Part Two: Significant Figures
T.2 Significant Figures The use of significant figures is a method of designating the reliability of measured quantities. Since experimentation in physics involves very many measured quantities, it is important to know how to express the results of the measurements and the calculations based on those measurements. Do the SelfCheck to see if you need to refresh your knowledge of the use of significant figures.
Self- Check Indicate the number of many significant figures in each of the following measured quantities. 1.
6 cm
2.
4.0 m
3.
0.02 km
4.
8000 nm
5.
0.06000
In the following problems all of the numbers can be assumed to be the result of a measurement. Solve each of the problems and give the answer with the correct number of significant figures. 6.
(0.101)(1.03 ) (0.025 )
7.
( 4.5 × 10 3 )(2.5 × 10 −2 ) (2.15 × 10 − 4 )
8.
6.27 + 0.2 + 0.410
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Part Two: Significant Figures
Determining the Number of Significant Figures The number of significant figures in a recorded number is the number of digits that are certain, plus the first digit that is uncertain. For example, a length given as 1.2 m means that the length lies somewhere between 1.15 m and 1.25 m. The distance 1.2 m has two significant figures: the numeral 1 is certain and the numeral 2 is the first digit that is uncertain. The limitations of the device used to make the measurement usually determine the decimal place of the first uncertain digit in a measurement. For example, a graduated cylinder marked as 25 ml ± 1 ml is certain for the tens of milliliters, but the uncertainly of estimation of the liquid level is in the units of milliliters. Thus if this graduated cylinder were used to measure a volume estimated to be at the 21 ml there are two significant figures in the measure: the certain numeral, 2, and the numeral 1, the first uncertain digit. Measurements that have zeroes to the left of an understood decimal and to the right of a non-zero digit are not significant. If the decimal is expressed then the zeroes are significant. In the number 4000 m there is only 1significant figure. The measure is just to the nearest kilometer. If the measure was written as 4000 m, then the measurement was made to the nearest meter, and there are 4 significant figures. This confusion can be eliminated by using scientific notation. The measurement reported as 4 × 10 3 m clearly has only 1 significant digit, while 4.000 × 10 3 m has 4 significant digits, since all zeros expressed to the right of the decimal point in scientific notation are significant. Zeroes to the right of an expressed decimal and to the right of a non-zero digit are significant. Again, putting the measurement into scientific notation helps with the decision as to what is significant and what is not. Suppose a measure of the thickness of the edge of a razor blade is made and recorded as 0.02 mm. This measurement has only 1 significant figure, which is seen clearly when the number is written in scientific notation as 2 × 10 −2 mm. If the person who made the measurement actually measured to the nearest one-thousandth of a millimeter, the measurement should be recorded as 0.020 mm, with 2 significant figures. This would be 2.0 × 10 −2 mm in scientific notation. As another example, the thickness of a piece of machined steel is expressed as 10.002 mm. This measurement has 5 significant figures, indicating the measurement was made to the nearest one-thousandth of a millimeter. How would this measurement be expressed in scientific notation?______________________
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Part Two: Significant Figures
Using Significant Figures in a Product or Quotient In multiplication and division calculations, the number of significant figures in a result usually cannot exceed the least number of significant figures used in the calculation, nor should it have fewer significant figures than the least number in the measured values. If the side of a square has been measured to be 1.2 × 10 2 cm, the number of significant figures (2) must be taken into account when calculating the area. A = L2 = (1.2 × 10 2 ) 2 = 1.44 × 10 4 = 1.4 × 10 4 cm 2 (2 significant figures) As another example, suppose you wish to determine the radius of a circle whose area is given as 46.3 m2. You will need to use the formula for the area of a circle, A = π r 2 , and the value of the ratio π . You must use the same number of significant figures in the value of π as in the given quantity (3). This gives A = πr 2 46.3 = 3.14 r 2 46.3 r = 3.14
1 2
= (14.7 )
1 2
= 3.83 m (3 significan t figures )
Counted numbers and defined quantities are not measured and do not influence the number of significant figures in a calculation. Suppose that you need to know how much cheese to purchase for a party. You estimate that there will be 7 people present and that each will eat 0.125 kg of cheese. Your total cheese purchase should be (7 people )(0.125 kg of cheese ) = 0.875 kg of cheese (3 significan t figures ) In this example, the 7 (as in 7 people) did not affect the number of significant figures in the final answer, because it was a counted quantity.
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Part Two: Significant Figures
Using Significant Figures in Sums or Differences In addition and subtraction, the number of significant figures in the result is determined by the position of the first uncertainty of the measure with the largest uncertainty. For example, four different carpet installer trainees were told to use the instrument of their choice to determine the length of one side of a closet. The four distances were then used to find the perimeter of the closet. The measurements they reported were 5.06 m, 5 m, 11.1 m and 11.20 m. To find the perimeter, the measurements were added together: 5.06 + 5 + 11.1 + 11.20 = 32.36 = 32 m (2 significan t figures ) From the measurements given, the value 5 m has the greatest uncertainty, since it was measured only to the nearest meter. Thus the sum must be rounded to the nearest meter, which gives the answer 2 significant figures.
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Part Two: Significant Figures
Practice Problems in Significant Figures 1.
Indicate the number of significant figures in each of the following measured quantities.
2.
3.14 s
3.
86 000
4.
3.1558 × 10 7
5.
6 × 10 7 yr
6.
0.324 × 10 1 hr
s
day
s
yr
Complete the following calculations involving measured quantities and express the answers with the correct number of significant figures. )(8.1× 10 3 s)
7.
(35.8
8.
(2.100 m) (9.81 m s2 )
9.
Determine the distance of tall, d, for the situation where d = ½ at2, a = 9.80 m/s2, and t = 4.0 s.
10.
1002.00 ml + 248.5 ml + 8.80 ml
m
s
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Part Two: Significant Figures
Answers to Self- Check 1.
1 significant figure
2.
2 significant figures
3.
1 significant figure
4.
1 significant figure
5.
3 significant figures
6.
4.2 (2 significant figures)
7.
5.2 × 10 5 (2 significant figures)
8.
6.9 (2 significant figures)
Answers to Practice Problems 1.
3 significant figures
2.
3 significant figures
3.
5 significant figures
4.
1 significant figure
5.
3 significant figures
6.
290 or 2.9 × 10 2 m (2 significant figures)
7.
0.214 or 2.14 × 10 −1 s2 (3 significant figures)
8.
d = ( 12 )(9.81)( 4.0) 2 = 78 m (2 significant figures)
9.
1259.3 ml The measure with the largest uncertainty is the 248.5 ml. The answer must be rounded to the nearest tenth milliliter. (5 significant figures)
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