass of one liter (10-3 cubic
meter) of pure water.
•  France holds the world's standard for the kilogram
weight. It's a cylinder-shaped object that has dictated
all other kilogram weights for 130 years now.
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What is one second
•  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 cesium 133 atom.
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Unit prefixes
•  Table 1.1 shows some larger and smaller units for the
fundamental quantities.
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More prefix for powers of ten
yotta- (Y-)
zetta- (Z-)
exa- (E-)
peta- (P-)
tera- (T-)
giga- (G-)
mega- (M-)
kilo- (k-)
hecto- (h-)
deka- (da-)
deci- (d-)
centi- (c-)
milli- (m-)
micro- (µ-)
nano- (n-)
pico- (p-)
femto- (f-)
atto- (a-)
zepto- (z-)
yocto- (y-)
1024
1021
1018
1015
1012
109
106
103
102
10
10-1
10-2
10-3
10-6
10-9
10-12
10-15
10-18
10-21
10-24
1 septillion
1 sextillion
1 quintillion
1 quadrillion
1 trillion
1 billion
1 million
1 thousand
1 hundred
1 ten
1 tenth
1 hundredth
1 thousandth
1 millionth
1 billionth
1 trillionth
1 quadrillionth
1 quintillionth
1 sextillionth
1 septillionth
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British System vs. SI
Length
kilometer (km) = 1,000 m
1 km = 0.621 mi 1 mi = 1.609 km
meter (m) = 100 cm
1 m = 3.281 ft
1 ft = 0.305 m
centimeter (cm) = 0.01 m
1 cm = 0.394 in. 1 in. = 2.540 cm
millimeter (mm) = 0.001 m 1 mm = 0.039 in.
micrometer (µm) = 0.000 001 m
nanometer (nm) = 0.000 000 001 m
Area
square kilometer (km2) = 100 hectares 1 km2 = 0.386 mi2 1 mi2 = 2.590 km2
hectare (ha) = 10,000 m2
1 ha = 2.471 acres
1 acre = 0.405 ha
square meter (m2) = 10,000 cm2
1 m2 = 10.765 ft2 1 ft2 = 0.093 m2
square centimeter (cm2) = 100 mm2
1 cm2 = 0.155 in.2 1 in.2 = 6.452 cm2
Volume
liter (L) = 1,000 mL = 1 dm3
1 L = 1.057 fl qt 1 fl qt = 0.946 L
milliliter (mL) = 0.001 L = 1 cm3
1 mL = 0.034 fl oz 1 fl oz = 29.575 mL
microliter (µL) = 0.000 001 L
Mass
kilogram (kg) = 1,000 g
1 kg = 2.205 lb
1 lb = 0.454 kg
gram (g) = 1,000 mg
1 g = 0.035 oz
1 oz = 28.349 g
milligram (mg) = 0.001 g
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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.”)
•  Always carry units through calculations.
•  Convert to standard units as necessary. (Follow Problem-Solving
Strategy 1.2)
•  Follow Examples 1.1 and 1.2.
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Uncertainty and significant figures—Figure 1.7
•  The uncertainty of a measured quantity
is indicated by its number of
significant figures.
•  For multiplication and division, the
answer can have no more significant
figures than the smallest number of
significant figures in the factors.
•  For addition and subtraction, the
number of significant figures is
determined by the term having the
fewest digits to the right of the decimal
point.
•  Refer to Table 1.2, Figure 1.8, and
Example 1.3.
•  As this train mishap illustrates, even a
small percent error can have
spectacular results!
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Ways to express accuracy
•  56.47 ± 0.02
•  47 ± 10% (fractional/percent error, fractional/
percent uncertainty)
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Estimates and orders of magnitude
•  An order-of-magnitude estimate of a quantity
gives a rough idea of its magnitude.
•  Can you count 1 million in your life?
•  Follow Example 1.4.
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Vectors and scalars
•  A scalar quantity can be described by a single
number.
•  A vector quantity has both a magnitude and a
direction in space.
•  In this book, a vector quantity is represented
in
è
boldface italic type with an arrow over it: A.
è
è
•  The magnitude of A is written as A or |A|.
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Drawing vectors—Figure 1.10
•  Draw a vector as a line with an arrowhead at its tip.
•  The length of the line shows the vector’s magnitude.
•  The direction of the line shows the vector’s direction.
•  Figure 1.10 shows equal-magnitude vectors having the same
direction and opposite directions.
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Adding two vectors graphically—Figures 1.11–1.12
•  Two vectors may be added graphically using either the parallelogram
method or the head-to-tail method.
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Adding more than two vectors graphically—Figure 1.13
•  To add several vectors, use the head-to-tail method.
•  The vectors can be added in any order.
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Subtracting vectors
•  Figure 1.14 shows how to subtract vectors.
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Multiplying a vector by a scalar
•  If c is a scalar, the
è
product cA has
magnitude |c|A.
•  Figure 1.15 illustrates
multiplication of a vector
by a positive scalar and a
negative scalar.
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Addition of two vectors at right angles
•  First add the vectors graphically.
•  Then use trigonometry to find the magnitude and direction of the
sum.
•  Follow Example 1.5.
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Components of a vector—Figure 1.17
•  Adding vectors graphically provides limited accuracy. Vector
components provide a general method for adding vectors.
•  Any vector can be represented by an x-component Ax and a ycomponent Ay.
•  Use trigonometry to find the components of a vector: Ax = Acos θ and
Ay = Asin θ, where θ is measured from the +x-axis toward the +y-axis.
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Positive and negative components—Figure 1.18
• The components of a vector can
be positive or negative numbers,
as shown in the figure.
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Finding components—Figure 1.19
•  We can calculate the components of a vector from its magnitude
and direction.
•  Follow Example 1.6.
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Calculations using components
•  We can use the components of a vector to find its magnitude
Ay
2
2
A = Ax + Ay and tanθ =
and direction:
A
x
•  We can use the components of a
set of vectors to find the components
of their sum:
Rx = Ax + Bx + Cx +L , Ry = Ay + By + Cy +L
•  Refer to Problem-Solving
Strategy 1.3.
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Adding vectors using their components—Figure 1.22
•  Follow Examples 1.7 and 1.8.
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Unit vectors—Figures 1.23–1.24
•  A unit vector has a magnitude
of 1 with no units.
•  The unit vector î points in the
+x-direction, ĵ points in the
$k and points in
+y-direction, $
k
the +z-direction.
•  Any vector can be expressed
in terms of its components as
è
$k.
A =Axî+ Ay $$jj + Az $
k
•  Follow Example 1.9.
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The scalar product—Figures 1.25–1.26
•  The scalar product
(also called the “dot
product”) of two
vectors is
•  Figures 1.25 and
1.26 illustrate the
scalar product.
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Calculating a scalar product
•  In terms of components,
•  Example 1.10 shows how to calculate a scalar product in two
ways.
[Insert figure 1.27 here]
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Finding an angle using the scalar product
•  Example 1.11 shows how to use components to find the angle
between two vectors.
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The vector product—Figures 1.29–1.30
•  The vector
product (“cross
product”) of
two vectors has
magnitude
r r
| A× B| = ABsinφ
and the righthand rule gives
its direction.
See Figures
1.29 and 1.30.
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Calculating the vector product—Figure 1.32
•  Use ABsinΦ to find the
magnitude and the right-hand
rule to find the direction.
•  Refer to Example 1.12.
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