
"Physics is Fun" Feimer's Physics Page 
__________________________ Physics Topics
Introduction to Math
and Measurement
Displaying (and Processing) Data
Making Tables and Graphs
with Excel Spreadsheet
Study
of Planetary Motion and the Law of Universal Gravity
Study
of Momentum and the Conservation of Momentum
Study of Work,
Power, and Energy
Study of Waves, Energy, Light, and Sound __________________________ Astronomy Topics 
An Introduction to Math and Measurement "The fundamentals: 'scientific skills, tools, and processes used by
Table of Contents and Links by topic below. Intrinsic and Extrinsic Properties Working with Metric Measurements: 
Measurements in science and engineering are made in either
the Metric or the British (English) system. The engineering people in the
United States still often use the British system.
Regardless of what system is used measurements fall into one of two categories. These are called the intrinsic and extrinsic properties of measurements: In physics and chemistry an intrinsic property (or intensive property) of a system is a physical property of the system which does not depend on the system size or the amount of material in the system. By contrast, an extrinsic property (or extensive property) of a system does depend on the system size or the amount of material in the system. Examples of intrinsic properties are temperature, pressure and density. Examples of extrinsic properties are mass, volume and energy. 
1. Measurement in the science of Physics and Astronomy,
as in all of the sciences, today, are made in the metric system.
It is used as the basis for all measurements.
a) The metric system: A system of measurement that uses "standardized" definitions of the base unit for each category of measurement The metric system is measuring system defined in terms of the decimal system. Each category of measurement, such as length, has a base unit of measurement. (i.e. The base unit for length is the meter, the base unit for mass is the gram, the base unit for volume is the liter, and the base unit for time is the second.) For measurement values larger or smaller than each category's
base unit of measure the metric system uses prefixes. The more common prefixes
are:

b) Fundamental units of measure: The fundamental
units of measure are the measurements from which all other measurements
are derived. Or to put it another way, all other units of measure are defined
in terms of one or more of these measurements. Here is a list of the fundamental
units of measure.

c) Derived units of measure: The
derived units of measure are all of the units of measure other than the
fundamental units of measure. Derived units of measure are themselves derived
from one or more fundamental units of measure. Some examples of derived
are described here in the following table.

d) Working with metric measurements: In
any measurement system it is often necessary to be able to work with more
than one kind of measurement unit while doing math calculations. Units
of measurements are handled and manipulated in calculations by essentially
the same rules used for handling variable symbols. They can be added together,
multiplied by one another, subtracted from each other, and divided by one
another, just as any variable symbols such as x and y. Similarly, other
operations, such as raising measurement values to powers or finding their
roots, require that you treat the measurement units in the same manner
as you would any variable symbol as you do the calculation
> Rule for addition and subtraction: When adding or subtracting measurements, the units of measure must be the same. i.e. Two masses, a 5 kg mass and a 10.5 kg mass, are being
added together. These two measurements have the same unit so the numerical
values can be added resulting in a sum having the same unit as the individual
measurements.
> Rule for multiplication and division: When multiplying or dividing measurements, the units do not have to be the same. However, all units, except those that legitimately divide out (sometimes referred to as canceling), must be retained. i.e. An object travels 3,000 m in a time interval of 15
seconds. Using the equation for speed of an object, v = d / t, calculate
the speed of the object.
> Other operations: Always perform other math operations, when using measurement unit symbols, as you would any variable symbol values such as x and y. All units must remain in the answer along with the numerical value arrived at, unless they actually divide out, and are legitimately gone from the answer. An Example: A rectangular shaped object is placed
upon a scale and found to have a mass of 25 grams. The dimensions of the
object, measured with a ruler, were found to be 5 cm long, 3 cm high, and
2 cm tall. Calculate (1) the volume of the object and (2) the density of
the object.
(1st) Volume is found by the equation V = L x W x H (length,
width, and height, respectively). so the calculation looks like:
(2nd) Density is found by dividing the mass of the object
by its volume. The equation is D = M / V.
*The problem solving format places emphasis on displaying the solution to a problem in a logical series of steps. This approach emphasizes critical thinking skills and over time will improve your ability to solve problems quickly. Always remember that, if you cannot get the units of measurement in your solution (not the correct one) to work out correctly, there is no point picking up your calculator, because when the units of measurement do not work out correctly, there is something wrong in your choice of math operations you have chosen to solve the problem. This applies to all kinds of problems involving the use of math, not just problems using the metric system. 
2. Working with LARGE and small
measurements: Measurements can have reasonably
manageable numbers and be relatively easy to manipulate. On the other hand,
especially in the sciences, numbers associated with measurements can be
quite large, as in the case of Astronomical data, or numbers associated
with measurements can be quite small, as in the case of Quantum Theory.
To deal with the very large and the very small, mathematics has given us
exponential notation. Exponential notation as used in the sciences is often
called scientific notation. Scientific notation is defined as "A method
of writing or displaying numbers in terms of a decimal number between 1
and 10 multiplied by a power of 10. The scientific notation of 10,492,
for example, is 1.0492 × 10^4. The basics of scientific notation
is modeled in the following table.
*The number 0, itself, is not a power of ten. It can't be written as 10 to the something power. 
3. Displaying and processing data: Gathering,
recording, and processing data are skills associated with serious research
into any human endeavor. An important data chart compiled by using accurate
measurements produced by astronomers is the Planetary Data Chart. A version
of this chart appears below.
Data Charts such as the one above are referred to as data tables. A table provides a means of organizing and displaying data that has been collected through observations. 
4. Categories of Observations: Observations fall
into two categories: One of the types of observation is the qualitative
observation, while the other is the quantitative observation.
> The qualitative observation is a descriptive
observation. It involves a clear and concise description of an object or
circumstance. It does not involve the making of measurements (quantities).
Some examples of qualitative measurements are:
> The quantitative observation is a measurement.
It involves an accurate measurement expressed with as much precision as
the measuring tool being used will allow. Some examples of quantitative
measurements are:

