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Physics Phundamentals

     

Essential Questions:

  • What is the difference between accepted value and measured value?
  • What are the appropriate measurement units in physics?
  • How do you calculate using significant figures?
  • What is the importance of graphical relationships?
  • What is the difference between scalar quantities and vector quantities?
  • How do you calculate the resultant of vector quantities in one and two dimensions?
 

Table of Contents

Uncertainty and Significant Figures
Scientific Notation
Significant Figure Mathematics
Metric System and Physics Measurement
Unit Conversions
Graphical Relationships
Vectors

 

Uncertainty and Significant Figures

Uncertainty and Sig Fig Notes


Trial of Pyx

Dimensional Lumber has a tolerance (uncertainty) of ± 1/32 "

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Number of Sig Figs
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The concept of uncertainty in measurement has been documented as far back as 1000 AD with the Trial of Pyx at which it was all minted coins in Britain had to meet specific standards. In particular, it stated that a particular coin had an allowance of 5 grains1. (0.324 grams). Now many scientists have stated that nonnumerical knowlege is unimportant; Lord Kelvin went so far as to label it "meagre and unsatisfactory." This view was shared by many scientists including Leonardo DaVinci and Francis Bacon.

In terms of science, however, uncertainty simply states how accurately a particular measuring tool is. Sometimes, uncertainty is stated in the form:

0.500 m ± 0.005 m

So for example a piece of wood is measured it would be exactly 50 cm long with an uncertainty of ± 0.005 m or 0.5 millimeters. this means that the measuring tool (for example, a meter stick can not be read any more accurately than one half a millimeter. Now consider the Swipe Ruler to the left:

A millimeter is the distance between the two smallest lines on the lower side of the ruler. It is difficult to get an accurate measurement between those two lines, so it is safest to say that the accuracy of the tool is 0.5 millimeters.

The other interesting thing about uncertainty is that a measurement can only be one decimal point beyond the smallest physical measurement. This means that for the case of a millimeter, the only possible smallest measurements are 0.1 millimeter ( 0.0, 0.1, 0.2, etc), 0.2 millimeters (0.0, 0.2, 0.4, etc.) and 0.5 millimeters ( 0.0, 0.5, etc.). You cannot use any of the other divisors within a millimeter because they are not evenly divisible with a single decimal.

Uncertainty is really the important factor in determining the number of significant figures. Significant figures for a number is determined by the number of digits physically shown on a measuring tool plus one estimated one. Using the picture below:

The gray bar is 11.66 cm ± 0.02 cm. Now the gray bar could also be 11.65 cm ± 0.05 cm. Some people might even say that it is 11.65 cm ± 0.01 cm, however, based on the size of the space between 11.60 cm and 11.70 cm, it seems unlikely that it can be subdivided 10 times equally using the naked eye. So for this gray bar, the number of significant figures in the measurement is 4. With the least significant figure being either a 6 or a 5 depending on the uncertainty of the measurement.

There are 4 basic rules for determining the number of significant figures. These rules are based on the type of number and its position within the actual measurement. The rules for the number of significant figures are:

  1. Nonzero numbers (numbers from 1-9) are always significant.
  2. Sandwiched zeros (zeros between non-zero numbers) are significant.
  3. Trailing zeros (zeros at the end of a number) are significant if there is a decimal point.
  4. Leading zeros (zeros at beginning of a number) are NOT significant.

Examples:

Number
Signicant Figures
Rule
12345
5
1
1020304
7
2
12.40
4
3
0.00565
3
4

 

Scientific Notation

Scientific Notation

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The term Scientific Notation was first used in 1961, however, the concept of base ten numbers have been around for as far back as the time of the ancient Greeks. In particular, Archimedes developed a base ten (exponential) measuring system in order to count the number of grains of sand needed to fill the universe as a part of the Sand Reckoner. Rene DeCartes also created a similar number system in 1631.

The important thing about scientific notation is its ability to easily specify very large or very small numbers. The general form of scientific notation is:

n.dddd x 10y

where n is the most significant figure,
d represents all the other digits in the number, and
y represents the exponent of of the number.

A number greater than zero will have a positive exponent and a number less than zero will have a negative exponent. In physics numbers are high as x 1024 and as small as x 10-6 are not unreasonable in particular concepts. As a physics student it is necessary to know how to convert a number into and out of scientific notation as well as put numbers in a common exponent to accurately perform significant figure mathematics.

To convert into scientific notation, you want to move a decimal point either to the left or right in order to get the number into the format specified above. The rules for conversion are:

  1. If you move the decimal point (whether it originally exists or not) to the left, the number gets smaller and the exponent must increase.
  2. If you move the decimal point to the right (typically it already is in the number), the number gets larger and the exponent must decrease.
  3. The number of significant figures in the original number must be included in the number in scientific notation.

For example:

1234609000 => 1.234609 x 1010
0.040508900 => 4.0508900 x 10-2


Significant Figure Mathematics

Sig Fig Math  

Metric System and Physics Measurements

   

Unit Conversions (Dimensional Analysis)

Unit Conversions Notes

Unit Conversion Practice

Lab - Do You Measure Up?

Significant Figure and Unit Conversions

 

Graphical Relationships

Graphical Relationship Notes

Graphing using Open Office Calc

 

Vectors