A Physics Toolkit: Basic Math

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A Physics ToolkitBasic Math Science Skills

• Chapter 1

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Mathematics and Physics
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What is Physics?

• branch of science that studies the physical world
• involves the study of energy, matter, and how the
  two are related

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Scientific Methods

• Scientific Law

• Scientific Theories

• A rule of nature that sums up related
  observations to describe a pattern in nature.
• Laws do not explain WHY these phenomena occur,
  they simply describe them.

• An explanation based on many observations
  supported by experimental results.
• Theories may serve as explanations for laws.

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SI Units
Base Quantity Base Unit Symbol
Length meter m
Mass kilogram kg
Time second s
Temperature kelvin K
Amount of a Substance mole mol
Electric Current ampere A
Luminous Intensity candela cd

• The 7 base units are listed in the table to the
  right.
• You need to know these!

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Prefixes Used with SI Units
Prefix Symbol Multiplier Scientific Notation
nano- n 0.000000001 10-9
micro- µ 0.000001 10-6
milli- m 0.001 10-3
centi- c 0.01 10-2
deci- d 0.1 10-1
kilo- k 1,000 103
mega- M 1,000,000 106
giga- G 1,000,000,000 109
You need to know these too!
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Dimensional Analysis

• The method of treating units as algebraic
  quantities that can be cancelled.
• How? Choose a conversion factor that will make
  the units you dont want cancel, and the units
  you do want stay in the answer.

• Example
• How many meters are in 30 kilometers?
• Conv. Factor? 1 km 1000 m
• 30 km x
• Try This One
• Convert 36 km/hr to m/s.

1000 m 1 km
30,000 m
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Significant Figures

• Sig figs are the valid digits in a measurement.
• answers cannot be more precise than the least
  precise measurement in calculations
• All answers on tests, quizzes, labs, etc. must
  have the proper amount of sig figs.

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Sig Fig Rules

• Determining the Number
• of Sig Figs in a Measurement

• Remember these four rules
• Nonzero digits are always significant.
• All final zeros after the decimal point are
  significant.
• Zeros between two other significant digits are
  always significant.
• Zeros solely used a placeholders are NOT
  significant.

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Operations Using Sig Figs

• Addition Subtraction

• Example

• To add or subtract measurements
• perform the operation
• round off the result to correspond to the least
  precise value involved

• Add 24.686 m 2.343 m 3.21 m.
• Just add the measurements.
• 24.686 m 2.343 m 3.21 m 30.239 m
• Round to the least precise measurement.
• 3.21 m is the least precise, so round to two
  decimal places 30.24 m

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Operations Using Sig Figs

• Multiplication Division

• Example

• To multiply or divide measurements
• perform the operation
• note measurement with the least number of sig
  figs
• round the product or quotient to this number of
  digits

• Multiply 3.22 cm by 2.1 cm.
• Just multiply the measurements.
• 3.22 cm x 2.1 cm 6.762 cm2
• Round the product to the same number of digits as
  the measurement with the least amount of sig
  figs.
• 3.22 cm has 3, 2.1 cm has 2, so, round to 2
  digits ? 6.8 cm2

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Measurement
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Measurement

• A comparison between an unknown quantity and a
  standard.

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Characteristics of Measured Values

• Precision

• Accuracy

• The degree of exactness of a measurement.
• Depends on the instrument and the technique used
  to make the measurement.

• Describes how well the results of a measurement
  agree with the accepted value as measured by
  skilled experimenters.

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Techniques of Good Measurement

• Know how to use the instrument you are using to
  obtain measurements.
• Use the instrument correctly.
• Handle instruments with care, to avoid damage.
• Always zero the instrument if necessary.
• Look straight at the markings at eye-level to
  avoid a parallax.
• Parallax the apparent shift in the position of
  an object when it is viewed from different
  angles.

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Graphs in Physics
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Linear Graphs

• You can see if a relationship exists between two
  quantities, also called variables, by graphing
  the data.
• If two variables show a linear relationship they
  are directly proportional to each other.

Examine the following graph
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Linear Graphs
Dependent Variable
Independent Variable
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Linear Graphs Slope of a Line

• The slope of a line is a ratio between the change
  in the y-value and the change in the x- value.
• This ratio tells whether the two quantities are
  related mathematically.
• Calculating the slope of a line is easy!

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Linear Graphs Slope of a Line
y2
y1
Run ?x x2 x1
x2
x1
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Linear Graphs Equation of a Line

• Once you know the slope then the equation of a
  line is very easily determined.

Slope Intercept form for any line
y mx b
y-intercept (the value of y when x 0)
slope
Of course in Physics we dont use x y. (We
could use F and m, or d and t, or F and x etc.)
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Linear Graphs Area Under the Curve

• Sometimes its what is under the line that is
  important!

Work Force x distance
W F x d
How much work was done in the first 4 m?
How much work was done moving the object over the
last 6 m?
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Non Linear Relationships

• Not all relationships between variables are
  linear.
• Some are curves which show a square or square
  root relationship

In this course we use simple techniques to
straighten the curve into a linear
relationship. This is called linearizing.
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Non Linear Relationships
This is not linear. It is an exponential
relationship. Try squaring the x-axis values to
produce a straight line graph
Equation of the straight line would then be y
x2
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Non Linear Relationships
This is not linear. It is an inverse
relationship. Try plotting y vs 1/x.
Equation of the straight line would then be y
1/x
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Meaning of Slope from Equations

• Often, in physics, graphs are plotted and the
  calculation and meaning of the slope becomes an
  important factor.

We will use the slope intercept form of the
linear equation described earlier.
y mx b
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Meaning of Slope from Equations

• Unfortunately physicists do not use the same
  variables as mathematicians!

d ½ x a x t2
For example
is a very common kinematic equation.
where d displacement, a acceleration and t
time
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Meaning of Slope from Equations
Physicists may plot a graph of d vs t, but this
would yield a non-linear graph in this case
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Meaning of Slope from Equations

• But what would the slope of a d vs t2 graph
  represent?

Lets look at the equation again
d ½at2
d is plotted vs t2
y mx b
d is y and t2 is x so whatever is before t2
must be equal to the slope of the line!
slope ½ a
and dont forget about the units! m/s2
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Meaning of Slope from Equations

• Now try These.

A physics equation will be given, as well as what
is initially plotted.
Tell me what should be plotted to linearize the
graph and then state what the slope of this graph
would be equal to.
Plot a vs v2 to linearize the graph
Example 1
a v2/r
Lets re-write the equation a little
a (1/r)v2
Therefore plotting a vs. v2 would let the slope
be
Slope 1/r
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Meaning of Slope from Equations

• Example 2

F 2md/t2
Slope 2md

Go on to the worksheet on this topic
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Classwork

• Start the Linearization Worksheet in class and
  finish for homework.