SPHU/4C REVIEW MEASUREMENT & ANALYSIS NOTES
CONTENTS Scientific Notation & SI. Scientific Notation. SI Accuracy. The Accuracy of Measured Quantities Precision. The Precision of Measured Quantities 4 Rounding Off 4. Rounding Off Numbers 4 5 Calculations 5. Calculations Involving Measured Quantities 5 6 Error 6. Expressing Error in Measurement 6 7 Measurements 7. Measuring and Estimating 7 7. Making Precision Measurements 7 7. Steel Rules 7 7.4 Vernier Calipers 8 8 Trigonometry 8. Right Triangles 9 9 Answers Sections 4 Sections 5 8
Scientific Notation & SI. Scientific Notation In science we frequently encounter numbers which are difficult to write in the traditional way - velocity of light, mass of an electron, distance to the nearest star. Scientific notation, or standard notation, is a technique, using powers of ten, for concisely writing unusually large or small numbers. In scientific notation, the number is expressed by writing the correct number of significant digits with one non-zero digit to the left of the decimal point, and then multiplying the number by the appropriate power (+ or -) of ten (). Therefore, if you had a number like 94 and wanted to use scientific notation, the answer would be.94. Scientific notation also enables us to show the correct number of significant digits. As such, it may be necessary to use scientific notation in order to follow the rules for certainty (discussed later).. Express each of the following in scientific notation. (a) 6 87. 5 (c) 9 879 8 (d). 8 (e).7 4 (f) 4 (g).8 (h) 68. Express each of the following in common notation. (a) 7 5. (c) 8. 9 (d). - (e) 6.86 8 (f) 4.86 - (g) 6.. SI (h) 5. - Over hundreds of years, physicists (and other scientists) have developed traditional ways (or rules) of expressing their measurements. If we can t trust the measurements, we can put no faith in reports of scientific research. The Systeme International d Unites (SI) is a revised and modernized descendant of the metric system of measurement with a few exceptions. For example, atmospheric pressure in SI is measured in kilopascals (kpa) whereas in the metric system it is measured in millimetres of mercury (mmhg). In the SI system all physical quantities can be expressed as some combination of fundamental units, called base units (refer to Appendix C Some SI Derived Units on page 577 of Nelson Physics or page 57 of Nelson Physics ). The SI system is used for scientific work throughout the world. Everyone accepts and uses the same rules, and understands that there are limitations to the rules. The SI convention includes both quantity and unit symbols. Note that these are symbols (e.g., 6 km/h) and are not abbreviations (e.g., 4 mi./hr.). When converting units the method most commonly used is multiplying by conversion factors (equalities), which are memorized or referenced (e.g., m = cm, h = 6 min = 6 s). It is also important to pay close attention to the units, which are also converted by multiplying by a conversion factor (e.g., m/s =.6 km/h).. Use the Factor/Prefix chart given to the right to convert each of the following measurements to their base unit. (a) 5.7 GW (f) 67 MJ 7 cm (g) 6 C (c) 5 km (h).5 cm (d) 6.8 mm (i) 548 m (e).75 km (j) mm 4. Convert the following measurements to the units indicated. (a) year (s) (c). m/s (km/h) 8. s (yr) (d) 8 km/h (m/s) 5. An athlete completed a 5-km race in 9.5 min. Convert this time into hours. 6. A train is travelling at 95 km/h. Convert 95 km/h into metres per second (m/s). FACTOR PREFIX SYMBOL 9 giga G 6 mega M kilo k hecto h deka da ----- ----- - deci d - centi c - milli m -6 micro -9 nano SCIENCE IS DOWNRIGHT UPLEFTING! MEASUREMENT AND ANALYSIS -
