IB Physics Self Study Unit Read the following sections and
IB Physics Self Study Unit Read the following sections and complete the exercises. How to read a scale Significant figures Errors & uncertainties Part 1. How to read a scale A scale is usually divided into units and subunits. Before attempting to make a reading with the scale one must find what the smallest graduation on the scale represents. The smallest graduation is the distance between consecutive lines or notches on the scale. For example consider your own ruler, graduated in centimetres. Between two consecutive centimetre marks we have 10 smaller graduations, which means that the smallest graduation is one tenth of a centimetre, or a millimetre. Try now to give the smallest graduations of the following scales. (Remember to count the sub-divisions between two known readings): Example 1 (Part (a) is done for you) (a) Smallest graduation = cm. (b) (c) (d) [answers: b = mm, c = 0.1mm, d = 0.02cm] Significant figures: How many figures to include in a reading. An accurate statement is one which tells exactly what is known, no more and no less. When you read a scale, you have to be accurate and include as many figures (digits) as you have an idea about, and no more. These figures will include all the digits you are sure of, plus one more: an estimated digit. Every one of those digits is then significant, or meaningful. (However, no zero digit to the left is significant, as we shall see later.) Thus an accurate measurement includes all the significant figures. So, for example 4.7 cm Sure figures = 4 Estimated figure = 0.7 (7/10 of the distance between 4 and 5) Accurate reading = 4.7 cm (2 significant figures) The following activity will illustrate these points: Activity. Measure the thickness of your textbook using a ruler graduated in mm, then try to answer these questions: 1. 2. 3. 4. 5. 6. How many sure figures do you have in the measurement? How many estimated figures? How many significant figures does your measurement contain? (You may need to read the above section again before answering this question). How does the magnitude (i.e. "size") of the last digit you have written compare to the graduation of your scale? Is the last figure exact, or is it uncertain? Is the whole measurement of the length exact or is it uncertain? We arrive at the following conclusions; 1. A measurement should be accurate. Therefore it should include all the sure figures plus one estimated figure, and all these figures are significant or meaningful. 2. The last figure that can be included in a measurement is usually half the smallest division and this last figure is the estimated figure. 3. The figures or digits that come after the estimated digit are unknown, and not zeros. 4. Every number obtained by measuring is uncertain. It is important to note that numbers not obtained by measuring may be exact. For example, by definition a quadrilateral has 4 sides. In this case the four is a mathematical number and it is exactly 4, which can be written as 4.000... etc. This number has as many significant figures as we want it to have. Also numbers obtained by counting are exact. Thus if a man says that he has 3 children, then he has exactly 3, and not 3.2 or 2.9 children. Another point to note is that the number of significant figures in a measured length depends on the magnitude (size) of the length and the graduations of the ruler. Thus if a length is measured to be 25.73 cm, then 1. This number has 4 significant figures. 2. If this number is written as 0025.73 cm, the number still contains only 4 significant figures, because all the digits to the left of 25.73 are already known to be zeros, and including them will not change the meaning of the measurement. Hence, zeros to the left of a number are not significant. 3. The digit after the 3 in 25.73 cm is not zero, but is unknown. We cannot write the number as 25.730, because by including more digits than we should we will be inaccurate, and we will mislead people into thinking that our measurement gave us five significant figures. Zeros written to the right are significant. 