SPH4C COLLEGE PHYSICS

SPH4C COLLEGE PHYSICS REVIEW: MATH SKILLS L Scientific Notation (P.547) 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. Expression Common decimal notation Scientific notation 14.5 million kilometres 154 thousand picometres 60 sextillion molecules 14 500 000 km 1.45 x 10 8 km 154 000 pm 1.54 x 10 5 pm 60 000 000 000 000 000 000 000 molecules 6.0 x 10 molecules August 15, 01 4CR - Scientific Notation 1 Scientific Notation SCIENTIFIC NOTATION uses powers of ten to write large/small numbers Expression Common decimal notation Scientific notation 14.5 million kilometres 154 thousand picometres 60 sextillion molecules 14 500 000 km 1.45 x 10 8 km 154 000 pm 1.54 x 10 5 pm 60 000 000 000 000 000 000 000 molecules 6.0 x 10 molecules August 15, 01 4CR - Scientific Notation 1

Scientific Notation In scientific notation, the number is expressed by: 1. 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 (10). August 15, 01 4CR - Scientific Notation Scientific Notation For example, 94 0.067 =.94 x 1000 = 6.7 x 0.01 =.94 x 10 = 6.7 x 10 - NOTE! 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). August 15, 01 4CR - Scientific Notation 4 Scientific Notation 1. Express each of the following in scientific notation. (a) 6 807 (b) 0.000 05 (c) 9 879 80 000 (d) 0.000 000 81 (e) 0.070 40 (f) 400 000 000 000 (g) 0.80 (h) 68 6.807 x 10 5. x 10-5.98798 x 10 10 8.1 x 10-7 7.040 x 10-4 x 10 11 8.0 x 10-1 6.8 x 10 1 August 15, 01 4CR - Scientific Notation 5

Scientific Notation. Express each of the following in common notation. (a) 7 10 1 (b) 5. 10 (c) 8. 10 9 (d) 10.1 10 - (e) 6.86 8 10 (f) 4.086 10 - (g) 6. 10 (h) 5.0 10-70 5 00 8 00 000 000 0.101 6 86.8 0.004 086 60 0.05 0 August 15, 01 4CR - Scientific Notation 6 Scientific Notation With Calculators On many calculators, scientific notation in entered using a special key, labelled EXP or EE. This key includes x 10 from the scientific notation; you need to enter only the exponent. For example, to enter 7.5 x 10 4 press 7.5 EXP 4.6 x 10 - press.6 EXP +/- NOTE! Depending on the type of calculator you have, the +/- signs may need to be entered after the relevant number. August 15, 01 4CR - Scientific Notation 7

SPH4C COLLEGE PHYSICS REVIEW: MATH SKILLS L International System of Units (SI) (P.57) SI 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. As such, the International System of Units (SI) is used for scientific work throughout the world everyone accepts and uses the same rules, and understands that there are limitations to the rules. August 15, 01 4CR - SI 1 SI SI RULES In the SI system all physical quantities can be expressed as some combination of fundamental units, called base units. (i.e., mol, m, kg, EC, s,...). For example: 1 N = 1 kg@m/s 7 unit for force 1 J = 1 kg@m /s 7 unit for energy August 15, 01 4CR - SI 1

SI SI RULES The SI convention includes both quantity and unit symbols. Note that these are symbols (e.g., 60 km/h) and are not abbreviations (e.g., 40 mi./hr.). August 15, 01 4CR - SI SI SI RULES When converting units the method most commonly used is multiplying by conversion factors (equalities), which are memorized or referenced (e.g., 1 m = 100 cm, 1 h = 60 min = 600 s). August 15, 01 4CR - SI 4 SI SI RULES It is also important to pay close attention to the units, which are converted by multiplying by a conversion factor (e.g., 1 m/s =.6 km/h). August 15, 01 4CR - SI 5

SI USEFUL CONVERSION FACTORS! H H H H H H H G 1000 M 1000 k 1000 base 100 c 10 m 1000 : 1000 0 ) ) ) u ) ) ) ) n it H H H H H m/s.6 km/h yr 65 d 4 hr 60 min 60 sec ) ) ) ) ) August 15, 01 4CR - SI 6 SI 1. Use the chart to convert each of the following measurements to their base unit. (a) 5.7 GW (b) 7 cm (c) 6 µc (d) 0.50 MJ (e) 6.8 ml (f) 548 ηm (g) 0.75 kg 5.7 x 10 9 W 7 x 10 - m 6 x 10-6 C 0.50 x 10 6 J 6.8 x 10 - L 548 x 10-9 m 0.75 x 10 g Power Prefix Symbol 10 9 giga G 10 6 mega M 10 kilo k 10 0 ----- ----- 10 - centi c 10 - milli m 10-6 micro : 10-9 nano 0 NOTE! This is only a partial list - refer to P.661 for a complete list. August 15, 01 4CR - SI 7 SI. An athlete completed a 5-km race in 19.5 min. Convert this time into hours. 19.5 min x = 0.5 hours 1 hour 60 min. A train is travelling at 95 km/h. Convert 95 km/h into metres per second (m/s). 1000 m 1 hour 95 km/h x x 1 km 600 s = 6.4 m/s August 15, 01 4CR - SI 8