Accuracy. The Accuracy of Measured Quantities Every measurement has a degree of certainty and uncertainty. As such, there is an international agreement about the correct way to record measurements Record all those digits that are certain plus one uncertain digit, and no more!. These certain-plus-one digits are called significant digits. Thus, the certainty or accuracy of a measurement is indicated by the number of significant digits. Exact Numbers All counted quantities are exact and contain an infinite number of significant digits. For example, if we count the students in a class, and get, we know that. or.9 are not possible. Only a whole-number answer is possible. Numbers obtained from definitions are considered to be exact and contain an infinite number of significant digits. As such, they do not influence the accuracy of any calculation. For example, m = cm, kwh = 6 kj, and =.459654 are all definitions of equalities. has an infinite number of decimal places as do numbers in equations such as C = r. When Digits Are Significant All non-zero digits are significant; e.g., 59.69 has five significant digits. Any zeros between two non-zero digits are significant; e.g., 66 has three significant digits. Any zeros to the right of both the decimal point and a non-zero digit are significant; e.g., 7. has four significant digits. All digits (zero or non-zero) used in scientific notation are significant. When Digits Are Not Significant Any zeros to the right of the decimal point but preceding a non-zero digit (i.e., leading zeros) are not significant; they are placeholders. For example,.9 and. both have two significant digits. Ambiguous case: Any zeros to the right of a non-zero digit (i.e., trailing zeros) are not significant; they are placeholders. For example, 98 and 5 both have two significant digits. If the zeros are intended to be significant, then scientific notation must be used. For example,.5 has two significant digits and.5 has four significant digits. (An exception to this statement is the following: unless the number of significant digits can be assessed by inspection. For example, a reading of 5 km on a car s odometer has four significant digits.). How many significant digits are there in each of the following measured quantities? (a) 5 g 865.7 cm (g).45 ml (h) 4.7 nm (m). m (n) 6 5. mm (s) 46. m (t). 68 m (c) 96.66 L (i). MW (o) 47. m (u).7 m (d) 5 s (j).6 ns (p) 4.6 kg (v) 98 s (e) 76 6 g (k). 4 GW (q). 67 s (w) 7.6 L (f) 7.5 kg (l).5 km (r) 6. cm (x).5 mm. Express each of the numbers above in scientific notation with the correct number of significant digits. MEASUREMENT AND ANALYSIS -
Precision. The Precision of Measured Quantities Measurements are always approximate. They depend on the precision of the measuring instruments used, that is, the amount of information that the instruments can provide. For example,.86 cm is more precise than 58.86 cm because the three decimal places in.86 makes it precise to the nearest one-thousandth of a centimetre, while the two decimal places in 58.86 makes it precise only to the nearest one-hundredth of a centimetre. Precision is indicated by the number of decimal places in a measured or calculated value. All measured quantities are expressed as precisely as possible. All digits shown are significant with any error/ or uncertainty in the last digit; e.g., in 87.64 cm the uncertainty is with the digit 4. The precision of a measuring instrument depends on its degree of fineness and the size of the unit being used. Using an instrument with a more finely divided scale allows us to take a more precise measurement. Any measurement that falls between the smallest divisions on the measuring instrument is an estimate. W e should always try to read any instrument by estimating tenths of the smallest division; e.g., for a ruler calibrated in centimetres, this means estimating to the nearest tenth of a centimetre or to mm. The estimated digit is always shown when recording the measurement; e.g., in 6.7 cm the 7 would be the estimated digit. Should the object fall right on a division mark, the estimated digit would be ; e.g., if we use a ruler calibrated in centimetres to measure a length that falls exactly on the 6 cm mark, the correct reading is 6. cm, not 6 cm.. Use the three