4. Changing the units of the measurement should not change the accuracy (i.e. the number of significant figures) of the number. Nothing done later (like changing the unit) can affect something done in the past (like actually making the measurement). Thus: 2m = 2 x 102 cm NOT 200cm 25.73 cm = 257.3 mm = 0.2573 m [these all have four significant figures.] QUESTIONS AND ANSWERS Example 1: (The last reading tells us about the smallest division of the measuring tape). A scientist uses a ruler to measure the length of a straight line. He gives his reading of the length as 23.91 cm. What is the smallest division on the ruler? Answer: The last digit given is estimated: the 1. We can only include one estimated digit. This digit is obtained by reading between two closest marks, so it is a fraction of the smallest division. Since the smallest division is one place-value higher than the 1, this means that the smallest division is the mm. Example 2: (The measuring tape determines the last reading). A ruler is graduated in cm (the smallest division is the centimetre). What is the least value (last) digit you can include in a measurement made by this ruler? Answer: The last digit should be estimated. We can only include one estimated digit. This digit is obtained by reading between two closest marks, so it is a fraction of the smallest division. Since the smallest division is a cm, a fraction of a division is millimetres. Therefore the reading should include centimetres, (sure figures), and a digit indicating millimetres (estimated). If we obtain a reading of 13.8 cm, The 13 cm are sure figure. The .8 cm (i.e. 8 mm) is estimated The number of significant figures is 3 (two sure and one estimated figures) Example 3: (The measuring tape determines the last reading). A ruler is graduated in cm and mm (the smallest division is the millimetre). What is the least value (last) digit you can include in a measurement made by this ruler? Answer: The last digit is obtained by reading between two closest marks, so it is a fraction of the smallest division. Since the smallest division is a mm, a fraction of a division is tenths of a millimetre. Therefore the reading should include centimetres, (sure figures), a figure indicating millimetres (also sure), and a figure indicating a fraction of a millimetre, and this is estimated. If we obtain a reading like 113.84 cm, The 113.8 are sure figures. The 0.04 is estimated The number of significant figures is 5 (4 sure and one estimated figures) Example 4: (The reading seems to be “exact”). A ruler is graduated in cm and mm (the smallest division is the millimetre). Using this ruler, a scientist makes out a reading to be 15.3 cm “exactly” (on the line), with no fractions of a mm. How should he write his reading? Answer: The last estimated digit, a fraction of a millimetre, is zero. This zero should be included in the reading, so the reading should be written as 15.30 cm It is wrong to write this reading as 15.3 cm, because this would give the impression that the smallest division on the ruler is a cm, and the last figure (3) is estimated. In fact, the 3 is sure, and the estimated digit is ‘0’. The reading 15.30 has four significant figures, the underlined digits in “15.30” are sure and the underlined digit in “15.30” is estimated. Example 5: (The reading seems to be “exact”). A ruler is graduated in cm and mm (the smallest division is the millimetre). Using this ruler, a scientist makes out a reading to be 15.00 cm “exactly”, with no fractions of a cm or mm. How should he write his reading? Answer: The last two digits, one sure and one estimated, are zeros. Both zeros should be included in the reading, so the reading should be written as 15.00 cm It is wrong to write this reading as 15 cm or 15.0 cm, because this would give the impression that the smallest division on the ruler is 10 cm or 1 cm respectively, and that the ‘5’ is estimated. In fact, the digit 5 is sure, the zero to its left is sure, and the last ‘0’ is estimated. The reading 15.00 has four significant figures, the underlined digits in 15.00 are sure and the underlined digit in 15.00 is estimated. Example 6: (When the units are changed, too many zeros appear). A tape graduated in m and cm, when used to measure the length of a football field, gives a reading of 93.54 m. Find the length in mm. 1mm Answer: 93.54 m = 93.54 m 3 = 93.54 103 mm = 93,540 mm 10 m The answer as written above is wrong, because it gives the impression that the answer has 5 significant figures, when it should have only 4. This is where the scientific notation helps. The answer should be written as 9.354 104 mm Example 7: (Zeros to the left are never significant). Using a tape graduated in m and cm the length of a room was found to be 93.54 m. Find the length in km. 1km Answer: 3.54 m = 3.54 m 3 = 3.54 103 km = 0.00354 km = 3.54 103 km 10 m The answer has 3 significant figures, no matter how it is written. Zeros to the left are not significant. Example 8: (Rounding gives a number ending with non-significant trailing zeros). Round the measurement 54,631 m to two significant figures. Answer: When we round to two significant figures we look at the digit giving the third significant figure. If it is a five or more we round up, otherwise we round down. 54,631 = 55,000 = 5.5 104 mm We should not leave the answer as 55,000 because this will give the false impression that the number has five significant figures, when it has only two. (Zeros to the right are significant.) Example 9: (Rounding a number ending with one non-significant zero). Round the measurement 54 m to one significant figure. Answer: When we round to one significant figure we look at the digit giving the second significant figure. If it is a five or more we round up, otherwise we round down. 54 = 50 = 5 10 m We should not leave the answer as 50. We must give it as 5 10, otherwise we will give the impression that it has two significant figures, when it has only one. (Zeros to the right are significant.) EXERCISE 1. What is meant by an accurate statement? 