UNIT: MATH REVIEW
Physics often uses math as its language. This chapter will review old techniques and introduce some new techniques (or tools). Central in problem solving is the use of graphs and equations to represent the results of observations and experiments. How are equations found? Many times a graph gives a clue to this by displaying the data in a very organized way.
The nature of physics demands that you understand strengths and limitations of the measurements you take. It is also essential for you to be familiar with interpreting and reporting data in the form of tables and graphs.
*It is required that you have access to a scientific calculator every day.
THE MEASURE OF SCIENCE
Metric System
The world-wide scientific community uses the SI system to make measurements so that system will be stressed here. SI is an adaptation of the metric system which was devised by the French in 1795. Presently NIST (National Institute of Standards and Technology) keeps the physical standards of length, mass, and time for the United States. These three quantities are considered fundamental because many other quantities can be described as functions of them.
The standard of time is the second (s). It was first defined as 1/86400 of the mean(average) solar day. Now (since 1967) it is defined as 9,192,631,770 oscillations of one type of radiations emitted by a Cesium-133 atom.
The standard SI unit of length is the meter (m). It was first defined as 1/ten millionth of the distance from the North Pole to the equator, measured along a line passing through Lyons, France. In 1960, it was redefined as a multiple of the wavelength of light emitted by Krypton-86. In 1982, it was redefined as the distance light travels in 1/299,792,458 seconds in a vacuum.
The third fundamental unit measures mass. The kilogram (kg) is the only unit not defined in terms of properties of atoms. It is the mass of a platinum-iridium alloy cylinder. A copy of the original is kept at NIST.
Four other fundamental units will be introduced in the book at appropriate times. They are:
current ampere A
temperature Kelvin K
amount mole mol
intensity of light candela cd
*The important thing is that units are obtained by physicals experiments. All other units are derived from these. A wide variety of other units will be encountered which will be combinations of these fundamental units. Ex. m/s, N = kgm/s 2 J = kgm 2/ s 2
The metric system is a decimal system. It is based on powers of ten and prefixes are used to indicate this.
Zepto 1024 Z zetta 10-24 z
Yocto 1021 Y yotta 10-21 y
exa 1018 E atto 10-18 a
peta 1015 P femto 10-15 f
tera 1012 T pico 10-12 p
giga 109 G nano 10-9 n
mega 106 M micro 10-6 µ
kilo 103 k milli 10-3 m
hecto 102 h centi 10-2 c
deka 101 da deci 10-1 d
*Raising an exponent by 1 means increasing the quantity by 10 times.
SCIENTIFIC NOTATION
The method of expressing numbers by writing them in a shortened form by writing decimal places as powers of ten is called exponential notation. Scientific notation is based on exponential notation. The numerical part of measurement is expressed as a number between 1 and 10 and then is multiplied by a whole-number power of 10. (2000 = 2 x 10 3)
Arithmetic Operations in Scientific Notation
Addition/Subtraction: Numbers must have the same exponent before doing operations. If not, then the numbers must be adjusted until they do have the same exponent of ten.
Multiplication: Multiply the numbers and add the exponents of ten.
Division: Divide the numbers and subtract the exponents of ten.
*Some calculator displays show numbers such as 2.3564 E8. This means 2.364 x 108.
CERTAINTY
No measurement is certain. All devices are subject to external influences. We can minimize uncertainties by following rules. There are also some errors for which to watch.
1. Changes in environment may affect measuring devices.
a. temperature
b. magnetic and electric fields
c. humidity
d. wind
2. Personal errors such as bias, insufficient expertise, insufficient care may also add to problems.
3. Parallax viewing of devices may be a problem. This is the apparent shift in position when an instrument is viewed from various angles.
4. There are also limitations on the device itself.
5. Errors due to method may make accuracy impossible.
Percentage error is the (actual error/standard value) x 100. Actual error is the amount by which the experimental value differs from the true one. Alone it does not give a true index of the precision of the measurement. The ratio of the actual error to the true value does indicate the precision of the measurement.