centimetre rulers to measure and record the length of the pen graphic. (a) Child s ruler Elementary ruler (c) Ordinary ruler. An object is being measured with a ruler calibrated in millimetres. One end of the object is at the zero mark of the ruler. The other end lines up exactly with the 5. cm mark. W hat reading should be recorded for the length of the object? W hy?. Which of the following values of a measured quantity is most precise? (a) 4.8 mm,.8 mm, 48. mm,.8 mm.54 cm,.65 cm, 6 cm,.54 cm,.4 cm 4. Refer to page. State the precision of the measured quantities (a) to (x) in practice question #. MEASUREMENT AND ANALYSIS -
4 Rounding Off 4. Rounding Off Numbers When making measurements, or when doing calculations, the final answer should have the same number of significant digits as the least accurate number in the calculation. For example,.6 +. =.9 not.9. The procedure for dropping off digits is called rounding off. Rounding Down When the digits dropped are less than 5, 5, 5, etc., the remaining digit is left unchanged. Example: 4. becomes 4. rounding based on the " 4. rounding based on the " Rounding Up When the digits dropped are greater than 5, 5, 5, etc., the remaining digit is increased or rounded up. Example: 4.756 becomes 4.76 rounding based on the 6" 4.8 rounding based on the 56" Rounding with 5, 5, 5, etc. When the digits dropped are exactly equal to 5, 5, 5, etc., the remaining digit is rounded to the closest even number. This is sometimes called the Even-Odd Rule. Example: 4.85 becomes 4.8 rounding based on the 5" 4.75 becomes 4.8 rounding based on the 5". Round off the following numbers to two significant digits. (a) 7.86 (f).75 (k) 74 85.5 (g) 6.75 (l).599 (c).85 (h) 49. (m) 45 (d) 9.865 (i).6 7 (n). 6 (e) 4.67 (j) 648 7 (o). 45. Round off each of the following to the number of significant digits specified. -5 (p) 6 87 (q). 5 (r).7 4 (s) 68 (t).8 5 (a).786 L () (f) 67.8 5 L () 6.85 m () (g) 68 75 m () 7 (c) 6.49 98 s () (h).87 55 L () (d) 876 m () (i) 7.6 s () (e).77 5 g () (j) 7 6 g (). Convert a measured length of 897 cm to metres and show the correct number of significant digits in the answer. 4. Copy and complete the following chart. Measurement Precision # of Sig. Dig. Measurement Now Needed Rounded Off Measurement in Sci. Not. 6.479 km (example) 5 6.5 km 6.5 km a 46 597. cm b.58 L c 5 kg d.67 8 mm e 485. kw 4 MEASUREMENT AND ANALYSIS - 4
5 Calculations 5. Calculations Involving Measured Quantities If measurements are approximate, the calculations based on them must also be approximate. Scientists agree that calculated answers should be rounded so they do not give a misleading idea of how precise the original measurements were. For multiple-step calculations, leave all digits in your calculator until you have finished all your calculations, then round the final answer. Otherwise you could be introducing error into your calculations. When multiplying and/or dividing, the answer has the same number of significant digits as the measurement with the fewest number of significant digits (e.g., 4.5 x 6.4 = 9.675 which rounds to 9.6 because 6.4 has three significant digits whereas 4.5 has four significant digits). When adding and/or subtracting measured values of known precision, the answer has the same number of decimal places as the measured value with the fewest decimal places (e.g., 6.6 + 8.74 +.766 = 6.6 which rounds to 6. because 6.6 has one decimal place whereas 8.74 has two decimal places and.766 has three).. Perform the following operations. Express your answer to the correct accuracy or precision warranted. (a) 67.8 + 968 +.87 5 - (j) (.75 )(.75 ) 46.66 + 9. +.7 - -4 (k) (.65 )(7. ) (q) (c) 6.6 + 87.45 + 68. (l) - 7 (.987 )(4.9 ) (d) 68.7 -.95 (e).9875 -.875 (f).97 -.5 - (m) (7.8 ) (.8 ) - (n) (8.875 ) (7.97 ) -4-7 (o) (5.9 ) (. ) (r) (s).5 (g) (.6)(4.) (t) (p) (h) (65)(.4)(5) (i) (.6)(6)(55.). Simplify each of the following, using scientific notation where appropriate. (a) 4 - (j) (.5 )(. ) 4-5 - (k) (5. )(4. ) (o) - 5 (c) -6 (d) 5 (e) (l) (p) 4 7 (f) (m) -6 (g) (q) -6-7 (h) (i) (.4 )( ) (n) (r). Each of the following questions has some measured data. Perform the required calculations with the data. Give your answer to the accuracy or precision warranted by the data. (a) Find the perimeter of a rectangular carpet that has a width of.56 m and a length of 4.5 m. Find the area of a rectangle whose sides are 4.5 m and 7.5 m. (c) A triangle has a base of 5.75 cm and a height of.45 cm. Calculate the area of the triangle. -9 (d) If a gold atom is considered to be a cube with sides.5 m, how many gold atoms could be stacked on -7 top of one another in a piece of gold foil with a thickness of. m? 5 (e) On the average,. kg of aluminum consists of. atoms. How many atoms would there be in a block of aluminum cm by. cm by 5.6 cm? ( Aluminum =.7 kg/m, or.7 g/cm.) 4. Solve each of the following word problems. Watch your accuracy/precision and show all your work! (a) On the planet Zot distances are measured in zaps and zings. If.9 zings equal 7.5 zaps, how many zings are equal to 9.5 zaps? 8 Neptune is 4.5 m from the Sun. If light travels at. m/s, how long does it take light from the Sun to reach Neptune? 4 7 (c) The Earth has a mass of 5.98 kg while Jupiter has a mass of.9 kg. How many times larger is the mass of Jupiter than the mass of the Earth? 9 (d) The average distance between the Sun and Pluto (the farthest planet) is 5.9 km, and the speed of light 5 is km/sec. Approximately how many hours does it take sunlight to reach Pluto? MEASUREMENT AND ANALYSIS - 5
6 Error 6. Expressing Error in Measurement In everyday usage, "accuracy" and "precision" are used interchangeably, but in science it is important to make a distinction between them. Accuracy refers to the closeness of a measurement to the accepted value for a specific quantity. Precision is the degree of agreement among several measurements that have been made in the same way. Think of shooting towards a bullseye target: No Precision No Accuracy Precision without Accuracy Precision with Accuracy Error is the difference between an observed value (or the average of observed values) and the accepted value. The size of the error is an indication of the accuracy. Thus, the smaller the error, the greater the accuracy. Every measurement made on every scale has some unavoidable possibility of error, usually assumed to be one-half of the smallest division marked on the scale. The accuracy of calculations involving measured quantities is often indicated by a statement of the possible error. For example, you use a ruler calibrated in centimetres and millimetres to measure the length of a block of wood to be.6 mm (6 is the estimated digit in this measurement). The possible error in the measurement would be indicated by.6 ±.5 mm. Relative error is expressed as a percentage, and is usually called percentage error. Sometimes if two values of the same quantity are measured, it is useful to compare the precision of these values by calculating the percentage difference between them.. Refer to page. State the possible error of measurements (a) to (c) in practice question #.. At a certain location the acceleration due to gravity is 9.8 m/s [down]. Calculate the percentage error of the following experimental values of g at that location. (a) 8.94 m/s [down] 9.95 m/s [down]. Calculate the percentage difference between the two experimental values (8.94 m/s and 9.95 m/s ) in question # above. MEASUREMENT AND ANALYSIS - 6
