SPH3U1 Lesson 03 Introduction. 6.1 Expressing Error in Measurement

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1 SIGNIFICANT DIGITS AND SCIENTIFIC NOTATION LEARNING GOALS Students will: 6 ERROR Describe the difference between precision and accuracy Be able to compare values quantitatively Understand and describe how error arises from measurement Review the trigonometry ratios 6.1 Expressing Error in Measurement In everyday usage, "accuracy" and "precision" are used interchangeably, but in science it is important to make a distinction between them. Accuracy refers to the closeness of a measurement to the accepted value for a specific quantity. Precision is the degree of agreement among several measurements that have been made in the same way. Think of shooting towards a bulls-eye target: Error is the difference between an observed value (or the average of observed values) and the accepted value. The size of the error is an indication of the accuracy. Thus, the smaller the error, the greater the accuracy. Every measurement made on every scale has some unavoidable possibility of error, usually assumed to be one-half of the smallest division marked on the scale. The accuracy of calculations involving measured quantities is often indicated by a statement of the possible error. For example, you use a ruler calibrated in centimetres and millimetres to measure the length of a block of wood to be 12.6 mm (6 is the estimated digit in this measurement). The possible error in the measurement would be indicated by 12.6 ± 0.5 mm. Relative error is expressed as a percentage, and is usually called percentage error. This same formula can also be used to express by what percentage a value has changed. Sometimes if two values of the same quantity are measured, it is useful to compare the precision of these values by calculating the percentage difference between them. 1

2 PRACTICE 1. At a certain location the acceleration due to gravity is 9.82 m/s 2 [down]. Calculate the percentage error of the following experimental values of g at that location. (a) 8.94 m/s 2 [down] (b) 9.95 m/s 2 [down] 2. Calculate the percentage difference between the two experimental values (8.94 m/s 2 and 9.95 m/s 2 ) in question #2 above. 7 MEASUREMENTS 7.1 Measuring Reliably Many people believe that all measurements are reliable (consistent over many trials), precise (to as many decimal places as possible), and accurate (representing the actual value). But there are many things that can go wrong when measuring. There may be limitations that make the instrument or its use unreliable (inconsistent). The investigator may make a mistake or fail to follow the correct techniques when reading the measurement to the available precision (number of decimal places). The instrument may be faulty or inaccurate or in need of calibration; a similar instrument may give different readings. To be sure you have measured correctly, you should repeat your measurements at least three times. If your measurements appear to be reliable, calculate the mean (average of the three measurements) and use that value. To be more certain about the accuracy, repeat the measurements with a different instrument. 7.2 Measurement Errors Systematic Errors If you use a measurement instrument that is worn or is not calibrated, you will introduce a systematic error into your measurement. A systematic error is one that makes your measurement always too small or too large by a certain amount. For example, a ruler that is worn at the zero end will make all your lengths a bit too large. A voltmeter that has the needle a bit below zero when not in use (this is what it means to be not calibrated) will make all your measurements a bit too small. Systematic errors can be fixed using a variety of techniques: If the amount of error is known, add or subtract it from your measurement. If you are measuring with a ruler that has a worn end, do not measure from the zero mark on the ruler. Start all your measurements from the 1.0 cm mark and then subtract 1.0 cm from each measurement. This should be done even if you think the start of your ruler is not worn it might be worn, but not visibly so. V expected result Use a graph of the data. For example, if you are measuring voltage and current over a resistor to result of V too low determine its resistance, and you suspect that the or I too high voltmeter or ammeter are introducing a systematic error that you cannot see, then plot a voltage versus current I graph. Such a graph will give you the resistance through its slope. According to Ohm s Law, it will pass through (0,0). However, if there is a systematic error, the graph will be shifted so that it will not pass through (0,0), but the slope will not be changed. If you simply determined resistance by dividing V by I, you would not get the correct result. However, the slope of the graph will still give the correct result in a case like this. 2

