1 Introduction to Measurement Physics 114 Eyres 6/5/2016 Module 1: Measurement 1 2 Significant Figures Count all non-zero digits Count zeros between non-zero digits Count zeros after the decimal if also after a non-zero digit Don t count zeros used only for column placement Try these: a. 34.607 b. 103.0 c. 23.01 d. 23.10 e. 230 f. 0.0002 g. 0.02070 It is important that you are able to count significant figures. In this class, you will need this primarily for your MPHW when it asks for a number to be input with a certain number of sig. figs. For example if you have 3.456 as your answer and MP wants the answer to 2 sig figs, then you need to type in 3.5. All non-zero digits are significant. Zero, however are only significant in certain situations (see slide). 3 Measured Values What is the Quantity & what Units? Position, Time, velocity, etc. Metric, US, small or large? How will you Measure it? Analog or Digital Directly: You have an instrument (ruler, clock) Indirectly: Calculate from direct (Also called: Derived) Reporting the Measured Quantity How accurate? (Uncertainty) Average +/- Uncertianty If Direction then the give angle in proper format Right Triangle Trig. may be necessary to find the angle. (More about this in chapter 3) When measuring and reporting values both the number (quantity) and unit must be considered To measure a quantity, different types of measuring devices are available: Analog and Digital. With these devices, direct measurements are read. When a new value is calculated from direct measurements, these are called indirect or derived values. When reporting these values, there are two considerations: First, the number of digits to report. Second the uncertainty. Finally, if the value has direction, then that must be reported as well.
4 Measurement Devices Analog: You must read from a scale. Digital: You have a display screen with the value given. This slide shows examples of both analog and digital measuring devices. Analog: ruler, dial, mass balance, bathroom scale Digital: Bathroom scale, watch, thermometer. Question: Are all thermometers digital? 5 How Many Digits to Write Analog: Write digits until the one you must interpolate Digital: Write what is shown on the screen 8.04 8.68 Write: 126.4 Write: 9.4 Question: If the arrow was directly over the 8.0 mark, what would be reported? When considering how many digits to write follow these rules: Analog: Write all digits including the 1 st on where interpolation is required. In the example, one must interpolate to see that the right arrow is 8 parts beyond 8.6 and before 8.7. So the digit representing those 8 parts of a tenth must be reported giving 8.68. With digital, the number of digits is exactly what is displayed. 6 Uncertainty Only one reading is possible or reasonable: Two different rules for finding uncertainty. Single measure digital: Values are rounded by the device before we see them. You must specify the range of values that might have been rounded to the number shown. Single measure analog: Values are read and the last digit is an interpolation between marks. You must make a judgement call about how others might reasonably read they same value. Multiple measure (both digital and analog follow the same rule): In this case, it is possible to take several measurements. Using the many values, the uncertainty must be calculated. When finding the uncertainty to report, the method is determined by what type of measurement: single reading or multiple reading. It is generally preferable to use multiple measure. However, sometimes, that is not possible or practical. For example, if a bathroom scale is reading your weight, multiple measures may not be practical. You look at the scale and it reads 120 s and you look again and it reads 120 s. You may not wish to cover your eyes and reread the value 20 times.
Or, when driving down the road, I check my speed as I pass a checkpoint. I cannot check the speed as I pass that checkpoint again because I m no longer there. There are different rules for digital and analog when there is a single reading. 7 Uncertainty: Single measure digital Consider all the values that would round to the number on the display. Write: value +/- uncertainty Draw the pdf (probability distribution function) and label its center and both sides. Upper side is value + uncertainty Lower side is value - uncertainty Write: (126.4 +/- 0.05) 125.35 125.4 125.45 Question: If a digital reading is 12.9. What is the uncertainty? For single reading-digital: Consider all the values that would round to the value displayed. In the example all the values from 126.41 to 125.4499999 (call it 125.45 would round down to 126.4. Likewise all the values from 126.35 up to 126.39 would round up. Thus all the values between 126.35 and 126.45 are equally possible values that would have been rounded to give us the 126.4 that is displayed. The rectangle pdf (probability density function) is drawn to show the equal probability. The uncertainty is the distance from the reading to the largest (or smallest) value. 8 Uncertainty: Single measure analog Single measure analog Consider all the values that would likely be read by a very careful person making the reading in the same situation as you. In the example I debated in my mind between 9.4 and 9.5 and decided 9.4 was better. The indecision helps me think of the uncertainty as 0.1 instead of something larger like 0.2. Write: value +/- uncertainty range Draw the pdf (probability distribution function) and label its center and both sides. Upper side is value + range Lower side is value - range Write: (9.4 +/- 0.1) Question: If an analog reading is 34.5. Is it possible that the uncertainty could be +/- 0/05? 9.3 9.4 9.5 For Single reading analog: In this case, the uncertainty is a judgement call. It is always in the last digit of the reading but can vary depending on how hard it is to interpolate between the marks. The triangle pdf indicates that readings farther away from the one at the center of the triangle are less likely than the ones closer.