7 Measurements 7. Measuring and Estimating Many people believe that all measurements are reliable (consistent over many trials), precise (to as many decimal places as possible), and accurate (representing the actual value). But there are many things that can go wrong when measuring. There may be limitations that make the instrument or its use unreliable (inconsistent). The investigator may make a mistake or fail to follow the correct techniques when reading the measurement to the available precision (number of decimal places). The instrument may be faulty or inaccurate; a similar instrument may give different readings. To be sure you have measured correctly, you should repeat your measurements at least three times. If your measurements appear to be reliable, calculate the mean and use that value. To be more certain about the accuracy, repeat the measurements with a different instrument. 7. Making Precision Measurements In order to make precise measurements you need to use a device that has a double scale. A double scale consists of a main scale that is an ordinary metric scale with centimetres and millimetres and a sliding or vernier scale. There are graduations on the vernier scale that occupy the same space as 9 graduations on the main scale. Therefore, only one graduation on the vernier can line up with a graduation on the main scale. A double scale can be placed on various types of instruments. One common instrument is the vernier caliper. It is used to measure the outside diameter of a cylinder or the inside diameter of a hollow cylinder. Another instrument with a double scale is the outside micrometer caliper. W hen using high-precision instruments, such as the vernier caliper or outside micrometer caliper, it is necessary to check the zero setting before taking a reading. If, for example, the instrument is supposed to read. cm but instead reads. cm, the error must be taken into consideration with each reading. NOTE: An error caused by a problem with the measuring device or the person using it is known as a systematic error. Some examples are a metre stick with worn ends or a dial instrument that is not properly zeroed. One way to reduce this type of error is to add or subtract the known error. Eliminating systematic error increases the accuracy of a measurement. In other words, it gives a reading closer to the true value. Notice that accuracy is different from precision. 7. Steel Rulers Metric rulers are graduated in millimetres and half millimetres. Whenever possible, it is advisable to butt the end of a ruler against a shoulder or step, to assure an accurate measurement. Through constant use, however, the end of a steel ruler becomes worn. Measurements taken from the end are therefore often inaccurate. Accurate measurements of flat work may be made by placing the cm graduation line on the edge of the work, taking the measurement, and subtracting cm from the reading. When measuring the diameter of a round object it is also advisable to start from the cm graduation line.. Use a ruler calibrated in centimetres to measure the distance from the centre of each symbol to the next. Include the error in your measurement. + + + + + + (a) (c) (d) (e) MEASUREMENT AND ANALYSIS - 7
7.4 Vernier Calipers Vernier calipers are precision measuring instruments used to make accurate measurements. The bar and movable jaw are graduated on both sides, one side for taking outside measurements and the other side for inside measurements. Vernier calipers are available in metric and in inch graduations, and some types have both scales. Parts Of A Metric Vernier Caliper The parts of the vernier caliper remain the same regardless of the measurement system for which the instrument is designed. Our metric calipers measure in centimetres with a precision of ±.5 cm (because the main scale on the bottom of the bar is graduated in millimetres). The graduations on the sliding or vernier scale, occupy the same space as 9 graduations on the main scale (9 mm). Therefore, only one graduation on the vernier scale can line up with a graduation on the main scale. How To Read A Metric Vernier Caliper Find the first line (the ZERO line) on the small sliding scale. Look on the stationary scale and record the number you just passed (or are currently on) as #.# cm. Find the FIRST pair of lines that match up perfectly. Read the line number off the small sliding scale as?. Determine the error (half of the smallest measurement possible). Record your measurement as #.#? ±.5 cm. What is the reading (including the error) of the following metric vernier caliper settings? (a) (d) (e) (c) (f) MEASUREMENT AND ANALYSIS - 8
