Measurements and Significant Figures Units of Measurement All measurements done in lab must be expressed using appropriate units. In any measurement, it is best to keep the data in original units and convert them to SI units or other appropriate units later in calculations. This practice saves you a lot of trouble. If you make a unit conversion error in calculation, you can always check the data and unit conversion. For example, if measurements are made with a meter stick, it is best to record the readings in centimeters, rather than in meters. The reason for this is that if you make a mistake in conversion and the reading was, for example, 25.62 cm, you may record the data incorrectly, as 0.02562 m, instead of correct, 0.2562 m. Once you make an error in recording your data, it is very difficult to find the error in the calculations. The most common mistake students make in calculations in the physics lab 1s, in fact, the conversion of units. Personal (Human) Error Personal errors, often called human errors, arise from carelessness in reading an instrument, performing an experiment, recording data, or calculations including unit conversion. Personal errors are not legitimate errors in experiments and can always be avoided with caution. You should always be careful not to make personal errors during an experiment. Uncertainty in Measurement Every measured number has some degree of uncertainty. What is the thickness of your lab table? Is it 4.7 centimeters, 4.73 centimeters, 4.730 centimeters or 4.7301 centimeters? You cannot state its exact measurement with absolute certainty. The uncertainty shows the margin of error in a measurement. The uncertainty in a measurement depends on the precision of the measuring device and the skill of the person who uses it. Errors due to uncertainty in laboratory measurements have a different meaning from “human error.” Uncertainties related to the precision of your measuring instruments cannot be avoided. Measurements convey two kinds of information: (1) the magnitude of the measurement and; (2) the precision of the measurement. The precision of the measurement depends on the measuring device and the method of measurement. Significant Figures Significant figures are the digits in any measurement that are known with certainty plus one digit that is estimated, and hence, is uncertain. The right-most digit is always an estimated digit. Only one estimated digit is ever recorded as part of a measurement. On the other hand, counting numbers are different from measured quantities. Counting numbers are exact. In other words, a counting number has an infinite number of significant figures. Rule 1: In numbers that do not contain zeros, all the digits are significant. e example 1: 3.5781 five significant figures 19 4 three significant figures 1621 four significant figures Rule 2: All zeros between significant digits are significant. e example 2: 4.028 four significant figures 4028 four significant figures 105 three significant figures 3002 four significant figures Rule 3: Zeros to the left of the first non-zero digit serve only to fix the position of the decimal point, therefore, these zeros are not significant. e example 3: 0.21 two significant figures 0.0567 three significant figures 0.0000031 two significant figure Rule 4: In a number with digits to the right of the decimal point, zeros to the right of the last non-zero digit are significant. sexamples 4: 75.0 three significant figures 75.00 four significant figures 0.00200 three significant figures 0.40050 five significant figures 1.0020 five significant figures Rule 5: In a number that has no decimal point and that ends in one or more zeros (such as 1800), the zeros that end the number may or may not by significant. The number is ambiguous in terms of significant figures. The confusion is avoided when numbers are expressed in scientific notations. All digits are taken to be significant when expressed using powers of ten. e example 5: 1.8x 10° two significant figures 1.80x 10° three significant figures 1.800x 10° four significant figures 2x 107 one significant figure 2.0% 10” two significant figures Rounding Off A calculator displays eight or more digits. How do you round off such a display of digits to, say, three significant figures? Three simple rules govern the process of deleting unwanted (not significant) digits from a calculator number. Rule 1: Tf the first digit to the right of the last significant figure is less than 5, that digit and all the digits that follow are simply dropped. o example 6: 43.549 rounded off to three significant figure becomes 43.5 Rule 2: If the first digit to be dropped is a digit equal to or greater that 5, the excess digits are dropped and the last retained digit 1s increased in value by one unit. e example 7: 54.36, 54.359, and 54.3598 rounded off to three significant figures all become 54.4. Significant Figures of Calculated Quantities Two separate rules apply for multiplication and division, and for addition and subtraction. Rule for Multiplication and Division For multiplication and division, an answer should have the number of significant figures found in the number with the fewest significant figures. e example 8: 1.356 x 0.452 = 0.612912 (calculator answer) The