Chapter
1
A Physics Toolkit
Chapter
1
A Physics Toolkit
In this chapter you will:
Use mathematical tools to
measure and predict.
Apply accuracy and
precision when measuring.
Display and evaluate data
graphically.
Section
1.1
Mathematics and Physics
In this section you will:
Be able to answer the question, what is physics?
Review the algebra required for this class.
Learn the GUESS problem solving strategy.
Use and do conversions with SI units.
Evaluate answers using dimensional analysis.
Perform arithmetic operations using scientific notation.
Learn about and use significant figures.
Section
1.1a
Mathematics and Physics
What is Physics?
Physics is a branch of science that involves the study of the
physical world: energy, matter, and how they are related.
Learning physics will help you to understand the physical
world.
Physics is considered the basis for all other sciences:
- Biology, Chemistry, Astronomy, Geology, etc.
Physics is the fundamental science.
Section
1.1
Mathematics and Physics
What does one do as a physicist?
Many research physicists work in environments where they
perform basic research in industry, research universities,
and astronomical observations.
Physicists who find new ways to use physics are often
employed by engineering, business, law, and consulting
firms.
Physicists are also extremely valuable in areas such as
computer science, medicine, communications, and
publishing.
Finally, many physicists who love to see young people get
excited about physics become teachers.
Section
Mathematics and Physics
1.1
What jobs do non-physicists hold that use physics
every day?
Every job has some relation to physics!
Athletes
- The laws of motion to lift, throw, push, hit, tackle, run, drag,
jump, and crawl.
- The more an athlete and coach understand and use their
knowledge of physics in their sport, the better the athlete
will become.
Section
1.1
Mathematics and Physics
Automotive mechanics
◦ areas such as optics, electricity and magnetism,
thermodynamics, and mechanics play greater roles as
vehicles become more and more complex.
CAT scan, computed tomography, and magnetic resonance
imaging (MRI)
◦ technicians in hospitals and medical clinics who use
such technology must have an understanding of what Xrays and magnetic resonance imaging are, how they
behave, and how such high-tech instruments are to be
used.
Section
1.1
Mathematics and Physics
Mathematics in Physics
Mathematics is the language of science.
In physics, equations are important tools for modeling
observations and for making predictions.
Section
Mathematics and Physics
1.1
Order Of Operations
Simply: 2 + 5 x
√
Correct: 2 + 5 x
X
Incorrect: ( 2 + 5 ) x = 7 x
Section
Mathematics and Physics
1.1
Order Of Operations : PEMA
P arentheses and Brackets
E xponents
M ultiplication and Division (from left to right)
A ddition and Subtraction (from left to right)
Section
Mathematics and Physics
1.1
Order Of Operations : Example 1
Step 1:
parentheses
Step 2:
exponent
Step 3:
multiplication
Step 4:
addition
Step 5:
solution
Section
1.1
Order Of Operations : Example 2
Step 1:
Step 2:
distribute 8 into (x + 1)
Step 3:
remove 1st parenthesis
Step 4:
combine like terms
Step 5:
within parenthesis, left to right, first
comes division
Step 6:
then multiplication
Step 7:
simplify exponent
Step 8:
solution
Section
1.1
Mathematics and Physics
Algebra Review with Physics
Variables are used to represent concepts.
ex: d for displacement, t for time, p for momentum
Units also are abbreviated.
ex: m for meters, s2 for seconds squared
Do not confuse variables with units.
Subscripts are used to give more information about a variable.
ex: vave : average velocity
vi : initial velocity
Section
Mathematics and Physics
1.1
Problem Solving Strategy
Given: Write down given or known quantities. Draw a picture.
Unknown: Write down the unknown variable.
Equation: Find an applicable equation. Isolate the unknown
variable.
Substitute numbers with units into the equation. Solve.
Sense: Are the units correct? Does the answer make sense?
Section
1.1
Mathematics and Physics
Example: Electric Current
The voltage across a circuit equals the current multiplied by the
resistance in the circuit. That is, V (volts) = I (amperes) × R (ohms).
What is the resistance of a light bulb that has a 0.75 amperes current
when plugged into a 120-volt outlet?
Step 1: Given - Write the given quantities with units.
Step 2: Unknown - Identify unknown variable.
Given:
Unknown:
I = 0.75 amperes
R=?
V = 120 volts
Section
Mathematics and Physics
1.1
Electric Current
Step 3: Equation. Rewrite the equation so that the unknown value is
alone on the left.
V = IR
IR = V
Reflexive property of equality.
Divide both sides by I.
Step 4: Substitute 120 volts for V, 0.75 amperes for I. Solve.
R=
120 volts
0.75 amperes
R = 160 
Resistance will be measured in ohms.
Section
Mathematics and Physics
1.1
Electric Current
Step 5: Sense
Are the units correct?
1 volt = 1 ampere-ohm, so the answer in volts/ampere is in
ohms, as expected.
Does the answer make sense?
120 is divided by a number a little less than 1, so the answer
should be a little more than 120.
Section
Mathematics and Physics
1.1
SI Units
Units are CRITICAL in physics.