8 Trigonometry 8. Right Triangles Trigonometry deals with the relationships between the sides and angles in right triangles. For a given angle in a right triangle, there are three important ratios. These are called the primary trigonometric ratios. The primary trigonometric ratios can be used to find the measures of unknown sides and angles in right triangles. NOTE: The values of the trigonometric ratios depend on the angle to which the opposite side, adjacent side, and hypotenuse correspond. If the value of a trigonometric ratio is known, its corresponding angle can be found on a scientific calculator using the inverse of that ratio. - - - (i.e., use nd SIN for sin, nd COS for cos, and nd TAN for tan ).. Determine the value of each ratio to four decimal places. (a) sin 5 cos 6 (c) tan 45 (d) cos 75 (e) sin 8 (f) tan 8 (g) cos 88 (h) sin 7. Determine the size of A to the nearest degree. (a) sin A =.599 cos A =.46 (c) tan A = 4.5 (d) cos A =.5. Solve for x to one decimal place. (a) sin 5 = x/8 cos 7 = x/5 (c) tan = x/9 (d) tan 55 = 8/x 4. Solve for B to the nearest degree. (a) cos B = /8 sin B = 7/8 (c) tan B = 5/9 (d) cos B = 6.8/.5 (e) sin A =.49 (f) tan A =.875 (e) sin = /x (f) sin 75 = 5/x (e) tan B = 5/ (f) sin B = ½ (g) cos A =.77 (h) sin A =.889 5. Use two different methods to find the value of the unknown in each triangle. Round your answer to one decimal place. (a) (c) MEASUREMENT AND ANALYSIS - 9
9 Answers Scientific Notation & SI Page -. (a) 6.87 (e) 7.4-5 5. (f) 4 (c).987 98 (g) - 8. (d) -7 8. (h) 6.8. (a) 7 (e) 6 86.8 5 (f).4 86 (c) 8 (g) 6 (d). (h).5. (a) 9 5.7 W = 5 7 W - 7 m =.7 m (c) 5 m = 5 m (d) - 6.8 m =.68 m (e).75 m = 75 m (f) 6 67 J = 67 J (g) -6 6 C =. 6 C (h) -.5 m =.5 m (i) -9 548 m =. 548 m (j) - m =. m 4. (a) 7 ~.5 s (c).6 km/h ~. years (d) 5 m/s 5..5 hr 6. ~6 m/s Accuracy Page. (a) (m) 4 4 (n) 6 (c) 6 (o) (d) (p) 4 (e) (q) (f) (r) (g) 5 (s) 4 (h) 5 (t) (i) (u) (j) (v) (k) (w) (l) (x). (a).5 g (m). m 8.657 cm (n) 6.5 mm (c) 9.666 L (o) 4.7 m (d) 4.5 s (p) 4.6 kg (e) 7 7.66 g (q) 5 6.7 s (f) 7.5 kg (r) 6. cm (g).45 ml (s) 4.6 m (h) 4.7 nm (t) 6.8 m (i) MW (u) 7 m (j) 6 ns (v) 9.8 s (k).4 GW (w) 7.6 L (l) 5. km (x) 5. mm Precision Page. (a) 4.4 cm (c) 4.5 cm 4. cm. 5. cm. (a).8 mm.4 cm 4. (a) (m) (n) (c) (o) (d) (p) (e) (q) 6 (f) (r) (g) (s) (h) (t) (i) (u) (j) (v) (k) 5 (w) (l) (x) 4 4 Rounding Off Page 4. (a) 7.8 (k) 7. 86 (l).6 (c).8 (m) 44 (d) 9.9 (n).6 (e) 4.7 (o). (f).8 (p) 68 (g) 6.8 (q).5 (h) 49 (r).7 (i).6 (s) 68 (j) 65 (t).8. (a) 5.8 L (f) 67 L 6.8 m (g) 687 m (c) 7 6. s (h).87 L (d) 88 m (i) 7. s (e).77 g (j) 7 g. 8.97 m, 4 significant digits 4. (a), 6, 47 cm, 4 4.7 cm 4, 4,.6 L, - 6 L (c),, kg,. kg (d) - 5, 4,.678 mm, 6.78 mm (e), 6, 485. kw, 4.85 kw MEASUREMENT AND ANALYSIS -
5 Calculations Page 5. (a) 4 (k). 6 49. (l).9 5 (c) 985.94 (m).6 5 (d) 44.8 (n).46 5 (e). (o).6 (f).58 (p) 6 (g) (q) 8 (h) 87 (r) 9 (i) 8.6 (s) (j).98 (t).. (a) 6 (c) (d) 4 (e) (f) (g) 8 (h) (i) 4 or 4 (j) 7. or 7 (k) 7. or. (l) 4.7 or 7 (m).4 5 (n).8 8 (o). 7 (p) 5 8. or.8 (q) 6. can t use 6 (only sig.dig.) so have to use sci.not. (r) 4. or 4.. (a) 6. m 4 m (c) 5.8 cm (d) 4 or 4. atoms (e) 5. atoms 8 Trigonometry Page 9. (a).576 (e).9.5 (f).78 (c). (g).49 (d).588 (h).9. (a) (e) 4 65 (f) 5 (c) 77 (g) 45 (d) 6 (h) 6. (a) 4.6 (d) 5.6 5. (e) 69. (c) 6.9 (f) 5. 4. (a) 68 (d) 9 6 (e) 64 (c) 59 (f) 5. (a) x = 7 + sin 5 = 7/x cos 5 = /x x =. mm 8.8 = x +.6 cos 4 = x/8.8 tan 4 =.6/x x = 8. km (c) 6 = x + 6 sin = x/6 cos = 6/6 = 64 x = 54 cm 4. (a) 49 zings 4.5 s (c) 8 times (d) 5 hr 6 Error Page 6. (a) ±.5 cm (c) ±.5 cm ±.5 cm. (a) 8.96%.%. (a).7% 7 Measurements Page 7 & 8. Done in class!. (a).69 ±.5 cm.8 ±.5 cm (c).77 ±.5 cm (d).68 ±.5 cm (e).9 ±.5 cm (f) 7.5 ±.5 cm MEASUREMENT AND ANALYSIS -