2. When you read a scale, how many sure and how many estimated figures should you include in the reading? Which of those digits is/are significant, or meaningful? 3. A student gives an accurate reading of the length of a pencil as 17.6 cm. Which of these digits are sure and which is estimated? How many significant figures does this measurement contain? How do you think the magnitude (i.e. "size") of the last digit written compares to the graduation of the scale used? Is the whole measurement of the length exact or uncertain? 4. Using the scale shown below, you measure an object around 15 cm long, and you write its accurate length as (i) 15 cm (ii) 15.4 cm (iii) 15.428 cm. a) Which of these readings is most accurate? b) What is the number of significant figures in the most accurate measurement? c) If the number is written as 0015 cm, how many significant figures will it have? d) True or false? The digit after the 5 in 15 cm is not zero, but is unknown. e) True or false? Changing the units of the measurement should not change the number of significant figures of the number: 15 cm = 0.15 m 5. Before answering this question remember that: (i) The number of significant figures in a measurement is determined by the measurer according to the rules you have learnt earlier. (ii) When you change units you cannot change the number of significant figures in the measurement. A scientist uses a ruler to measure a line and as far as he can tell the length of the line is twenty cm. How should he write this number (with the correct number of significant figures) if his ruler was graduated in (a) m? (b) cm? (c) mm? 6. Each of the following sentences has a number in it. Is each of these numbers exact or is it uncertain? (a) The length of a line is 13.42 cm. (b) A field was measured to be 123 m long. (c) A field was measured to be 123.0 m long. (d) A triangle has 3 sides. (e) “In the 8th grade of a school there are 65 girls.” 7. Referring to question 6: (a) How was the number in each of the sentences obtained? (b) Where a measuring rule was used, what was the smallest division on the rule? (c) How many significant figures are there in each of these numbers? (d) In each number, which figures are certain and which are not? 8. How many significant figures are in each of the following measurements? (a) 7.5 cm (b) 7.500 cm (c) 0.50 m (d) 1.2 × 103 mm. 9. A person reports the length of a fish to be 7.4 cm. The digit after the "4" is: (a) five (b) a zero (c) unknown 10. Change the following measurements into m, and write the correct number of significant figures in your answer. (a) 123 cm (b) 61.8 cm (c) 1.8 mm (d) 5.1km [Hint: express all answers in the scientific notation. 11. A room is measured to be 3.27m long. (a) How many significant figures are in this measurement? (b) How many sure figures are in this measurement? (c) How many estimated figures are in this measurement? (d) What is the smallest division on the scale used? 12. Assuming the lower extremity to be at the zero graduation in each case, write down the measurement of the line and the smallest graduation. Smallest graduation _________ cm Measurement ______________ 5 6 Smallest graduation _________ cm Measurement ______________ 2.5 2.6 Smallest graduation _________ m Measurement ______________ 0.01 0.02 13. Fred measured a line and wrote down its length as 9.876 cm. (a) What is the smallest graduation on a ruler which could be used to make this measurement? (b) Do you think that Fred's ruler was graduated that way? Explain. (c) Write down the measurement he would make if he used a ruler with (i) graduations every 1 cm; (ii) graduations every 1 mm. How to read a scale – answers The following answers refer to the questions at the end of the ‘How to Read a Scale’ booklet 1. One that has the correct number of significant figures. 2. 1 estimated digit; all digits to the left of the estimates digit should be included as these are ‘sure’. All digits are significant. 3. Sure = 17, estimated = 6; smallest graduation = 1 cm 4. (a) 15.4 cm 5. 20.0 x 10-2 m 6 (a) U (b) U (c) U (d) E (e) E 7. (a) Uncertain by measurement, exact by counting. (b) (i) mm (ii) 10m (iii) m (iv) (v) 4 significant figures 3 significant figures 4 significant figures infinite significant figures infinite significant figures 8. (a) 2 (c) 2 9. Unknown 10. (a) 1.23 m (b) 0.618 m 11. (a) 3 (c) 1 12. 