Chapter 1
Units, Physical Quantities, and Vectors
Disclaimer:
• All images copyright of Pearson Education unless otherwise stated.
• Lecture slides format and content adapted from Lectures by Jason Harlow
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The nature of physics
• Physics is an experimental
science
• Physicists seek patterns that
relate the phenomena of
nature.
• The patterns are
physical theories.
called
• A very well established or
widely used theory is called a
physical law or principle.
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Solving problems in physics
The following four steps can be useful in forming ProblemSolving Strategies in physics:
• Identify the relevant concepts, target variables, and
known quantities, as stated or implied in the problem.
• Set Up the problem: Choose the equations that you will
use to solve the problem, and draw a sketch of the
situation.
• Execute the solution: This is where you “do the math.”
• Evaluate your answer: Compare your answer with your
estimates, and reconsider things if there’s a discrepancy.
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Idealised models
Use simplified idealized model as
aid to understanding e.g.
•
frictionless surface and light as
a wave or a particle.
• Use language of mathematics
• Quantitative (numbers) not just
qualitative
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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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Unit prefixes
• Prefixes can be used to create larger and smaller units for
the fundamental quantities. Some examples are:
• 1 µm = 10−6 m (size of some bacteria and living cells)
• 1 km = 103 m (about 10-minute walk)
• 1 mg = 10−6 kg (mass of a grain of salt)
• 1 g = 10−3 kg (mass of a paper clip)
• 1 ns = 10−9 s (time for light to travel 0.3 m)
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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.
• Always carry units through calculations,
e.g. 𝑑𝑑 = 𝑣𝑣𝑣𝑣 =
𝑚𝑚
2.0
𝑠𝑠
× 5.0 𝑠𝑠 = 10 𝑚𝑚.
• Convert to standard units as necessary, by forming
a ratio of the same physical quantity in two
different units, and using it as a multiplier.
e.g. , 3 min to seconds:
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Scientific Notation
• To deal with big and small numbers we use
scientific notation:
– Number = mantissa×10exponent
• The mantissa is usually chosen so that it has one digit
preceding the decimal point:
– e.g., 3.00×108.
– but not always.
• The exponent is an integer representing the powers of 10.
• Multiplication and division using scientific notation:
– (7×1027 atoms/person)(7×107 persons) = 49×1036 = 4.9×1037 atoms
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Uncertainty and significant figures
• The uncertainty of a measured quantity is indicated by its
number of significant figures.
• The lengths of two rods, A and B are measured as shown in
diagram below.
a) Are the rods of the same lengths?
b) How confident are you of your answer to part a)?
c) Which length measurement ( A or B ) are you more
certain of?
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Estimating Significant Figures
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Significant figures (s.g.f) in calculations
• For multiplication/division, the answer can have
no more significant figures than the smallest
number of sgf in the factors.
e.g. 2.3 m/s × 0.293 s = 0.67 m/s
• For addition/subtraction, the number of significant
figures is determined by the term having the
fewest digits to the right of the decimal point.
e.g. 210.23 m+60.1 m= 270.3 m
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Vectors and scalars
• Scalar quantity can be described by a single number
(magnitude or size).
• Vector quantity has both a magnitude and a direction in
space.
• In this course, a vector quantity is represented in italic
⃗
type with an arrow over it in boldface or normal: 𝑨𝑨 or 𝐴𝐴.
⃗
• The magnitude of 𝑨𝑨 is written as A, |𝑨𝑨| or |𝐴𝐴|.
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Drawing vectors
• 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.
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Adding two vectors graphically
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Adding two vectors graphically
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Adding more than two vectors graphically
• To add several vectors, use the head-to-tail method.
• The vectors can be added in any order.
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Adding more than two vectors graphically
• To add several vectors, use the head-to-tail
method.
• The vectors can be added in any order.
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Subtracting vectors
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Multiplying a vector by a scalar
• If c is a scalar, the product c
has magnitude |c|A.
• The figure illustrates multiplication
of a vector by
(a) a positive scalar and
(b) a negative scalar.
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Addition of two vectors at right angles
• To add two vectors that are at right angles, first
add the vectors graphically.
• Then use trigonometry to find the magnitude and
direction of the sum.
• In the figure, a crosscountry skier ends up
2.24 km from her
starting point, in a
direction of 63.4° E of N
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Components of a vector
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Positive and negative components
• The components of a vector may be positive
or negative numbers, as shown in the figures.
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Finding components
• We can calculate the components of a vector
from its magnitude and direction.
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Calculations using components
• We can use the components of a vector to find its
magnitude and direction:
• We can use the components of a set of vectors to find
the components of their sum:
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Vector addition using components
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Unit vectors
• A unit vector has a magnitude
of 1 with no units.
