since percent, the ratio5/540 reduces to 0.0093 (rounded off) in decimal form.
Significant Digits
The accuracy of a measurement is often described in t e r m s of the number of significant digits used in expressing it. If the digits of a number resulting from measurement a r e examined one by one, beginning with the left-hand digit, the f i r s t digit that i s not 0 is the f i r s t significant digit. F o r example, 2345 has four significant digits and 0.023 has only two significant digits. The digits 2 and 3 in a measurement such as 0.023 inch signify how many thousandths of an inch comprise the measurement. The 0 s a r e of no significance in specifying the number of thousandths in the measurement; their presence is required only as "place holders" in placing the decimal point. A rule that is often used s t a t e s that the significant digits in a number begin with the f i r s t nonzero digit (counting f r o m left to right) and end with the l a s t digit. This implies that 0 can be a significant digit if i t is not the f i r s t digit in the number. F o r example, 0.205 inch is a measurement having three significant digits. The 0 between the 2 and the 5 is significant because i t is a part of the number specifying how many thousandths a r e in the measurement. The rule stated in the foregoing paragraph fails to classify final 0 s on the right. For example, in a number such as 4,700, the number of significant digits might be two, three, o r
four. If the 0 s merely locate the decimal point (that is, i f they show the number to be approximately forty -seven hundred r a t h e r than forty seven), then the number of significant digits is two. However, if the number 4,700 represents a number such as 4,703 rounded off to the nearest hundred, there a r e three significant digits. The last 0 merely locates the decimal point. If' the number 4,700 represents a number such as 4,700.4 rounded off, then the number of significant digits is four. Unless we know how a particular number was measured, it is sometimes impossible to determine whether right-hand 0 s a r e the result of rounding off. However, in a practical situation it is normally possible to obtain information concerning the instruments used and the degree of precision of the original data before any rounding was done. In a number such as 49.30 inches, it is r e a sonable to assume that the 0 in the hundredths place would not have been recorded a t all if i t were not significant. In other words, the instrument used for the measurement can be read to the nearest hundredth of an inch. The 0 on the right is thus significant. This conclusion can be reached another way by observing that the 0 in 49.30 is not needed as a place holder in placing the decimal point. Therefore its p r e s ence must have some other significance. The facts concerning significant digits may be summarized as follows: 1. Digits other than Oare always significant. 2. Zero is significant when i t falls between significant digits.
3. Any final 0 to the right of the decimal point is significant. 4. When a 0 is present only as a place holder for locating the decimal point, i t is not significant. 5. The following categories comprise the significant digits of any measurement number: a. The f i r s t nonzero left-hand digit is significant. b. The digit which indicates the precision of the number is significant. This is the digit farthest to the right, except when the right-hand digit is 0. If it is 0 , it may be only a place holder when the number is an integer. c. A l l digits between significant digits a r e significant. P r a c t i c e problems. Determine the percent of e r r o r and the number of significant digits in each of the following measurements: 1. 5.4 feet 2. 0.00042 inch Answers: 1. P e r c e n t of e r r o r : 0.93% Significant digits: 2 2. P e r c e n t of e r r o r : 1.19% Significant digits: 2 3. 4.17 s e c 4. 147.50 miles
CALCULATING WITH APPROXIMATE NUMBERS
The concepts of precision and accuracy form the b a s i s for the r u l e s which govern calculation with approximate numbers (numbers resulting from measurement).
Addition and Subtraction
A sum o r difference can never be more p r e cise than the least precise number in the calculation. Therefore, before adding o r subtracting approximate numbers, they should be rounded to the same degree of precision. The more precise numbers a r e a l l rounded to the precision of the least precise number in the group to be combined. F o r example, the numb e r s 2.95, 32.7, and 1.414 would be rounded to tenths before adding as follows:
Multiplication and Division
3. P e r c e n t of e r r o r : 0.128 Significant digits: 3
4. P e r c e n t of e r r o r : 0.0034% Significant digits: 5
When two numbers a r e multiplied, the result often has several more digits than either of the original factors. Division a l s o frequently produces more digits in the quotient than the original data possessed, if the division i s "carried out1' to several decimal places. Results such as these appear to have more significant digits than the original measurements from which they came, giving the false impression of greater
accuracy than is justified. In order to correct this situation, the following rule is used: In order to multiply o r divide two approximate numbers having an equal number of significant digits, round the answer to the same number of significant digits as a r e shown in the original data. If one of the original factors has more significant digits than the other, round the more accurate number before multiplying. I t should be rounded to one more significant digit than appears in the l e s s accurate number; the extra digit protects the answer from the effects of multiple rounding. After performing the multiplication o r division, round the result to the same number of significant digits as a r e shown in the l e s s accurate of the original factors. Practice problems:
'
mals is a measuring instrument known a s a micrometer. The ordinary micrometer is capable of measuring accurately to one -thousandth of an inch. One-thousandth of an inch is about the thickness of a human hair o r a thin sheet of paper. The parts of a micrometer a r e shown in figure 6-1.
