Transcript Chapter 1 Introduction    Length (m) Mass (kg) Time (s) ◦ other physical quantities can be constructed from these three.

Chapter 1
Introduction
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Length (m)
Mass (kg)
Time (s)
◦ other physical quantities can be constructed from
these three
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To communicate the result of a measurement
for a quantity, a unit must be defined
Defining units allows everyone to relate to the
same fundamental amount
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Standardized systems
◦ agreed upon by some authority, usually a
governmental body
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SI - Systéme International
◦ agreed to in 1960 by an international committee
◦ main system used in this text
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Units
◦ SI – meter, m
◦ cgs – centimeter, cm
◦ US Customary – foot, ft
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Defined in terms of a meter – the distance
traveled by light in a vacuum during a given
time
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Units
◦ SI – kilogram, kg
◦ cgs – gram, g
◦ USC – slug, slug
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Defined in terms of kilogram, based on a
specific cylinder kept at the International
Bureau of Weights and Measures
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Units
◦ seconds, s in all three systems
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Defined in terms of the oscillation of
radiation from a cesium atom
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A significant figure is one that is reliably
known
All non-zero digits are significant
Zeros are significant when
◦ between other non-zero digits
◦ after the decimal point and another significant
figure
◦ can be clarified by using scientific notation
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Accuracy – number of significant figures
When multiplying or dividing two or more
quantities, the number of significant figures
in the final result is the same as the number
of significant figures in the least accurate of
the factors being combined
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When adding or subtracting, round the
result to the smallest number of decimal
places of any term in the sum
If the last digit to be dropped is less than 5,
drop the digit
If the last digit dropped is greater than or
equal to 5, raise the last retained digit by 1
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When units are not consistent, you may
need to convert to appropriate ones
Units can be treated like algebraic
quantities that can “cancel” each other
See the inside of the front cover for an
extensive list of conversion factors
Example:
2.54 cm
15.0 in 
 38.1 cm
1 in
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All physical quantities encountered in this
text will be either a scalar or a vector
A vector quantity has both magnitude (size)
and direction
A scalar is completely specified by only a
magnitude (size)
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When handwritten, use an arrow: A
When printed, will be in bold print with an
arrow: A
When dealing with just the magnitude of a
vector in print, an italic letter will be used: A
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Equality of Two Vectors
◦ Two vectors are equal if they have the same
magnitude and the same direction
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Movement of vectors in a diagram
◦ Any vector can be moved parallel to itself without
being affected
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Negative Vectors
◦ Two vectors are negative if they have the same
magnitude but are 180° apart (opposite directions)
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 
A  B; A  A  0
Resultant Vector
◦ The resultant vector is the sum of a given set of
vectors
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R  A B
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When adding vectors, their directions must be
taken into account
Units must be the same
Geometric Methods
◦ Use scale drawings
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Algebraic Methods
◦ More convenient
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When adding vectors, their directions must be
taken into account
Units must be the same
Geometric Methods
◦ Use scale drawings
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Algebraic Methods
◦ More convenient
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Choose a scale
Draw the first vector with the appropriate
length and in the direction specified, with
respect to a coordinate system
Draw the next vector with the appropriate
length and in the direction specified, with
respect to a coordinate system whose origin
is the end of vector A and parallel to the
coordinate system used for A
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Continue drawing the
vectors “tip-to-tail”
The resultant is
drawn from the origin
of A to the end of the
last vector
Measure the length of R
and its angle
◦ Use the scale factor to
convert length to actual
magnitude
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When you have many
vectors, just keep
repeating the process
until all are included
The resultant is still
drawn from the origin
of the first vector to
the end of the last
vector
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Vectors obey the
Commutative Law
of Addition
◦ The order in which
the vectors are
added doesn’t
affect the result
◦
A B B A
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Special case of
vector addition
◦ Add the negative of
the subtracted
vector
A  B  A  B
 
Continue with
standard vector
addition
procedure
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The result of the multiplication or division
is a vector
The magnitude of the vector is multiplied or
divided by the scalar
If the scalar is positive, the direction of the
result is the same as of the original vector
If the scalar is negative, the direction of the
result is opposite that of the original vector
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A component is a
part
It is useful to use
rectangular
components
◦ These are the
projections of the
vector along the xand y-axes
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The x-component of a vector is the
projection along the x-axis
Ax  A cos 
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The y-component of a vector is the projection
along the y-axis
Ay  A sin
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Then,
A  Ax  Ay
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The components are the legs of the right
triangle whose hypotenuse is A
A
2
x
2
y
A A
and
 Ay 
  tan  
 Ax 
1
◦ May still have to find θ with respect to the
positive x-axis
◦ The value will be correct only if the angle lies in
the first or fourth quadrant
◦ In the second or third quadrant, add 180°
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Choose a coordinate system and sketch the
vectors
Find the x- and y-components of all the
vectors
Add all the x-components
◦ This gives Rx:
Rx   v x
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Add all the y-components
◦ This gives Ry: R y   v y
Use the Pythagorean Theorem to find the
magnitude of the resultant:
Use the inverse tangent function to find the
direction of R:
R  R2x  R2y
  tan
1
Ry
Rx