Significant Figures 2019 PDF

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Significant Figures Lec Notes...

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Short Tutorial on Significant Figures July 15, 2019 As engineers we will be frequently engaged in measurements. We may use sensors and/or measuring devices for these measurements. Students are to be mindful that such measurements are never perfect - there is always uncertainty associated with these measurements. This tutorial will deal with how engineers cope with such uncertainties. Suppose you measure a length of a line with a ruler and report it as 4.2cm. This indicates that the length has an uncertainty of 0.1cm. In other words, we have measured the length to the nearest 0.1cm. Now suppose you measure the same line with a different ruler that has much finer scale graduations, and suppose also that you have perfect eyesight, and you report the length of the line as 4.20cm. This implies that the uncertainty in our measurement is 0.01cm and that the length has been measured to the nearest 0.01cm.

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Precision and Accuracy

At this stage, the student needs to be informed of two terms that are associated with uncertainties, namely, precision and accuracy. The most commonly used analogy to describe these terms is with the shooting of a target with arrows. The target is a round one with concentric circles. The smallest circle located at the center of the target is called the bullseye. The arrow is analogous to a measurement. Suppose three arrows were shot in sequence and that they all land very close together, say at the top left of the target. This is analogous to making three measurements that are very close to each other. In this case, we say that the measurements are precise but not accurate because the arrows are not at the bullseye. Since the three sequential measurements are close, we sometimes refer to these measurements as being repeatable. Now suppose, that a different shooter shot three arrows in sequence and they all landed in the bullseye. This situation is analogous to making three measurements with a new instrument and we say that it is both precise and accurate. 1

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Calibration

When an instrument records inaccurate readings, it has to be calibrated to improve its accuracy. During calibration, the instrument measures an object whose value is accurately known beforehand. The instrument’s measurement is compared with the known value and the difference between the two is used to adjust the measuring instrument. Most of us will be familiar with bathroom weighing scales. We have often zero-adjusted the weighing scale by turning a knob (or an adjusting button on digital scales) to zero. Alternatively, we can also calibrate the weighing scale with someone standing on it. This person’s weight must be known accurately (perhaps by weighing this person on a different weighing scale that is accurate).

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What are Significant Figures?

The number of significant figures in any measurement is the number of digits that are known with some degree of reliability. For example, in our first measurement of a line, we reported the length as 4.2cm. This measurement is represented using two significant figures. While 4.20cm is a measurement with three significant figures.

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Recording Measurements

When using a measuring instrument (examples will include the humble ruler, vernier caliper, thermometer, barometer etc), the person making the measurement will express the reading as one which is of reasonable reliability. For example, let us use a ruler to measure a line. In the first example as depicted in Figure 1, measurement A is the measurement of a line. The length of this line is in between the 2.6cm and 2.7 cm marks. We estimate the length of the line to be half of the distance between 2.6 and 2.7 cm. We therefore express the reading as 2.65cm, which is accurate to three significant figures. In Figure 2, we make a similar measurement of the length of a second line. Measurement B is spot on the 3.5cm mark. As in the previous example, we can also estimate an additional digit to our reading. We estimate this additional digit to be zero, as the reading is spot on 3.5cm. Therefore we report our reading to be 3.50cm, and as before, this measurement is also accurate to three significant figures.

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Figure 1: Measurement A

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Figure 2: Measurement B

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Rules for deciding the number of Significant Figures 1. All non zero digits are significant. For example, • 1.234m has four significant figures. • 12.3m has three significant figures. 2. Zeros between non-zero digits are significant. For example, • 502K has three significant figures. • 2.3kg has two significant figures. 3. Leading zeros to the left of the first non-zero digit are not significant. These zeros indicate the position of the decimal point. For example, • 0.0005m has only one significant figure, even if it has four decimal places. • 0.015m has two significant figures, even if it has three decimal places. • 0.1012m has four significant figures. 4. Trailing zeros to the right of a decimal point are significant. For example, • 0.0540kg has three significant figures. • 0.30m has two significant figures. 5. When a number is expressed in scientific notation, For example, • 112, 000 = 1.12 × 105 1.12 is referred to as the mantissa and 5 is referred to as the exponent. 4

• The number of significant figures are identified from only the mantissa. Therefore, 1.12 × 105 is said to have three significant figures. • On the other hand, if we had written 112, 000 = 1.120 × 105 then, 1.120 × 105 has four significant figures.

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Rules for Deciding the number of significant figures in a mathematical operation. 1. For addition and subtraction • When two numbers are added or subtracted, the result will be of the precision of the number with the least precision. For example, 210m + 6m = 216m (Is this correct?) The first number, 210, has two significant figures and implies that we can measure it up to the nearest 10m. The second number, 6, is one significant figure and we can measure it to the nearest m. Since the precision of the nearest 10m is less than to the nearest m, the answer should be shown to the nearest 10m. Therefore, 216m should be rounded to the nearest 10m and written as 220m. • for adding or subtracting two or more numbers with decimal places, the final answer will have decimal places that are the least of the decimal places of the input numbers. For example, 5.27kg + 2.1kg = 7.37kg (Is this correct?) since 2.1 has only one decimal place and 5.27 has two decimal places, the answer must be written as 7.4 as 7.37 is rounded to the nearest number with one decimal place. Note that 2.1 has less precision that 5.27. 2. For multiplication and division • When two numbers are multiplied or divided, the result will have the same number of significant figures as the number with the least number of significant figures. For example,

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1.35m × 2.62m = 3.537m (Is this correct?) should be written as 3.54m rounded to three significant figures as both 1.35m and 2.62m have three significant figures. • Another example, 3.27kg × 1.1kg = 3.597kg (Is this correct?) should be written to only two significant figures and hence the answer is 3.6kg . • How many tiles will be required to fit a rectangle of sides 10.5m and 1.3m, if the area of a tile is 1.01 m2 ? Answer: We first calculate the area of the rectangle. 10.5m × 1.3m = 13.65m2 Note that we do not round our answer to the most appropriate significant figures at intermediate stages of our calculations. Doing so, will introduce errors. We will round our answer only in the final stage of the calculation. To find the number of tiles, we divide the area of the rectangle by the area of one tile, i.e., 13.65 1.01

= 13.5148514851 · · · tiles The numbers used in the calculations are 10.5m, 1.3m and 1.01m2 , and they have three, two and three significant figures respectively. Therefore, the number of tiles should be rounded to two (the least) significant figures and written as 14 tiles.

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