It is the units that give meaning to the numbers. It is helpful to
use units that everyone understands.
Scientific institutions have been created to define and regulate
measures.
The worldwide scientific community and most countries currently
use an adaptation of the metric system to state measurements.
Section
Mathematics and Physics
1.1
SI Units
The Système International d’Unités, or SI, uses seven base
quantities, also called the fundamental units.
Section
Mathematics and Physics
1.1
SI Units
The base quantities were originally defined in terms of direct
measurements. Other units, called derived units, are created by
combining the base units in various ways.
ex: speed is measured in meters per second (m/s)
The SI system is regulated by the International Bureau of Weights
and Measures in Sèvres, France.
This bureau and the National Institute of Science and Technology
(NIST) in Gaithersburg, Maryland, keep the standards of length,
time, and mass against which our metersticks, clocks, and
balances are calibrated.
Section
Mathematics and Physics
1.1
SI Units
Measuring standards for kilogram and meter are shown below.
Section
Mathematics and Physics
1.1
SI Units
You probably learned in math class that it is much easier to
convert meters to kilometers than feet to miles.
The ease of switching between units is another feature of the
metric system.
To convert between SI units, multiply or divide by the
appropriate power of 10.
Section
1.1
Mathematics and Physics
Prefixes are used to change SI units by powers of 10, as shown
in the table below.
Section
1.1
Mathematics and Physics
Scientific Notation
A number in the form a x 10n is written in scientific notation
where 1 ≤ a < 10, and n is an integer. (An integer is a whole
number, not a fraction, that can be positive, negative, or zero.)
When moving the decimal point to the right, you reduce the
exponent when using scientific notation.
Right – REDUCE
When moving the decimal point to the left, you make the
exponent larger when using scientific notation.
LEFT – LARGER
Common powers of ten include 100 = 1, 101 = 10, 102 = 100, etc.
Section
1.1
Mathematics and Physics
Scientific Notation: Example 1
Write 7,530,000 in scientific notation.
The value for a is 7.53
(The decimal point is to the right of
the first non-zero digit.)
So the form will be 7.53 x 10n.
7,530,000. = 7.53 x 106
(Move the decimal point 6 places to
the left; exponent gets larger.)
Section
1.1
Mathematics and Physics
Scientific Notation: Example 2
Write 0.000000285 in scientific notation.
The value for a is 2.85
(The decimal point is to the right of
the first non-zero digit.)
So the form will be 2.85 x 10n.
0.000000285 = 2.85 x 10-7
(Move the decimal point to the right
7 places; exponent gets smaller.)
Section
1.1
Mathematics and Physics
Dimensional Analysis
You often will need to use different versions of a formula, or use
a string of formulas, to solve a physics problem.
To check that you have set up a problem correctly, write the
equation or set of equations you plan to use with the appropriate
units.
Before performing calculations, check that the answer will be in
the expected units.
For example, if you are finding a speed and you see that your
answer will be measured in s/m or m/s2, you know you have
made an error in setting up the problem.
This method of treating the units as algebraic quantities, which
can be cancelled, is called dimensional analysis.
Section
1.1
Mathematics and Physics
Dimensional Analysis
example: Calculate the distance a car travels when it is moving at a
velocity of 20 meters per second for 10 seconds.
Use the formula:
Use dimensional analysis:
distance = velocity
?
meters = meters
second
x time
x seconds
Treat the units as if they were algebraic quantities.
?
meters = meters x seconds
second
Seconds in the numerator cancel seconds in the denominator.
The formula is therefore dimensionally correct. meters = meters 
Section
1.1
Mathematics and Physics
Dimensional Analysis
Dimensional analysis also is used in choosing conversion
factors. This is also known as the factor-label method.
A conversion factor is a multiplier equal to 1. For example,
because 1 kg = 1000 g, you can construct the following
conversion factors:
Section
Mathematics and Physics
1.1
Factor-Label Method
Choose a conversion factor that will make the units cancel,
leaving the answer in the correct units.
For example, to convert 1.34 kg of iron ore to grams, do as
shown below:
1.34 kg
1000 g
1 kg
= 1,340 g
Section
Mathematics and Physics
1.1
8760 hours/year
Factor-Label Method: Example 2
Convert 1 year to hours.
1 year
365 days
24 hours
= 8,750 hours
1 year
1 day
Section
1.1
Mathematics and Physics
8760 hours/year
Warmup Problem
Pressure is defined as force divided by area of contact ( P = F / A).
What pressure must you apply to your pen in order to create a force
of 0.25 N on a piece of paper, if the tip of the pen has a surface
area of 3 mm2 touching the paper?
Section
1.1
Mathematics and Physics
Rules for Rounding Off
1. In a series of calculations, carry the extra digits through to the final
answer, then round off. Rounding only occurs ONCE in a calculation!
2. If the digit to be removed is:
 <5, the preceding stays the same.
example: 1.33 rounds to 1.3
 5 or greater, the preceding digit increases by 1.
example: 1.36 rounds to 1.4.
Example: Round 62.5347 to four significant figures.
