1º EM - Physical Measures

 PHYSICAL MEASURES

A - SCIENTIFIC NOTATION 

In Physics, you will have to cope with very small and very large numbers. Numbers are written using powers of ten to make them less awkward. This is known as SCIENTIFIC NOTATION. This type of representation, in addition to being more compact, allows us a quick comparison with other known values, facilitating its understanding. Any number can be written as the product of a number between 1 and 10 by an integer power of 10.

B - Significant Figures

Significant Figures of a measure: We call significant figures all the correct figures (read on the measuring device) and one more doubtful figure (evaluated by the operator).

PRACTICE: Measure the 3 straight segments using devices with different accuracy. Express these measurements using significant figures.

Straight segment	Scale 1	Scale 2	Scale 3
AB
CD
EF

NOTE: The number of significant figures depends on the device used in the measurement. As the last digit is ALWAYS doubtful (evaluated by the operator), the previous digit shows the smallest scale division of the device. If we have a measurement like 2.4 m, this indicates that the “ruler” used was divided every 1 meter. If it were 2.40 m, the doubtful number is in the cm and therefore the smallest division is dm. Therefore, in the case of measurements, 2.4 m is different from 2.40 m.

C - OPERATIONS WITH SIGNIFICANT FIGURES

Calculations involving measurements should only contain significant figures in the results. For that, it is necessary to follow some rules that, if not followed, may generate figures that are not significant.

1.  ADDITION AND SUBTRACTION: suppose you want to add the following portions:

2807.5 + 0.0648 + 83.645 + 523.35

In order for the result of the addition to contain only significant figures, you should initially observe which or which of the plots has the least number of decimal places. In our example it is the portion that has only one decimal place. This portion will be kept as is. The remaining plots must be modified in order to keep the same number of decimal places as the first chosen installment, leaving as many digits in them as necessary.

Thus, in the 0.0648 portion we must abandon the digits 6, 4 and 8. When we abandon digits in a number, the last digit maintained must be added by one unit if the first digit abandoned is greater than or equal to 5 or remains as it is. if the first digit left is less than 5 (rounding rule). Let's see how the example portions look like:

2807.5 remains unchanged 2807.5

0.0648 is replaced by 0.1

83.645 is now written 83.6

535.35 is replaced by 525.3

The correct result is 3416.5

In SUBTRACTION, the same procedure must be followed.

2. MULTIPLICATION AND DIVISION: Suppose we want, for example, to multiply 6.85 by 3.2. Normally performing the operation, we find

6.85. 3.2 = 21.92

However, proceeding in this way, figures appear that are not significant. To avoid this, we must observe the following rule: check which factor has the lowest number of significant figures and, in the result, keep only a number of figures equal to this factor. Thus, in the previous example, since the factor with the fewest significant figures is 3.2, we must maintain, in the result, only two figures, that is, the result must be written as follows:

6.85. 3.2 = 22

In the application of this rule, when abandoning figures in the product, we must follow the same rounding criterion clarified in the addition. A similar procedure must be followed when making a DIVISION.

COMMENTS:

- when counting the significant figures of a measurement, we must note that the figure zero is only significant if it is located to the right of a significant figure. So,

0.00035 has only two significant digits (3 and 5) because the zeros are not significant

35000 has five significant digits, because zeros here are significant

0.00305 has three significant figures, since the zeros to the left of the figure 3 are not significant.

- when making a change of units, we must be careful not to write zeros that are not significant. For example, suppose that we want to express, in grams, a measure of 8.5 kg. Note that this measure has two significant figures,  and the figure 5 is doubtful. If we were to write

8.5 kg = 8500 g

we would be giving the erroneous idea that 5 is a correct number, the last zero being added to the doubtful number. To avoid this misinterpretation, we use scientific notation and write

8.5 kg = 8.5x10³ g

In this way, the change of units was made and we continue to indicate that the number 5 is doubtful.

(text adapted from COURSE OF PHYSICS volume 1 by Beatriz Alvarenga and Antônio Máximo Ed. Harbra SP / 1985).

Other sources for operations with significant figures:

https://chem.libretexts.org/Courses/Eastern_Mennonite_University/EMU%3A_CHEM_155_-_Matter_and_Energy_(Yoder)/Unit_1/A%3A_Preliminary/2.5%3A_Significant_Figures_in_Calculations (access on 03/15/21) 

https://www.khanacademy.org/math/arithmetic-home/arith-review-decimals/arithmetic-significant-figures-tutorial/v/significant-figures (access on 03/15/2021)