Units and Measurement Introduction to Measurement One of the most important steps in applying the scientific method is experiment: testing the prediction of a hypothesis. Typically we measure simple quantities of only three types: mass, length, and time. Occasionally we include temperature, electrical charge or light intensity. It is amazing, but just about everything we know about the universe comes from measuring these six quantities. Most of our knowledge comes from measurements of mass, length, and time alone. We will use the standard used by the international scientific community for measuring these quantities: the SI metric system. A measurement without a unit is meaningless! For more information, check out these websites: SI Base Units: http://physics.nist.gov/cuu/units/units.html Metric prefixes: http://physics.nist.gov/cuu/units/prefixes.html Conversion of Units Often it is necessary to convert from one unit to another. To do so, you need only multiply the given quantity by a conversion factor which is a ratio equal to 1, derived from definitions. For example, there are 100 centimeters in a meter. You can make a conversion factor out of that definition so that if you need to convert 89cm to meters, simply multiply by the conversion factor, which is equal to 1: You can make conversion factors out of the definitions listed in the tables above and in your book. Please always have leading zeros on decimals!! (0.89 and NOT.89) Precision & Accuracy 1m 100cm=1m =1 100cm 1m 89cm = 0.89m 100cm By their nature, measurements can never be done perfectly. Part of the error in making measurements may be due to the skill of the person making the measurement, but even the most skillful among us cannot make the perfect measurement. Basically this is because no matter how small we make the divisions on our ruler (using distance as an example) we can never be sure that the thing we are measuring lines up perfectly with one of the marks. Therefore the judgment of the person doing the measurement plays a significant role in the accuracy and precision of the measurement. Accuracy: Accuracy describes the nearness of a measurement to the standard or true value, i.e., a highly accurate measuring device will provide measurements very close to the standard, true or known values. Example: in target shooting a high score indicates the nearness to the bull's eye and is a measure of the shooter's accuracy. Precision: Precision is the degree to which several measurements provide answers very close to each other. It is an indicator of the scatter in the data. The lesser the scatter, the higher the precision. Ideally, we want to make measurements that are both accurate AND precise. However, we can never make a perfect measurement. The best we can do is to come as close as possible within the limitations of the measuring instruments.
Uncertainty Since we can never make a perfect measurement, every measurement is approximate. Therefore it is important to always report the amount of confidence we have in our measurements, what we call experimental uncertainty. For example, you may estimate the length of the lab bench to be 5 meters give or take a meter. The give or take part is an expression of your confidence in your estimate. In scientific measurements we say plus or minus but it means the same as give or take. We write that our measurement of the length, represented by L is : L 5 m /-1m If you are using a scale such as a ruler to measure the length of an object, then your uncertainty is usually estimated to be one tenth the smallest division. For example, this bug has a length between 1.54 and 1.56 in or L 1.55 in /-0.01in. The 1.55in is the average measure and the 0.01in is the uncertainty. Discrepancy/Error An experimental error is not a mistake! It is the difference between a measurement and an accepted value or theoretically predicated value of something. For example, if you determine from an experiment that the acceleration due to gravity is 10 m/s 2 then the error is the difference or DISCREPENCY between that value and the accepted value of 9.8m/s 2, or 0.2m/s 2. The error can also be expressed as a percent: 10 9.8 % error 100% 2% 9.8 Comparing or discussing your results means to use discrepancy and percent discrepancy. You also want to ask if the true or predicted value falls within your experimental value +/- you uncertainty!! Here is a sample: The experimental value of g (8.6+/-0.5)m/s^ is off from the accepted value of 9.80 m/s^2 by 1.2 m/s^2, or 12.2%. The true value does not fall within our experimental range of values. Sources of Experimental Uncertainty Systematic error the idea that a piece of equipment may not be calibrated or functioning correctly, or that conditions are not as you assume. All data points will vary in the same direction (all too high or too low). The way to deal with systematic error is to examine possible causes inherent in your equipment and/or procedures, and whether they may be corrected. Human error (aka mistakes) the idea that humans mess up. The way to deal with human error is to repeat the measurement. Uncertainty can be expressed in two ways: absolute or fractional. Absolute: uses same units as the measured quantity: (21.3 +/- 0.2) cm Fractional: the uncertainty is expressed as a fraction or a percent of the measured quantity: 0.2cm/21.3cm =.009 =.9% Final measure: 21.3cm +/-.9%
Statistical Uncertainty Random fluctuation the idea that the actual quantity may vary slightly around some central value. It is assumed that the variations go in both directions equally. The way to deal with random fluctuation is to take several measurements of the same quantity and calculate the average. The uncertainty is given by the average deviation.