input with the fewest significant figures is 0.452, which has three significant figures. Therefore, the calculator answer 0.612912 must be rounded off to 0.613. The next example is a rare case when a calculator gives a less number of significant figures than expected. e example9: 17.612 — 1.258 = 14 (calculator answer) The inputs contain five and four significant figures. Therefore, the correct answer must contain four significant figures and must be written as 14.00. > e example 10: calculation involving a counting number. Suppose you measured the time for a pendulum to make 10 complete oscillations and want to find the period (time for one oscillation) of the pendulum: 15.6s-10=156s The time 15.6 s has three significant figures and 10 is a counting number (exact number), so the answer should have three significant figures, same as that of the measured value. e example 11: calculation involving a constant. When a constant is used in calculation, you should use one extra number of significant figures than that of a measured value so that you do not have a round-off error in the result. All other rules mentioned earlier apply in determining the number of significant figures in the result. Area of circle of radius 5.25 cm: A = m“ =3.142 x (5.25 ст)? = 86.6 ст” Addition and Subtraction For addition and subtraction, the answer should not have digits beyond the last digit position common to all the numbers being added or subtracted. e example 12: 7.26 +28.1 +33 2 68.56—68.6 The last digit position common to all numbers is the first decimal place. The answer must be rounded off to 68.6. e example 13: 345.293 - 215.15 130.143—130.14 The last digit position common to both numbers is the second decimal place. Therefore, the answer should be rounded off to 130.14. e example 14: percent error. À percent error 1s defined as follows: | measured value - accepted value | р percent error = x 100% accepted value If the measured value of the acceleration of gravity is found to be 9.73 m/s”, and the accepted value is 9.80 m/s”, then the percent error is | 9.73 т/ $ - 9.80 т/ 5” | ZZ > sr x 100% = EU = 0714% = 0.7% 980 m/ s 9.80 percent error = Note that since the absolute value of the difference of the measured value and the accepted value has only one significant figure in this case, the result should be rounded off to 0.7%. 6 * example 15: If the number of significant figures of an experimental result is different from that of an accepted value, one of them should be rounded off so that both have the same number of significant figures in calculating percent errors. 267g / em” —2.7g/ em: 2.78 /em* —2.7g ! em” %error = Е x 100% = - -x 100% = 0% 2.78 / cm 2.781 em This is a percent error calculation for the density of aluminum. In this case, the accepted value has only two significant figures, so the experimental value of 2.67 g/cm’ was rounded off to 2.7 g/cm’ and the percent error is 0%. ® example 16: Percent difference. In order to compare two experimental values, percent difference 1s used in stead of percent error. Percent difference = —————— x100% (E,+E,)I2 IE, - E, where E, and F, are two experimental values. The percent difference between two measured values of 9.71 m/s? and 9.9 m/s? is 9.71-9.9| 0.2 percent difference =—————x 100% = —x 100% = 2% (9.71+9.9)/2 9.8 Note that in this case, the two values have different number of significant figures, so the sum and the difference of the two values are rounded off to the first decimal place to b= 0.2 and 9.8, respectively. Since the numerator 0.2 has only one significant figure, the result has also one significant figure, 2%. = Standard Deviation G The precision of the measurement can be determined using a quantity called the standard deviation, o,_,, which is defined by 1 y AT Li | 2 я (1) where N is the number of repeated measurements, x, is the ith value of the measured quantity, and X is the mean (average) of the measurements. Using the standard deviation, it is possible to determine the probability that the measurements fall within any specific range about the mean (average) of the data. For example, approximately 68.3% of all repeated measurements fall within a range of one c from the mean. Furthermore, 99.73% of all measurements should fall within 3c of the mean. This indicates that if one of the measurements is more than 36 farther from the mean, it is likely the result of a personal or systematic error. The precision of the repeated measurements can be expressed by using a quantity called the standard error, & , which is defined by Per 2 de 2) If several groups of N measurements are made, 68.3% of the means produced from each group should fall in the range x +a. Or, one can say that there is a 68.3% probability that the true value lines in this range. = Example Assume five measurements are made of some quantity x, and the measured values are 12.6, 13.4, 11.9, 12.0, and 13.1. The mean (average) of the values x is given by x =(12.6+13.4+11.9+12.0+13.1)/5 =12.6 The standard deviation for these values 1s 1 2 2 2 42 + Tn = E (12.6-12.6)° + (13.4 -12.6)* + (11.9-12.6)? + (12.0-12.6)* + (13.1-12.6)* ]=0.7 _ 0.66 _ JN AS We can write the mean of the data with the standard error a as 0.3 x=126+1L3

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