• The unit vector points in the
+x-direction, points in the +ydirection, and points in the
+z-direction.
• Any vector can be expressed
in terms of its components as
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The scalar product
Scalar (dot) Product : is the magnitude of the
vectors multiplied by the cosine of the angle
between them.
The result of the dot operation is a scalar
quantity e.g. Energy and work-done
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The scalar product
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The scalar product
The scalar product can be
positive, negative, or
zero, depending on the
angle between and .
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Scalar product using components
• In terms of components:
• The scalar product of two vectors is the sum
of the products of their respective
components.
Chapter 6 to describe work done by a force.
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The vector product
• The vector (cross) product of two vectors give a
vector quantity.
• The magnitude of the vector product is the
product of the magnitudes of the vectors
multiplied by the sine of the angle between them
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The vector product
The direction of the vector product can be
found using the right-hand rule:
Examples of cross products are Torque and angular momentum
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The vector product is anti-commutative
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Vector product using components
This is 2 D version; full 3D version in textbook
𝐴𝐴⃗ × 𝐵𝐵 = 𝐴𝐴𝑥𝑥 𝚤𝚤̂ + 𝐴𝐴𝑦𝑦 𝚥𝚥̂ × 𝐵𝐵𝑥𝑥 𝚤𝚤̂ + 𝐵𝐵𝑦𝑦 𝚥𝚥̂
= 𝐴𝐴𝑥𝑥 𝐵𝐵𝑥𝑥 𝚤𝚤̂ × 𝚤𝚤̂ + 𝐴𝐴𝑥𝑥 𝐵𝐵𝑦𝑦 𝚤𝚤̂ × 𝚥𝚥̂ + 𝐴𝐴𝑦𝑦 𝐵𝐵𝑥𝑥 𝚥𝚥̂ × 𝚤𝚤̂ + 𝐴𝐴𝑦𝑦 𝐵𝐵𝑦𝑦 𝚥𝚥̂ × 𝚥𝚥̂
Now 𝚤𝚤̂ × 𝚤𝚤̂ = 0; 𝚥𝚥̂ × 𝚥𝚥̂ = 0 (vectors in same direction)
� 𝚤𝚤̂ × 𝚥𝚥̂ = 𝑘𝑘�
Also 𝚤𝚤̂ × 𝚥𝚥̂ = 𝑘𝑘,
for a right-handed co-ordinate system
(the only kind we use;
�
it is defined by 𝚤𝚤̂ × 𝚥𝚥̂ = 𝑘𝑘)
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Vector product using components
𝐴𝐴⃗ × 𝐵𝐵 = 𝐴𝐴𝑥𝑥 𝐵𝐵𝑦𝑦 𝑘𝑘� + 𝐴𝐴𝑦𝑦 𝐵𝐵𝑥𝑥 −𝑘𝑘� = 𝐴𝐴𝑥𝑥 𝐵𝐵𝑦𝑦 − 𝐴𝐴𝑦𝑦 𝐵𝐵𝑥𝑥 𝑘𝑘�
Meaning vectors 𝐴𝐴⃗ and 𝐵𝐵 are in xy-plane,
then 𝐴𝐴⃗ × 𝐵𝐵 is in the ±𝑧𝑧 direction
• 𝐴𝐴⃗ × 𝐵𝐵 is in +𝑧𝑧 direction
if 𝐴𝐴𝑥𝑥 𝐵𝐵𝑦𝑦 > 𝐴𝐴𝑦𝑦 𝐵𝐵𝑥𝑥
• 𝐴𝐴⃗ × 𝐵𝐵 is in −𝑧𝑧 direction
if 𝐴𝐴𝑥𝑥 𝐵𝐵𝑦𝑦 < 𝐴𝐴𝑦𝑦 𝐵𝐵𝑥𝑥
• (this is the simple case of 2D vectors)
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Vector product using components
More generally, 3D (i.e. if 𝐴𝐴⃗ and 𝐵𝐵 have 𝑥𝑥, 𝑦𝑦
and z components) we have the result
𝐴𝐴⃗ × 𝐵𝐵 = 𝐴𝐴𝑦𝑦 𝐵𝐵𝑧𝑧 − 𝐴𝐴𝑧𝑧 𝐵𝐵𝑦𝑦 𝚤𝚤̂
+ 𝐴𝐴𝑧𝑧 𝐵𝐵𝑥𝑥 − 𝐴𝐴𝑥𝑥 𝐵𝐵𝑧𝑧 𝚥𝚥̂
+ 𝐴𝐴𝑥𝑥 𝐵𝐵𝑦𝑦 − 𝐴𝐴𝑦𝑦 𝐵𝐵𝑥𝑥 𝑘𝑘�
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