Chapter 1 – Units, Physical Quantities, and Vectors
I.
Measurements - scientists need to agree on the size of various quantities that will be used in
measurements
A.
Measurement: number and unit
B.
Three primary quantities on which all measurements in mechanics are based:
C.
Standardized and agreed upon units and their values assigned to each of these quantities:
Length [L] –
Mass [M] –
Time [T] –
D.
Systems of Units - When the unit of meters is used for length, kilograms for mass, and
seconds for time, this grouping is referred to as the International System of Units or the SI
system of units. There are other groupings of units or other systems of units that are
commonly used.
System of Units
SI
(International
System of Units)
Length [L]
Mass [M]
cgs
British Engineering
System
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Time [T]
E.
Derived Units – we have other units in physics besides length, mass, and time. These units
are referred to as derived units and , in mechanics, are defined in terms of the units of
length, mass, and time. For example:
What are units of the following quantities in the various systems of units?
F.
1.
speed or velocity - [speed] = [L]/[T]
2.
force - [force] = [M][L]/[T]2
Converting Units - need conversion factors
1.
30 miles/hour = ? m/s; 1 mile = 1609 m and 1 hour = 3600 sec
2.
10 slug ft/s2 = ? kg m/s2; 1 slug = 14.59 kg, 1 ft = 0.3048 m
3.
9.8 m/s2 = ? km/hr2
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II.
Significant Figures - important to know how accurate a measurement is and how the
accuracy affects combinations of measurements
A.
Measurements, Significant Figures and Uncertainties in Measurements
1.
How do you make and record a measurement? Record every value you know for sure plus
one more estimated or extrapolated value.
For example: Measure the length of a sheet of paper with the meter stick.
2.
a.
Scale one:
b.
Scale two:
c.
Scale three:
How do you find the number of significant figures in a measurement?
In the measurement, count all digits in the measurement you know for sure plus the last
uncertain digit.
Suppose the following numbers are measurements. How many significant figures are
there in each of the measurements?
B.
4.07
612.3
0.123
482
0.00378
5480
12300.
1.2300 x 104
Keeping track of significant figures when performing mathematical operations:
1.
Adding or subtracting - the number of decimal places in the sum or difference = the
least number of decimal places in any of the terms.
Find the sum.
1234.5
5.672
0.0123
23.4673
.012
586.
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2.
Multiplying or dividing - the number of significant figures in the result = the least
number of significant figures in any of the factors.
Evaluate the following:
C.
a.
(12.36)(4.28) =
b.
the circumference of a circle of radius 2.56 cm
c.
the volume of a right circular cylinder with R = 1.95 cm and h = 2.4 cm
Since the last digit in a measurement is an estimate, there must be an uncertainty associated
with this digit. This then means that there is an uncertainty associated with the
measurement.
How do you go about determining what the uncertainty is for a measurement?
There are two primary contributions to the uncertainty - the instrument used to make the
measurement and your ability to read and estimate the reading.
1.
Instrumental Uncertainty:
2.
Reading Uncertainty:
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D.
Propagation of Uncertainties – when the measurements are used in calculations, the
uncertainty associated with the measurements propagates through and affects the result.
The following examples demonstrate a method of determining the uncertainty in the result.
This is referred to as a way to find the “maximum possible error.”
1.
Find the area and its uncertainty if the length and width and their uncertainties are: L
= 12.36 cm and L =  0.05 cm; W = 6.79 cm and W =  0.04 cm.
2.
Find the density and the uncertainty in the density of a right circular cylinder: M =
58.45 g and M =  0.52 g; R = 1.95 cm and R =  0.06 cm; h = 2.54 cm and h =
 0.10 cm.
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III. Vectors
A.
Vectors vs scalars – properties
scalar quantity –
vector quantity –
B.
Representation of Vectors
1.
In prose, a vector quantity is indicated using boldface type.
2.
In lecture, a vector quantity is indicated by drawing an arrow above the variable

representing the vector, e.g., A . Unit vectors have carats drawn above them, e.g., î
and read as "i hat."
3.
Graphical representation is with an arrow whose length is proportional to the
magnitude and pointing in the direction of the vector quantity.
4.
C.
D.


The magnitude of a vector A is written as A or simply as A.
Properties of Vectors
1.
 
Equality of vectors: A  B
2.


Negative of a vector: given the vector A , the negative of the vector is - A .

 

Adding Vectors: For example, find the sum of A and B . That is, find A + B . The sum of
 


vectors is usually called the “resultant” and is denoted by R , so that A + B = R .
1.
Graphical addition - parallelogram method: Can only be done for two vectors at a time.
Their tails are connected together and a parallelogram is drawn using the two vectors as
two sides of the parallelogram. The resultant is the diagonal drawn from where the
tails are connected to the opposite vertex.
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2.
Graphical addition - polygon method: The resultant of any number vectors can be
found. Draw the first vector, then connect the tail of the second to the head of the first, then
connect the tail of the next to the head of the previous, and so on. The resultant is the
vector drawn from the tail of the first to the head of the last. The order in which the
vectors are added does not affect the result, i.e., the addition of vectors is commutative.

Example: Graphically determine the resultant of the following displacements: s1 = 100


m at 30o north of east, s2 = 150 m at 42o west of north, and s3 = 200 m due south. What
is your scale factor?
3.
Mathematical addition - rectangular resolution:
a.
Vectors are resolved into components along the x and y axes.
b.
The x-components are added to each other and the y-components are added to
each other.
c.
The components that are mutually perpendicular are then vectorially added
together.
d.
The resultant is a vector whose magnitude is the length of the hypotenuse of a
right triangle.
e.
The direction is the angle the resultant makes with the x-axis.
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Digression: Find the legs of right triangles.

Example: Find the resultant of the following four forces: F1 = 100 N at 30o above the +x


axis, F2 = 150 N at 48o above the -x axis, F3 = 200 N in the negative y direction,

and F4 = 50 N in negative x direction.
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Example: A person undergoes the following displacements and ends up 20 m to the


west of the starting position: s1 = 100 m at 30o south of east, s2 = 150 m at 20o


west of south, and s3 which is unknown. Find s3 .
E.
 

  


Subtracting Vectors: For example A  B  R is the same as A   B  R . That is,




subtracting B from A is the same as adding the negative of B to A .
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F
Multiplying Vectors – three different multiplications
1.


Scalar times a vector: Given the vector A , what is 3 A ?
2.
 
Scalar Product or Dot Product: What is A  B ? This is read as A “dot” B.
 
 
A  B  A B cos 
3.
 
Vector Product of Cross Product: What is A  B ? This is read as A “cross” B.
  

 

A  B  C where C  A B sin  and the direction of C is found using the “right-hand
rule.”
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IV. Unit vectors: î , ĵ , and k̂ have a magnitude of one (1) and point in the positive x, y , and z

directions, respectively. A force F = 100 N at 30o above the negative x-axis can be written as

F  (100cos30o î  100sin 30o ĵ ) N, in unit vector notation. Note that the two terms in the
parentheses are the x and y components of the force.
A.

Write the following displacement in unit vector notation: s  120 m at 37o below the positive
x axis.
B.

What is the magnitude and direction of the following velocity: v  (4.5î  6.3 ĵ ) m/s.
C.

Given: A  3iˆ  2 ˆj  kˆ and Bˆ  5iˆ  4 ˆj  6kˆ find the following:
1.
 
AB
2.
 
AB
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3.
 
AB
4.
 
A B
5.


The angle between A and B
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