9 Uncertainty: Single Measure Analog Consider all the values that would likely be read by a very careful person making the reading in the same situation as you. In the example the marks are close together and the line farther from the scale. The reading is larger than 1.7 and less than 1.8. I chose 1.75 as the best reading I can make and selected the range as 0.05. Notice how this is NOT the same as a single DIGITAL reading. Notice the last digit read is in the same column as the uncertainty. Another possible reading would have been 1.75 +/- 0.02 Write: (1.75 +/- 0.05) 1.70 1.75 1.80 Notice that in this case of an analog reading, the marks are very close together making it hard to see the space between. In this case, the reading has been given an uncertainty of +/- 0.05 instead of +/- 0.02 or 0.01. Question: Do you think an uncertainty of +/-0.07 would be reasonable? Why or why not? 10 Uncertainty: Multiple Reading Decide how many digits to measure. At least 10 are needed. Draw Histogram or number line to check for outliers Look at spread & keep >90% You may need to make more measurements to test for multiple outliers. If you think you have 3, you need to make more measurements to a total of at least 30. Calculate Average Calculate Uncertainty by taking the largest difference from the average Round the Uncertainty and then the Reading Uncertainty for multiple reading situations requires more steps. There are basically four steps after making the reading: Check for outliers, Calculate the Average, Calculate the Uncertainty, Round both the reading and uncertainty properly. 11 Uncertainty: Multiple Reading Check for outliers 2 readings on the far right are outliers More than 20 readings were taken. Calculate the average: Exclude outliers in the calculation 114.1 in this case (see arrow) Calculate Uncertainty Choose largest distance from average Highlights in green box Round Uncertainty to 2 sig figs. Reading to same decimal place. Min Avg Max 7 6 5 4 3 2 1 0 113.8 113.9 114 114.1 114.2 114.3 114.4 114.5 Avg-Min Max-Avg 114.6 114.7 114.8 Choose Largest Here is an example. Many length measurements were taken and all were read to the same number of digits. Now for the 4 steps to finish with the uncertainty. Step 1: histogram was drawn to check for outliners. If you do not know how to do a histogram, you should instead do a number line. Do NOT do a histogram in Excel unless you have been taught. The bar chart is NOT a histogram. Step 2: Calculate the average. Do not include the outliers.
Step 3: Subtract Max (not including outliers) Average. And Average Min (not including outliers) The larger of these two numbers is to be reported as the uncertainty. Step 4: Round the uncertainty to 2 sig. figs. Round the reading to the same decimal place as the uncertainty. Read your lab directions carefully for a complete example as well as more about rounding and checking for outliers. 12 Uncertainty: Multiple Reading N=24 N=22 (92%) 7 6 5 4 3 2 1 0 113.8 113.9 114 114.1 114.2 114.3 114.4 114.5 114.1 +/- 0.3 114.2 +/- 0.6 114.6 114.7 114.8 Notice that the green uncertainty is better. The yellow one is very wide due to the outliers. Two outliers are allowed because the students went back to take more measurements to make the total greater than 20. This slide shows the average and uncertainty calculated with the outliers removed and again with the outliers included. Notice that the outliers cause the average to be higher than the grouping of readings would indicate. Also, the uncertainty is very large. Strive for small uncertainties. If it looks like you have lots of outliers, you may have large uncertainty and need to think about how to make more accurate readings. If you cannot make better readings, then take more to see what readings are actually outliers.