1 mm, 5.53 cm 13. (a) 0.01 cm (b) 4 (b) 2 (b) 3 (c) 2 (d) True 20.0 cm (e)True 200mm (d) 2 (c) 1.8 x 10-3 m or 0.0018 m (d) 5.1 x 103 m (d) 10cm 0.1 mm 2.532 cm 10cm 0.0144 m (b) no because it is 0.1 mm !!! (c) 9.9 cm, 9.88 cm Part 2. Significant figures Estimation. Every measurement involves some estimation, so a measurement may have some certain digits and always has one uncertain digit. Measure the above line using a ruler graduated in mm. Your measurement must include only one estimated digit, which you obtain when you try to read between two millimetric divisions (e.g.; 10.03 cm). How many digits does your measurement contain? How many certain digits? How many estimated (uncertain)? All the above digits are significant. How many significant figures do you have in your measurement? Significant (and nonsignificant) figures: You may write as many zeros as you like to the left of the above measurement. They are all certain but not significant. However, you may not write as many zeros as you like to the right. Digits to the right of the estimated digit are totally unknown (they can be neither read nor estimated from your ruler), and it is wrong to think of them as zeros. They are unknown and nonsignificant. Example: 21.23 cm may be written as 00021.23 cm or as 0.0002123 km and remain as a 4 significant figure number. 21.23 cm may not be written as 21.2300 cm. 21.23 cm may be written as 2.123 × 105 micrometers or as 2.123 × 10-4 km. In each case it has 4 significant figures. One major reason for different people obtaining different measurements is that there are no perfect measuring instruments. When a copy is made of a standard unit, some estimation is involved. Manufacturing problems introduce errors as well, with the result that measuring the same quantity with different instruments may give different results. Another major reason for different people obtaining different measurements is that there are no perfect measurers. When a person takes a reading, (s)he has to estimate the last figure, and different people make different estimates. Accordingly, every measurement is uncertain. And since science is heavily dependent on measurement, every scientific statement is uncertain. Every measurement has units. Larger and smaller units can be obtained from the standard units by using prefixes, like c (centi = 102), m (milli = 103), k (kilo = 103) M (mega = 106) (micro = 106) and n (nano = 109). EXERCISE 1 1. Every measurement involves some estimation. Measure the line below and indicate which digits are sure and which digit is estimated. ________________________________________ 2. In the measurement of the line above, how many significant figures do you have? 3. Give two major reasons for uncertainty in measurements. 4. Some numbers are exact, like saying a square has four sides. How many significant figures are in 4 of the previous sentence? 5. How many significant figures do you have in each of: (a) 13.52 cm; (b) 35 students; (c) 0.035 kg (d) 100 km (e) 2.3 103 m. 6. What do the following prefixes mean? Find out and remember their values. (a) c as in cm (b) m as in mm (c) k as in km. (d) M as in Mton (e) as in g (f) n as in ns. 7. Express the number 23,400 to two significant figures in the scientific notation. 8. Simplify, giving the answer in scientific notation: 2.3 103 5.4 102 9. Simplify, giving the answer in scientific notation: (2.3 103) (5.4 102) Manipulating significant figures (a) ADDITION: In addition, the number of significant figures may increase. Example 1 72.53 cm + 93.13 cm 165.66 cm Example 2 72.57 cm + 9.1 cm 81.7 cm Notice that it is not allowed to add a zero to the right of the "1" in 9.1. Hence we cannot add 7 in 72.57 to an unknown digit, so we round up the .57 to a 0.6. But since the digit to the left of 9.1 is definitely a zero, it is possible to add 8 + 0 = 8. The answer has 3 significant figures, two are certain (81) and one uncertain (0.7). SUBTRACTION: The number of significant digits may decrease: Example 13.83 cm 13.1 cm 0.7 cm The answer has only one significant figure. MULTIPLICATION AND DIVISION: An answer will have as many significant figures as the operand with the fewer significant figures. Example: 19.23 cm × 3.2 cm = 62 cm2 EXERCISE 2 Give answers to the correct number of significant figures: (i) 13.83 cm + 13.1 cm (ii) 13.1 cm + 0.031 cm (iv) 19.23 cm × 7.2 cm (v) 73 cm × 81 cm (iii) (vi) 27.3 cm – 21.21 cm 150 cm3/bottle × 3 bottle Solutions Solutions for exercise 1 (first page). 4. infinite. 5. (a).4, (b) 2, (c) 2, (d) 3, (e) 2. 6. (a) centi, (b) milli, (c) kilo (d) mega (e) micro (f) nano. 7. 2.3 x 104 8. 1.8 x 103 9. 