3 Random Errors Another type of error might make your measurement either too large or too small in a random, unpredictable fashion. For example, using a stop-watch to time a 100 metre race requires you to judge when to start and when to stop the stopwatch. In any given race, your result might be a bit too high or a bit too small, but you cannot tell which. This is called a random error. In a case where you are experiencing random errors, these errors can be reduced. Repeat the measurement several times and average the results. If the errors are truly random, then some will be too high and some will be too small. The average should therefore cause some of these to cancel out and give a more accurate result. Repeating can take several forms. In the case of the 100 m race, you cannot have the race re-run, so you use several timers for each runner and average their measurements. In the case of measuring the period of a pendulum, repetition is achieved by measuring the period of say 10 periods and dividing the result by 10. Since the majority of the error in such a measurement comes at the start (first period) and at the end (last period), you are effectively spreading out this error over 10 periods and thus reducing it. Another method of repetition is to create a table of data an plot it. Using the voltage and current example again, random errors will cause the data points to vary sometimes above and sometimes below the trend. Analyzing the data with a best-fit line will produce a result that minimizes the random error. Overall, graphing is a powerful tool for analyzing data. It incorporates repetition with techniques for dealing with both random errors and systematic errors. If at all possible, data analysis should be done graphically. Using a graph has one further advantage. When you plot your data, you can usually see the trend it follows. Occasionally, you will have one or more data points that do not follow the trend. These data points are called outliers (they lie outside the trend). In the V- I example above, if you were to calculate resistance from each V-I pair and then average your results, you give the outlier an equal weight as the other data points in determining your average. The best fit line technique actually reduces the weight given to outliers and gives you potentially a more accurate result. PRACTICE 1. Use a ruler calibrated in centimetres to measure the distance from the centre of each plus sign to the next. Include the error in your measurement. 2. The following voltage and current data was gathered to determine the resistance of a resistor. a) First, fill the third column by dividing V by I. Once you have completed the column, average the values and record your result. b) Plot a graph of the values. Use the slope of a bestfit line to get your resistance. Record your result. c) Explain why your result in b) is more accurate than your result in a). Current (A) Voltage (V) V/I

4 8 TRIGONOMETRY 8.1 Right Triangles Trigonometry deals with the relationships between the sides and angles in right triangles. For a given acute angle in a right triangle, there are three important ratios. These are called the primary trigonometric ratios. The primary trigonometric ratios can be used to find the measures of unknown sides and angles in right triangles. (SOH CAH TOA) N O T E: The values of the trigonometric ratios depend on the angle to which the opposite side, adjacent side, and hypotenuse correspond. If the value of a trigonometric ratio is known, its corresponding angle can be found on a scientific calculator using the inverse of that ratio. (i.e., use 2nd SIN for sin -1, 2nd COS for cos -1, and 2nd TAN for tan -1 ). PRACTICE 1. Determine the value of each ratio to four decimal places. (a) sin 35 (b) cos 60 (c) tan 45 (d) cos 75 (e) sin 18 (f) tan 38 (g) cos 88 (h) sin 7 2. Determine the size of to the nearest degree. (a) sin = (b) cos = (c) tan = (d) cos = (e) sin = (f) tan = (g) cos = (h) sin = Solve for x to one decimal place. (a) sin 35 = x/8 (b) cos 70 = x/15 (c) tan 20 = x/19 (d) tan 55 = 8/x (e) sin 10 = 12/x (f) sin 75 = 5/x 4. Solve for to the nearest degree. (a) cos = 3/8 (b) sin = 7/8 (c) tan = 15/9 (d) cos = 16.8/21.5 (e) tan = 25/12 (f) sin = ½ 5. Find the value of the unknown in each triangle. Use trigonometry to solve and then check your answer with the pythagorean theorem. Round your answer to one decimal place. 4

5 9 EXTRA PRACTICE QUESTIONS Recall that every measurement has a degree of uncertainty. It is general practice in physics to take the uncertainty to be half the smallest division of the measuring device. Examine the following examples: the uncertainty length = 5.8 cm ± 0.5 cm length = 3.38 cm ± 0.05 cm Notice that the uncertainty corresponds to the last decimal of the measurement. 1. Record each measurement with the correct number of decimal places and the correct uncertainty. Assume the units are cm unless otherwise noted. a. b. c. 2. For each measurement, state the uncertainty. a. 2.3 m b km c ml d W e. 12 cm 3. For each of the measurements above, state which digit is the estimate. 5

6 4. Three lab groups measured the time taken for a ball to roll down a ramp. They did the experiment five times each with the same ramp. Their results are shown below. Time for Group 1 (s) Time for Group 2 (s) Time for Group 1 (s) a. Calculate the percent difference for each group. b. Which group had the most precise results? How does this relate to your answer from a? c. A photo-sensor timed the ball to have actually taken 2.22 s to go down the ramp. Which group would you say is the most accurate? Explain. d. Which group probably has a systematic error in their measurements? Explain. e. Calculate the percent deviation (percent error) of each group (use their average as the experimental value and the photo-sensor as the accepted value). How do your values relate to your answer for c? 6