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Manipulating Numbers A Manipulating Numbers A A.1 Units our unit of length. However, if we choose to use the foot as our unit of length, most adults have a height that is on the order of magnitude of ten Physics introduces many concepts that help us feet. Yet 10-ft is approximately 3 meters. understand and analyze physical processes— An atom has a diameter on the order of magni- sound and light phenomena, in particular. tude of 1-Ångstrom ≡ 1-Å≡ 10−8-cm. The size However, ultimately we come to trust the claims of the observable universe is 1027-cm. The weight of Physics by comparing observed numerical of most adults is an order of magnitude of 100- values of important parameters—such as sound lbs. The number of hairs on an adult with a full or light intensity and sound or light frequency— head of hair can have an order of magnitude as to how they are related to each other according high as 100-thousand.1 to the laws of Physics. We must note that a numerical value depends upon the choice of units. For example, my height is about 5-ft 10- A.3 Significant Figures in, or 70-in. In the metric system, my height is 1.78-m. Another unit of length is the Ångstrom, The order of magnitude is a very crude estimate abbreviated with the letter Å. Given that − of the value. We often have to express how 1-Å=10 10-m, my height is 1.78 × 1010- precisely we know the value of a parameter. Ångstroms. The relationship between various For this purpose, let us now introduce the term units for a given physical parameter is listed in significant figures. A value of my height as 70- Appendix D on the conversion of units. in provides me with two significant figures. A height expressed as 184-cm has three significant A.2 Order of Magnitude figures. Over the course of one day, my height certainly wouldn’t vary by as much as a centime- ter. Therefore, I might wake up with a height We begin by noting that while Physicists would of 181-cm and go to bed with a height of 180- like to know a value as precisely as possible, they are also extremely concerned with what 1 is referred to as the order of magnitude of a We can arrive at this number as follows: The dimen- sions of a scalp is on the order of 30-cm by 25-cm, parameter. Typically it is expressed by giving corresponding to an area of 750-cm2. For a thick head the value to the nearest power of ten. Thus, for of hair, the distance between hairs might be about 1-mm. example, any value ranging from 5 to any value Each hair takes up an area of about 1-mm2, equivalent to (10−1cm)2 = 10−2-cm2. Therefore the number of hairs is less than 50 is rounded off to ten. We say that − 750/10 2 = 75, 000. On the other hand, if the distance a value of 21 has an order of magnitude of ten. between neighboring hairs in 3-mm, the area per hair An adult has a height on the order of magnitude would be 9-mm2; in this case, the number of hairs would of one meter, if we choose to use the meter as be 75, 000/9 ∼ 8, 300 and the order of magnitude would be 10,000. © Springer Nature Switzerland AG 2019 409 L. Gunther, The Physics of Music and Color, https://doi.org/10.1007/978-3-030-19219-8 410 A Manipulating Numbers cm due to the compression of the disks in my Problem Confirm the above result. NOTE: One spine. If I were to express my height to only kilogram is equivalent to about 2.20-lbs and one two significant figures, I wouldn’t be taking into meter equals about 39.37 inches. account this change. Here is an ambiguity that can arise. Consider The lesson we learn from these examples is the following: If I were to tell you that my height that we must be careful not to take seriously is 180-cm, you would not know whether or not the number of significant figures listed for an the “0” digit is significant. The use of scientific important parameter. notation takes care of this problem as follows: If the zero is significant, I would express my height as 1.80×102-cm; this expression provides A.4 Relative Changes my height to three significant figures. If the zero is not significant, I would express my height as The relative change of a parameter gives us a 1.8 × 102-cm and we see my height expressed sense of the magnitude of the change that is to two significant figures. One foot is about one- independent of the units. Specifically third meter, which is, to one significant figure, 0.1-m. |change in value| relative change ≡ (A.1) Rounding off to a given number of significant |original value| figures can lead to some curious numbers. Con- sider the following: Normal body temperature is Here the vertical bars surrounding a number said to be 98.60 F. However, a body temperature represent the absolute value of the number.2 that is anywhere between about 970 F and 990 F is regarded as “normal” or acceptable. Given this Sample Problem A.1 Above we discussed the fact, how could such a strange number arise? change of my height over the course of a day. My Why should the temperature be specified to three height changed from 181-cm in the morning to a significant figures!? height of 180-cm at the end for the day. What is To answer this question note that 0C=(5/9)(0F- the relative change in my height? 32). Then we find that 98.60F=370Ctothree significant figures. In fact, doctors realize that Solution Letting the heights be h1 and h2,and people’s body temperatures are not expected to h be the absolute value of the change in height, be precisely 370C. This value was chosen as a we have to one significant figure rounded off number. h 181 − 180 Here is another example of how a choice relative change = = = 0.0.006 of units introduces uncanny constants. The so- h1 180 (A.2) called body mass index (BMI) is defined in British units (feet for length, seconds for time, or about 0.6%. and pounds for weight) as Suppose that you know that the frequency of × 2 703 weight in lbs/(height in inches) . a sound has changed by 4-Hz. Is this change sig- nificant? The relative change will be extremely Here again, we have a strange number informative. “703” occurring with three significant figures. Suppose that the original frequency is f1 and It wouldn’t seem to matter had the number been the new frequency is f2.Then chosen to be 700 precisely. Using metric system, with kilograms for weight and meters for length, |f − f | relative change ≡ 2 1 (A.3) changes the coefficient to unity (one). |f1| 2Thus, |2|=|−2|=2. A.4 Relative Changes 411 If we set f = f2 − f1, we can express the Here is an example wherein we can use al- relative change of the frequency as gebra to simplify a calculation. Suppose that we know x1 and its corresponding y1,sothaty1 = |f | bx1. Next, we know another value of x, say x2, relative change ≡ (A.4) |f1| and want to know its corresponding value y2.We could determine the constant b and then calculate Thus if the original frequency is 400Hz, the y2. Alternatively, we can use the following trick: relative change is 4/400 or one part in a hundred We have (0.01). [Note that the relative change as we have y2 bx2 x2 defined it would be 0.01 whether the change is = = (A.7) y bx x ±2 Hz, that is, from 400-Hz to 396-Hz or 400- 1 1 1 Hz to 404-Hz, since our definition of the relative or, change includes the absolute value.] y2 x2 = (A.8) In the context of music, if the relative change y1 x1 is much less than unity, the relative change in and finally sound frequency is closely related to the change in pitch: As we will learn in Chap. 12, a change = x2 in frequency by about 6% or one part in 16 y2 x1 . (A.9) x1 corresponds to a change by what is referred to 3 as a half-step. For example, if x1 = 4, x2 = 12, and y1 = 2, One might ask whether a change of one part y2 = (12/4) × 2 = 6. in a hundred significant. The answer depends We might want to know the relative change in upon the ear of the listener. For a person with y in terms of the relative change in x. We will a “good ear,” the change would certainly be now prove that ifthetwoareproportionalto significant. For many people the change would each other, the relative changes are equal. not be noticed. Then y (bx) x Problem Suppose that the change in frequency = = b . (A.10) is 4-Hz and the original frequency is that of y bx bx the lowest note on a double bass, about 41-Hz. Determine the relative change in frequency. Thus y x = . (A.11) A.4.1 Proportionalities y x Suppose that a variable y is proportional to a Sample Problem A.2 Here is a simple common variable x.Wehave example.
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