A Justification for Sig Digs
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- Clarence Nash
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1 A Justification for Sig Digs Measurements are not perfect. They always include some degree of uncertainty because no measuring device is perfect. Each is limited in its precision. Note that we are not talking about human errors here.
2 Precision We indicate the precision to which we measured our quantity in how we write our measurement. For example, which measurement is more precise? 15 cm 15.0 cm This one, obviously. Physicists are lazy. They wouldn t bother to write the.0 if they didn t mean it.
3 What we mean When we write 15 cm, we mean that we ve measured the quantity to be closer to 15 cm than to 14 cm or 16 cm BUT When we write 15.0 cm, we mean that we ve measured the quantity to be closer to 15 cm than to 14.9 cm or 15.1 cm.
4 Significance Digits that have been measured are said to be significant. 15 cm This measurement has cm This measurement has 3.
5 The following rules are used to determine if a digit is significant: All non-zero digits are significant e.g N has three significant digits
6 The following rules are used to determine if a digit is significant: All non-zero digits are significant Any zeroes placed after other digits and behind a decimal are significant e.g kg has three significant digits
7 The following rules are used to determine if a digit is significant: All non-zero digits are significant Any zeroes placed after other digits and behind a decimal are significant Any zeroes placed between significant digits are significant e.g m has four significant digits
8 The following rules are used to determine if a digit is significant: All non-zero digits are significant Any zeroes placed after other digits and behind a decimal are significant Any zeroes placed between significant digits are significant All other zeroes are not significant e.g. both 100 cm and kg each have only one significant digit
9 How can you say those digits are not significant? Both 100 cm and kg each have only one sig dig? The zeros here are placeholders they re just there to show in which place the non-zeros belong. If the measurements are rewritten 1 m and 4 g, it becomes apparent that there s only one sig dig. But what if you measured 100 cm exactly?
10 Making Zeros Significant But what if you measured 100 cm exactly? You can show that a zero is significant by either: underscoring or overscoring the zero: 100 cm (if the measurement is in a table) rewriting the measurement in scientific notation: 1.00 x 10 2 cm
11 Making Zeros Significant And yes, if you measure a zero, you must write it. Your lab tables should not look like this: Time (s) Distance (cm)
12 Making Zeros Significant They should look like this: Time (s) Distance (cm)
13 No ½ measurements Your tables also should not look like this: Time (s) Distance (cm)
14 No ½ measurements If you can clearly measure.5 in one case, surely you could measure to the tenths place in the other cases too? We don t use.5 to substitute for about ½ :.5 means closer to.5 than to.4 or.6. Be exact. Time (s) Distance (cm)
15 How many significant digits are there in each of the following? 25 W 60 km/h 305 K 3.6 kg 5.0 mt 0.1 N cal
16 How many significant digits are there in each of the following? 25 W 60 km/h 305 K 3.6 kg 5.0 mt 0.1 N cal 2 s.d. 1 s.d. 3 s.d. 2 s.d. 2 s.d. 1 s.d. 4 s.d.
17 How many significant digits are there in each of the following? 1.10 A h MHz 2100 kj W kg km
18 How many significant digits are there in each of the following? 1.10 A h MHz 2100 kj W kg km 3 s.d. 2 s.d. 4 s.d. 2 s.d. 4 s.d. 4 s.d. 5 s.d.
19 Round each measurement to the required significant digits: 4080 J to 1 s.d J to 2 s.d kg to 1 s.d kg to 2 s.d V to 1 s.d V to 2 s.d.
20 Round each measurement to the required significant digits: 4080 J to 1 s.d J to 2 s.d kg to 1 s.d kg to 2 s.d V to 1 s.d V to 2 s.d J 4100 J 3 kg 2.7 kg 1 V 0.99 V
21 Write each of the following in scientific notation cal 1200 N J km s 20.5 khz
22 Write each of the following in scientific notation cal 1200 N J km s 20.5 khz cal N J km s khz
23 The Final Answer When a measurement is used in a calculation, the final answer must take into consideration the uncertainty in the original measurements. Why? You don t want to suggest that you know something to a greater precision than you actually measured. For example, 4 m/3 s = m/s???
24 The Final Answer When a measurement is used in a calculation, the final answer must take into consideration the uncertainty in the original measurements. Why? You don t want to suggest that you know something to a greater precision than you actually measured. For example, 4 m/3 s = 1 m/s.
25 The Final Answer When a measurement is used in a calculation, the final answer must take into consideration the uncertainty in the original measurements. Note: Exact numbers used in calculations (e.g. a factor such as ½ in the equation K=½mv 2 ) are not measurements and do not have any uncertainty.
26 Addition and Subtraction When adding or subtracting measurements, the final answer should be rounded off to the least number of decimals in the original measurements. e.g cm (3 decimal places) cm (2 decimal places) 5.13 cm (2 decimal places)
27 Practice 7 m + 7 m = 7 m m = 7.0 m m =
28 Practice 7 m + 7 m = 14 m 7 m m = 14 m 7.0 m m = 14.0 m
29 Multiplication and Division When multiplying or dividing measurements, the final answer should be rounded off to the same number of sig digs as are in the measurement with the least number of sig digs. e.g cm (4 sig digs) x 0.01 cm (1 sig dig) 0.05 cm 2 (1 sig dig)
30 Practice 7 m x 7 m = 7 m x 7.0 m = 7.0 m x 7.0 m =
31 Practice 7 m x 7 m = 50 m 2 (Yes, your math teacher would not approve.) 7 m x 7.0 m = 7.0 m x 7.0 m =
32 Practice 7 m x 7 m = 50 m 2 7 m x 7.0 m = 50 m m x 7.0 m = 49 m 2
33 More Practice ½(5.0 kg)(20 m/s) 2 = ½(5.0 kg)(20.0 m/s) 2 = kg 120N / m kg N / m
34 More Practice ½(5.0 kg)(20 m/s) 2 = 1000 J ½(5.0 kg)(20.0 m/s) 2 = 1.0 x 10 3 J kg 0. 41s 120N / m kg s N / m
35 A Warning When solving multi-step problems, round-off error can occur if a rounded-off answer from one step is used in a later step. Record or store in your calculator all calculated digits for use in later steps. SERIOUSLY!!!! DON T ROUND TOO SOON
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