Look at the fifth figure. It is a 4, a number less than 5. Therefore, simply
drop every figure after the fourth, and the number becomes
62.53
Section
Mathematics and Physics
1.1
Significant Digits (Significant Figures or Sig Figs)
Definition: All the valid digits in a measurement, the number of which
indicates the measurement’s precision
(degree of
exactness).
Do not count sig figs for non-measurement quantities such as
counting (4 washers) or exact conversion factors (24 hrs. in 1 day).
Use the Atlantic & Pacific Rule to determine the sig figs.
PACIFIC
ATLANTIC
OCEAN
OCEAN
Section
Mathematics and Physics
1.1
The Atlantic /Pacific Rule for Sig Figs
If the…
• Decimal is Present
– Count all digits from the Pacific side from the first non-zero digit.
• Decimal is Absent
– Count from the Atlantic side from the first non-zero digit.
– Trailing zeros are indeterminate; they may or may not be
significant. Use scientific notation to remove the ambiguity.
Section
Mathematics and Physics
1.1
How many sig figs are there?
Count all the digits from the first non-zero digit.
PACIFIC OCEAN
ATLANTIC OCEAN
Decimal point present
Decimal point absent
5 sig figs 705.00 g
4 sig figs 523.0 g
5 sig figs 0.0098070 mm
3 sig figs (and 1 indeterminate)
2130 m
3 sig figs (and 3 indeterminate)
706, 000 g
9,010, 000 km 3 sig figs (and 4 indeterminate)
Section
1.1
Mathematics and Physics
Significant Digits
Using the Atlantic rule, we can’t be sure if trailing zeros are significant
or not. To specify the exact number of sig figs, use scientific notation.
Exponents do not count towards significant digits.
Example: 9,010, 000 km (3 sig figs and 4 indeterminate).
To write this number indicating:
3 significant digits: 9.01 x 106 km
4 significant digits: 9.010 x 106 km
5 significant digits: 9.0100 x 106 km
Section
Mathematics and Physics
1.1
Operations with Significant Digits
When you perform any arithmetic operation, it is important to
remember that the result never can be more precise than the leastprecise measurement.
Add / Subtract: Round to the least number of DECIMAL places as
determined by the original calculation. (All numbers must be the
same power of 10).
Example:
23.1
4.77
125.39
+ 3.581
156.841
Round to 156.8
(one decimal place)
Section
1.1
Mathematics and Physics
Significant Digits – Addition / Subtraction
Example: Add 5.861 dL + 2.614 L + 3.5 mL
Convert to the same units and power of 10. Add in column form.
5.861 dL = 5.861 x 10-1 L = 0.5861 L
2.614 L
3.5 mL = 3.5 x 10-3 L =
 fewest decimal places
0.0035 L
3.2036 L
Round to the least amount of decimal places
3.204 L
Section
1.1
Mathematics and Physics
Significant Digits – Multiplication / Division
Multiplication / Division: Round to the fewest number of SIGNIFICANT
FIGURES.
Example:
(3.64928 x 105) (7.65314 x 107)
(5.2 x 10-3) (5.7254 x 105)
least precise measurement
= (3.64928 x 105) x (7.65314 x 107) ÷ (5.2 x 10-3) ÷ (5.7254 x 105)
= 9.3808 x 109
= 9.4 x 109
because the least precise measurement has 2 sig figs.
Section
Mathematics and Physics
1.1
Significant Digits – Combination Operations
When doing a calculation that requires a combination of
addition/subtraction and multiplication/division, use the multiplication
rule.
Example:
slope = 70.0 m – 10.0 m = 3.3 m/s
29 sec – 11 sec
29 sec and 11 sec only have two significant digits each, so the
answer should only have two significant digits
Section
1.1
Mathematics and Physics
Multistep Calculations
Do not round to 580 N2 and 1300 N2
Do not round to 1800 N2
Final answer, so it should be rounded to two sig figs.
Section
Section Check
1.1
Question 1
The potential energy, PE, of a body of mass, m, raised to a height,
h, is expressed mathematically as PE = mgh, where g is the
gravitational constant. If m is measured in kg, g in m/s2, h in m, and
PE in joules, then what is 1 joule described in base unit?
A. 1 kg·m/s
B. 1 kg·m/s2
C. 1 kg·m2/s
D. 1 kg·m2/s2
Section
Section Check
1.1
Answer 1
Answer: D
Reason:
Section
Section Check
1.1
Question 2
A car is moving at a speed of 90 km/h. What is the speed of the car
in m/s? (Hint: Use Dimensional Analysis)
A. 2.5×101 m/s
B. 1.5×103 m/s
C. 2.5 m/s
D. 1.5×102 m/s
Section
Section Check
1.1
Answer 2
Answer: A
Reason:
Section
Section Check
1.1
Question 3
Which of the following representations is correct when you solve
0.03 kg + 3.333 kg?
A. 3 kg
B. 3.4 kg
C. 3.36 kg
D. 3.363 kg
Section
Section Check
1.1
Answer 3
Answer: C
Reason: When you add or subtract, round to the least number of
decimal places.