MICROMETER SCALES
The spindle and the thimble move together. The end of the spindle (hidden from view in figure 6-1) is a screw with 40 threads per inch. Consequently, one complete turn of the thimble moves the spindle one-fortieth of an inch o r
THIMBLE RATCHET STOP
I
n
1
1. Find the sum of the sides of a triangle in which the lengths of the three sides a r e as follows: 2.5 inches, 3.72 inches, and 4.996 inches.
2. Find the product of the length and width of a rectangle which is 2.95 feet long and 0.9046
foot wide. Answers:
1. 11.2 inches
FRAM E
'
2. 2.67 square feet
MICROMETERS AND VERNIERS
Figure 6-1.-(A) P a r t s of a micrometer; (B) micrometer scales.
Closely associated with the study of deci178
0.025 inch since
1
is equal to 0.025.
The
sleeve h a s 40 markings to the inch. Thus each space between markings the on the sleeve is a l s o 0.025 inch. Since 4 such spaces a r e 0.1 inch (that is, 4 x 0.025), every fourth m a r k is labeled in tenths of an inch for convenience in reading. Thus, 4 m a r k s equal 0.1 inch, 8 marks equal 0.2 inch, 12 m a r k s equal 0.3 inch, etc. T o enable measurement of a partial turn, the beveled edge of the thimble is divided into 25 equal parts. Thus each marking on the 1 1 1 thimble is 23 of a complete turn, or 25 Of 40 of an inch. Multiplying
J.
Thus, the reading is 0.227 inch. As explained previously, this is read verbally as "two hundred twenty-seven thousandths." A more skillful method of reading the scales is to read all digits as thousandths directly and to do any adding mentally. Thus, we read the major division on the scale as "two hundred thousandthsft and the minor division is added on mentally. The mental process for the above setting then would be "two hundred twenty-five; two hundred twenty-seven thousandths." Practice problems:
1. Read each of the micrometer settings shown in figure 6-2.
times 0.025 inch, we
find that each marking on the thimble r e p r e s e n t s 0.001 inch.
READING THE MICROMETER
It is sometimes convenient when learning to read a micrometer to write down the component p a r t s of the measurement as read on the s c a l e s and then to add them. For example, in figure 6-1 (B) t h e r e a r e two major divisions visible (0.2 inch). One minor division is showing clearly (0.025 inch). The marking on the thimble nearest the horizontal o r index line of the sleeve is the second marking (0.002 inch). Adding these p a r t s , we have
Figure 6-2.-Micrometer
settings.
Answers:
1 (A) 0.750 .
(F) 0.009
(G) 0.662
(H) 0.048
(B) 0.201
(C) 0.655
9 space is - inch. The vernier space is smaller 100 by the difference between these two numbers, as follows:
(D) 0.075
(I) 0.526
(E) 0.527
VERNIER
Sometimes the marking on the thimble of the micrometer does not fall directly on the index line of the sleeve. T o make possible readings even s m a l l e r than thousandths, an ingenious device is introduced in the form of an additional scale. This scale, called a VERNIER, was named a f t e r i t s inventor, P i e r r e Vernier. The vernier m a k e s possible accurate readings to the ten-thousandth of a n inch.
Principles of the Vernier
VERNIER
/
F i g u r e 6-3.-Vernier
scale.
