1. A Physics Toolkit: Basic Math & Science Skills
Chapter 1
A Physics Toolkit: Basic Math & Science Skills

2. Mathematics and Physics

3. What is Physics? branch of science that studies the physical world
involves the study of energy, matter, and how the two are related
The goal of this course is not to make you a physicist. It is to give you an idea of the way physicists view the world; to have the satisfaction of understanding and even predicting the outcome of the things that are happening all around you.

4. 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.
A law only describes what happens, not why. A theory is the best available explanation of why things work the way they do. A theory must be well-supported.
The Law of Universal Gravitation gives the relationship between the gravitational force the distance and masses of two objects. The theory of universal gravitation explains that all the mass in the universe is attracted to other mass. Laws and Theories may be revised or rejected over time.

5. SI Units
The 7 base units are listed in the table to the right. You need to know these!
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
To share results, it’s practical to use units that everyone recognizes. The worldwide scientific community (and most countries) presently use an adapted version of the metric system, called SI. Uses 7 base quantities. You are going to see many derived units also this year. Derived units come from the combination of these 7 base units in different ways.

6. Prefixes Used with SI Units
Symbol
Multiplier
Scientific Notation
nano- n
10-9
micro- μ
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 probably already know from chem. that its much easier to convert units in the SI system because all you have to do is multiply or divide by the appropriate power of 10. Prefixes are used to change SI units by powers of 10.
You need to know these too!

7. 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 don’t 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

8. 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.

9. Determining the Number of Sig Figs in a Measurement
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.

10. 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 m m m.
Just add the measurements.
m m m = m
Round to the least precise measurement.
3.21 m is the least precise, so… round to two decimal places: m

11. 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 = 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

12. Measurement

13. Measurement
A comparison between an unknown quantity and a standard.

14. 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.
Precision – How close measurements are to one another. Depends on tool – as you can probably guess, the device that has the finest division on its scale will produce the most precise measurements.
Accuracy – How close the measurements are to the correct value.

15. 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.

16. Graphs in Physics

17. 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:

18. Linear Graphs
Dependent Variable
Independent Variable

19. 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!

20. Linear Graphs – Slope of a Liney x y2
Rise =
Δy = y2 – y1
Slope =
Rise
Run y1
Run = Δx = x2 – x1
Slope =
y2 – y1
x2 – x1 x1 x2

21. 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 don’t use “x” & “y”. (We could use F and m, or d and t, or F and x etc.)

22. Linear Graphs: Area Under the Curve
Sometimes it’s 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?

23. 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.

24. 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

25. 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

26. 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

27. 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

28. Meaning of Slope from Equationsd t
Physicists may plot a graph of d vs t, but this would yield a non-linear graph in this case: d t2
To linearize the curve
Square the time

29. d = ½at2 Meaning of Slope from Equations
But what would the slope of a d vs t2 graph represent?
Let’s 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 don’t forget about the units! m/s2}

30. 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 a v
Let’s re-write the equation a little:
a = (1/r)v2
Therefore plotting a vs. v2 would let the slope be:
Slope = 1/r

31. Meaning of Slope from Equations
Example #2:
F = 2md/t2 F t
Plot F vs 1/t2
to linearize the graph F
1/t2
Slope = 2md
Go on to the worksheet on this topic

32. Classwork
Start the Linearization Worksheet in class and finish for homework.