4.3 x 104 Solutions for exercise 2 (i) 26.9 cm (ii) 13.1 cm (iii) 6.1 cm (iv) 1.4 x 102 cm2 (v) 5.9 x 103 cm2 (vi) 450 cm3 Uncertainties in Measurement There are two methods of quoting the uncertainty in a value; they are to state the absolute error and to state the percentage error. To illustrate the two methods consider the case of a voltmeter observed to read 13.8V. The 3 digits are certain, so we can add an estimated digit to give the reading as 13.80 volts. The standard uncertainty is usually given as ± half the smallest graduation, so in this case the uncertainty is 0.05 volts. The reading can therefore be quoted as: 13.80 0.05 V absolute error 13.80 V 4% percentage error* (0.05/13.80) = 4% *since Error calculations Error analysis goes far beyond the scope of any Advanced-level course. For advancedlevel work three simple rules are required for calculating experimental errors, these are: 1. When two quantities are to be added or subtracted, then the absolute errors are added to give the absolute error. 2. When two quantities are to be multiplied or divided then the percentage errors are added together to give the percentage error. 3. When a quantity is to be raised to a power n then the percentage error is multiplied by n. (Since X3 = X*X*X which is the same as rule 2 to give 3 times error.) IB Past paper questions 1. When a voltage V of 12.2 V is applied to a DC motor, the current I in the motor is 0.20 A. Which one of the following is the output power VI of the motor given to the correct appropriate number of significant digits? A. 2W B. 2.4 W C. 2.40 W D. 2.44 W (1) 2. Natalie measures the mass and speed of a glider. The percentage uncertainty in her measurement of the mass is 3% and in the measurement of the speed is 10%. Her calculated value of the kinetic energy of the glider will have an uncertainty of A. 30%. B. 23%. C. 13%. D. 10%. (1) 3. A student measures the current in a resistor as 677 mA for a potential difference of 3.6 V. A calculator shows the resistance of the resistor to be 5.3175775 Ω. Which one of the following gives the resistance to an appropriate number of significant figures? A. 5.3 Ω B. 5.32 Ω C. 5.318 Ω D. 5.31765775 Ω (1) 4. When a force F of (10.0 0.2) N is applied to a mass m of (2.0 0.1) kg, the F percentage uncertainty attached to the value of the calculated acceleration is m A. 2 %. B. 5 %. C. 7 %. D. 10 %. (1) 5. A student measures two lengths as follows: T = 10.0 0.1 cm S = 20.0 0.1 cm. The student calculates: FT, the fractional uncertainty in T FS, the fractional uncertainty in S FT–S, the fractional uncertainty in (T – S) FT+S, the fractional uncertainty in (T + S). Which of these uncertainties has the largest magnitude? A. FT B. FS C. FT–S D. FT+S (1) 6. The volume V of a cylinder of height h and radius r is given by the expression V = πr2h. In a particular experiment, r is to be determined from measurements of V and h. The uncertainties in V and in h are as shown below. V 7% h 3% The approximate uncertainty in r is A. 10 %. B. 5 %. C. 4 %. D. 2 %. (1) 7. The radius of a loop is measured to be (10.0 ± 0.5) cm. Which of the following is the best estimate of the uncertainty in the calculated area of the loop? A. 0.25 % B. 5% C. 10 % D. 25 % (1) Solutions to above IB questions 1. B [1] 2. B [1] 3. A [1] 4. C [1] 5. C [1] 6. B (accept D) [1] 7. C [1] Additional questions 1. (a) The capacity of a glass is measured, using a measuring cylinder, to be 285 cm3. Which of these figures are certain and which are estimated? (b) Do exact numbers exist in science? If yes, give an example. 2. (a) What are the rules concerning the number of significant figures in the answer when two numbers are: (i) multiplied. (ii) divided. (iii) added. (iv) subtracted. (b) Calculate the following to give the answer to the correct number of significant figures. (i) 1.89 cm + 0.085 cm. (ii) 98 cm + 1.65 cm. (iii) 1.8 L - 0.9 L (iv) 3.7 L - 0.0342 L (v) 12 cm x 100 cm (vi) 0.021 cm x 4000 cm (vii) The total volume of 5 bottles of 750.0 cm3. (viii) 12.3 L 6.0 (ix) The area of a square of grass if a field contains 200 identical squares and the area of the field is 16000 cm2. 3. Convert: (a) 245 cm into (i) mm (ii) m (b) 4.28 m2 into (i) cm2 (ii) km2 (c) 4.28 L into (i) cm3 (ii) m3 (d) 1.58 mg into (i) g (ii) kg (e) 7.20 106 s into (i) minutes (ii) ms (iii) h 4. (a) Find the speed of a train in ms1 if it is travelling at 144 km/h. (b) Find the speed in km/h of a horse running at 5.00 ms1. These are the solutions for the additional questions 1. (a) 28 is certain, 5 is estimated. 2. (a) Review from text (b) (i) 1.98 cm (vi) 84 cm2 3. 4. (ii) 100 cm (b) counting numbers are exact. (iii) 0.9 L (iv) 3.7 L (v) 12 x 102 cm2 (vii) 3.75 x 103 cm3 (viii) 2.1 L (a) 245 x 101 mm 2.45 m (b) 4.28 x 104 cm2 4.28 x 10-6 km2 (c) 4.28 x 103 cm3 4.28 x 10-3 m3 (d) 1.58 x 103 g 1.58 x 106 kg (e) 120 x 103 mins 7.20 x 109 ms (a) 40.0 m/s (b) 18.0 km/h (ix) 80.000 cm2 2.00 x 103 hr
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