Suppose a r u l e r h a s markings every tenth of a n inch but it is desired to read accurately to hundredths. A separate, freely sliding vernier scale (fig. 6-3) is added to the ruler. It has 10 markings on it that take up the same distance as 9 markings on the r u l e r scale. Thus, each 1 9 9 space on the vernier is -of - inch, o r 10 10 100 inch. How much s m a l l e r is a space on the v e r nier than a space on the r u l e r ? The r u l e r space is - inch, o r - inch, and the vernier 10 100
1
1 Each vernier space is - inch smaller than a 100 r u l e r space. As an example of the use of the vernier scale, suppose that we a r e measuring the steel b a r shown in figure 6-4. The end of the b a r almost reaches the 3-inch mark on the r u l e r , and we estimate that it is about halfway between 2.9 inches and 3.0 inches.
DECIMAL RULER (ENLARGED)
I
l I I I 1 . l I l l
I I l I 1 l I I I
2
I I I I l 1 I I I
3
4
10
VERNIER
F i g u r e 6-4.-Measuring
with a v e r n i e r .
BEING MEASURED
The 0 on the vernier scale is spaced the distance of exactly one r u l e r m a r k (in this case, one tenth of a n inch) from the left hand end of the vernier. Therefore the 0 is a t a position between r u l e r m a r k s which is comparable to the position of the end of the bar. In other words, the 0 on the vernier i s about halfway between two adjacent m a r k s on the r u l e r , just as the end of the bar i s about halfway between two adjacent m a r k s . The 1 on the vernier scale i s a little c l o s e r to alinement with a n adjacent r u l e r mark; in fact, it i s one hundredth of a n inch c l o s e r to alinement than the 0. This is because each space on the vernier i s one hundredth of a n inch s h o r t e r than each space on the r u l e r . Each successive m a r k on the vernier scale i s one hundredth of a n inch closer to alinement than the preceding m a r k , until finally alinement is achieved a t the 5 mark. This means that the 0 on the vernier must be five hundredths of a n inch f r o m the n e a r e s t r u l e r mark, since five increments, each one hundredth of an inch in s i z e , w e r e used before a m a r k was found in alinement. We conclude that the end of the bar i s five hundredths of a n inch from the 2.9 m a r k on the r u l e r , since its position between m a r k s i s exactly comparable to that of the 0 on the vernier scale. Thus the value of our measurement is 2.95 inches. The foregoing example could be followed through for any distance between markings. Suppose the 0 m a r k fell seven tenths of the distance between r u l e r markings. It would take
seven vernier markings, a loss of one-hundredth of an inch each time, to bring the marks in line a t 7 on the vernier. The vernier principle may be used to get fine linear readings, angular readings, etc. The principle i s always the same. The vernier has one more marking than the number of markings on an equal space of the conventional scale of the measuring instrument. For example, the vernier caliper (fig. 6-5) has 25 markings on the vernier for 24 on the caliper scale. The caliper is marked off to read to fortieths (0.025) of an inch, and the vernier extends the accuracy to a thousandth of an inch.
Figure 6-5.-A vernier caliper. Vernier Micrometer
By adding a vernier to the micrometer, i t is possible to read accurately to one ten-thousandth of an inch. The vernier markings a r e on the sleeve of the micrometer and a r e parallel to the thimble markings. There a r e 10 divisions on the vernier that occupy the s a m e space as 9 divisions on the thimble. Since a thimble space i s one thousandth of an inch, a vernier space is
9 1 9 -of inch, o r -inch. It i s 10 1000 10000 10:00 inch l e s s than a thimble space. Thus, as in the preceding explanation of verniers, it i s possible to read the n e a r e s t ten-thousandth of an inch by reading the vernier digit whose marking coincides with a thimble marking. I= I t I El=
The reading i s 0.3834 inch. With practice these readings can be made directly from the microme t e r , without writing the partial readings. Practice problems: 1. Read the micrometer settings in figure 6-6. Answers:
1. (A) See the foregoing example.
Figure 6 -6.-Vernier
micrometer settings.
$
(F)
I5
(B) 0.1539
(E) 0.4690
(F) 0.0552
1 0
(C) 0.2507
(D) 0.2500
5
In figure 6-6 (A), the l a s t major division showing fully on the sleeve index i s 3. The third minor division i s the l a s t m a r k clearly showing (0.075). The thimble division nearest and below the index i s the 8 (0.008). The vernier marking that matches a thimble marking is the fourth (0.0004). Adding them all together, we have,