5. Accuracy and Precision: Accuracy and
Precision
always come into play when anyone is making quantitative observations (measurements).
Accuracy is defined as "The ability of a measurement to match the actual (correct) value of the quantity being measured." It is a concept that deals with whether a measurement is correct when compared to the known value or standard for the particular measurement being made. When this comparison is being made using a percent, it is referred to a percent error. > Example of accuracy. Suppose that a carpenter is making a cabinet to fit in an exact space having a width of 1.15 m. Now what happens if in his effort to cut the materials he makes his cuts 1.14 m. What effect does this have on the making of the cabinet? Well on the first look, it may seem that this 0.01 difference is quite small. However, upon closer inspection, 0.01 m is a whole centimeter, which is close to one half of an inch. The cabinet will now show a gap of almost a half an inch, because the carpenter was not accurate in measuring. This why in terms of measurement, carpenters, etc., have always said "measure twice and cut once". In the above situation, more careful measurement would reveal the true (correct) width, either by measuring more carefully or by using a more precise ruler, perhaps one that measured to the 3rd or even the 4th decimal place should be used. However, when measurements are being made of the very large (astronomical distances) or of the very small (subatomic scale) for the very first time, accuracy is difficult to determine, because no one has access to correct answer. In this kind of a situation, we turn to precision and statistics to help us ascertain the usefulness of our measurements. Precision is defined as "The ability of a measurement
to be consistently reproduced." and "The number of significant digits to
which a value can be reliably measured." The first definition is the main
focus of scientists and engineers who are looking at the work of others.
After studying what the other people had done, they set out to test the
shared (often in the form of published reports) information to see whether
they can reproduce the results. Reproducible results are much more likely
to be accurate (correct) than results that cannot be reproduced. The second
definition is the main focus of anyone using a measuring device. How precise
or exact the measuring device is that you are using is based upon the degree
to which the measurement units are broken down (subdivided). Precision
is the concept which deals with the degree of exactness when using a specific
measuring tool and expressing a measurement made with that tool. The precision
of any measuring tool and the exactness of the measurement made with that
tool is limited by how precise the instrument is capable of being. the
precision of a measuring tool is based upon the smallest defined unit of
measure on the devise. The number of digits used to express a measurement
when using a tool whose degree of precision (smallest unit available) are
called significant digits. See the example below:

6. Handling Measurements in Calculations: The use
of significant digits in problem solving.
In all problem solving situations where measurements are
involved, not only should you carry units along with each math operation
performed, but you need to take into consideration the number of significant
digits being expressed in each measurement being used, so each and every
calculation using measurements is never expressed with more precision than
the measurements themselves. That would be absurd (or, if you prefer, ridiculous).
In working with measurements expressed in significant digits, there are
three things you always need to consider. They are:
The Rules of zeros (0's) used in identifying the
number of significant digits in a measurement. The rules are as
follows:
Significant Digits in Calculations: The precision in the answer to a calculation where new information is derived from existing information cannot be greater than the precision of the least precise measurement. Now the handling of this sort of situation where measurements are used in simple math operations is divided into two categories of math problems. They are the AdditionSubtraction problems and the MultiplicationDivision problems. > 1. Addition and Subtraction Problems: When
adding or subtracting measurements where precision and significant digits
are being used, the answer can be no more precise than the measurement
with the least amount of precision. In the following example you will be
rounding off to the second decimal place (the onehundredths column), because
nothing is known about the 3rd decimal value in the measurement 3.11 kg.
This makes the 3rd decimal in the final answer too uncertain to be recorded.
So we round off correctly to the 2nd decimal place. In Addition and subtraction
it is the measurement with the least number of decimal places that affects
how much you are to round off to.
> 2. Multiplication and Division Problems: When
multiplying or dividing measurements where precision and significant digits
are being used, the answer can be no more precise than the measurement
with the least amount of precision. In the following example you will be
rounding off to only two digits, because in multiplication (and division)
you round off to the number of digits in your answer equal to the value
with the least amount of significant digits being used in the problem.
In multiplication and division it is the measurement with the least number
of significant digits that affect how you are to round off.