Percentage difference is found by subtracting two measurements that one wishes to compare and then dividing the result by the average of the two.
Uncertainty may be obtained by calculating the average deviation. To do this take the absolute value of the difference between the mean and the individual values. Then average these deviations. The expression should look like this:
Q = M+ d
Otherwise the measuring instrument is used to determine the uncertainty.
Accuracy and Precision
Precision is the degree of exactness to which the measurement of a quantity can be reproduced. It is limited by the smallest division on the measurement scale.
Thus readings such as (3.001+0.001) x 108 may be seen if a student measures the speed of light at 3.000 x 108m/s, 3.001 x 10 8m/s, and 3.002 x 10 8m/s. The precision is 0.001 x 10 8m/s.
The accuracy of a measurement describes how well the result agrees with an accepted value. If the accepted value of c = 2.998 x 108m/s, then the accuracy is 0.003 x 108m/s.
It is possible to be precise but not accurate. Precision is based on a number of measurements and accuracy is a comparison of one measurement to an accepted value. Precision is based only on the smallest division of the instrument. Accuracy is based on the performance of the instrument ( and observer) so the calibration must be checked often ( how close it is to an accepted standard).
Because precision is limited, the number of valid digits is limited also. The valid digits are called significant digits. They include all of the digits which can be read from an instrument plus one estimated digit. The rules are:
1. All nonzero digits are significant.
2. All final zeros after the decimal are significant.
3. Captured zeros are significant.
4. Beginning zeros are not significant.
5. Ending zeros before a decimal may or may not be significant.
186000 has at least 3 but may have 6 ---undetermined
6. Adding or subtracting: The result cannot be more precise than the least precise number. The answer may have only the number of decimal places that the number with the fewest decimal places has.
7. Multiplying and dividing: The result may have only the number of significant figures as there are in the least precise number.
8. Significant figures are used in measurements not when counting. An exact number such as 12 dozen has an infinite number of significant digits.
Displaying Data
A graph is a way of displaying data that can help one see patterns, can help one make predictions, and can organize data to show trends in behaviors. There are several details that make a graph useful. The independent variable is the variable controlled by the experimenter. Other variables are dependent because they are controlled by the independent variable.
To plot data:
1. Independent variable is plotted on the x-axis.
2. Dependent variable is plotted on the y-axis.
3. Determine the range needed to display each variable. The plot should take up about 3/4 of the graph.
4. Decide if the origin is to be included or if you need only a section of a quadrant. Space data out by making spaces be convenient units.
5. Number and label axes.
6. Plot points with o or D or other symbol.
7. Draw the best straight line or a smooth curve. DO NOT connect the dots.
8. Give the graph a title.
Relationships
If the x and y values vary linearly with each other, then the relation is said to directly proportional and linear. The equation for such a relation is
y = mx + b
where m is the slope and b is the y-intercept. These two values are constants which come from the graph. The slope is the ratio of the vertical change to the horizontal change. To find it select 2 points (not data points) on the best straight line that you have drawn and substitute them into the point slope formula.
m = rise = Dy
run = Dx
The y-intercept is where the function crosses the y axis(x = 0).
*If y decreases with an increase in x then the slope will be negative.
If the graph is a smooth line curving upward then the relation is quadratic meaning that one variable depends on the square of another.
y = kx2
The shape of the curve is a parabola.
If the graph is a smooth line curving downward and over then the relation is an inverse one, meaning that one variable depends on the inverse of another.
y = k(1/x) or xy = k
Manipulating Equations
It is important the realize that any equation may be solved to yield that answer to any variable. Many times it is simpler to solve for the desired variable and then to substitute in the numbers to obtain a value. Other times it may be necessary to substitute in one set of symbols for another symbol in order to obtain a usable equation. This process is also called deriving.