SPH4C COLLEGE PHYSICS REVIEW: MATH SKILLS L Uncertainty in Measurements (P. 546) Uncertainty in Measurements There are two types of quantities used in science: exact values and measurements. Exact values include defined quantities (1 m = 100 cm)andcountedvalues(5beakersor10trials). Measurements, however, are not exact because there is always some uncertainty or error associated with every measurement. As such, there is an international agreement about the correct way to record measurements. 1 Significant Digits The certainty of any measurement is communicated by the number of significant digits in the measurement. In a measured or calculated value, significant digits are the digits that are known reliably, or for certain, andincludethelastdigitthatisestimatedoruncertain. Assuch,thereare a set of rules that can be used to determine whether or not a digit is significant(refer to P.650 of your text). SIGNIFICANT DIGITS digits that are certain plus one estimated digit indicates the certainty of a measurement rules L P.650 1

Significant Digits WHEN DIGITS ARE SIGNIFICANT 1. All non-zero digits(i.e., 1-9) are significant. For example: 59.69 has five significant digits 61. has three significant digits Significant Digits WHEN DIGITS ARE SIGNIFICANT. Any zeros between two non-zero digits are significant. For example: 606 has three significant digits 6006 has four significant digits 4 Significant Digits WHEN DIGITS ARE SIGNIFICANT. Any zeros to the right of both the decimal point and anon-zero digit are significant. For example: 7.100 has four significant digits 7.10 has three significant digits 5

Significant Digits WHEN DIGITS ARE SIGNIFICANT 4. All digits(zero or non-zero) used in scientific notation are significant. Forexample:.4x10 hastwosignificantdigits.400x10 hasfoursignificantdigits 6 Significant Digits WHEN DIGITS ARE SIGNIFICANT 5. All counted and defined values have an infinite number of significant digits. For example: 16 students has 4 significant digits B =.1415... has 4 significant digits 7 Significant Digits WHEN DIGITS ARE NOT SIGNIFICANT 1. If a decimal point is present, zeros to the left of other digits (i.e., leading zeros) are not significant they are placeholders. For example: 0. has two significant digits 0.000 has two significant digits 8

Significant Digits WHEN DIGITS ARE NOT SIGNIFICANT. Ifadecimalpointisnotpresent,zerostotherightofthelastnon-zero digit(i.e., trailing zeros) are not significant they are placeholders. For example: 98 000 000 has two significant digits 5 000 has two significant digits NOTE! Inmostcases,thevaluesyouwillbeworkingwithinthiscoursewillhave two or three significant digits. 9 Significant Digits 1. How many significant digits are there in each of the following measured quantities? (a) 5g (b) 9.66L (c) 76600000g (d) 0.405 ml (e) 0.MW (f) 0.000067s (g) 10.00m (h) 47.m (i).7x10 5 s (j).400x10 - m 4 5 1 4 4 10 Significant Digits. Express the following measured quantities in scientific notation with the correct number of significant digits. (a) 865.7cm (b) 5000s (c) 0.05kg (d) 40.070 nm (e) 0.000060ns (4) 8.657x10 cm ().5x10 4 s (1) 5x10 - kg (5) 4.0070x10 1 nm () 6.0x10-5 ns 11 4

Precision Measurements depend on the precision of the measuring instruments used, that is, the amount of information that the instruments can provide. For example,.861cmismoreprecisethan.86cmbecausethethreedecimal places in.861 makes it precise to the nearest one-thousandth of a centimetre, while the two decimal places in.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. PRECISION indicated by the number of decimal places in the number 1 Precision RULES FOR PRECISION 1. All measured quantities are expressed as precisely as possible. All digits shown are significant with any error or uncertainty in the last digit. For example, in the measurement 87.64 cm the uncertainty is with the digit 4. 1 Precision RULES FOR PRECISION. The precision of a measuring instrument depends on its degree of finenessandthesizeoftheunitbeingused. For example, a ruler calibrated in millimetres(ruler#) is more precise than a ruler calibrated in centimetres (ruler #1) because the ruler calibrated in millimetres has more graduations. #1 # 14 5

Precision RULES FOR PRECISION. Any measurement that falls between the smallest divisions on the measuring instrument is an estimate. We should always try to read any instrument by estimating tenths of the smallest division. Forexample,withruler#1wewouldestimatetothenearesttenthofa centimetre (i.e.. cm); with ruler # we would estimate to the nearest tenth of a millimeter(i.e..4 cm). #1 # 15 Precision RULES FOR PRECISION 4. The estimated digit is always shown when recording the measurement. Forexample,the7inthemeasurement6.7cmwouldbetheestimated digit. 16 Precision RULES FOR PRECISION 5. Should the object fall right on a division mark, the estimated digit wouldbe0. Forexample,ifweusearulercalibratedin centimetres tomeasurea lengththatfallsexactlyonthe5cmmark,thecorrect readingis 5.0 cm,not5cm. 17 6

Precision. Usethetwocentimetrerulerstomeasureandrecordthelengthofthe pen graphic. (a) Child s ruler ~4.9cm (b) Ordinary ruler ~4.9cm 18 Precision 4. An object is being measured with a ruler calibrated in millimetres. One endoftheobjectisatthezeromarkoftheruler. Theotherendlines upexactlywiththe5.cmmark. Whatreadingshouldberecordedfor thelengthoftheobject? Why? 5.0 cm should be recorded since the object falls right on division. Since theruler is calibrated in millimetres,we needto estimate tothe nearest tenth of the smallest division 19 Precision 5. Whichofthefollowingvaluesofameasuredquantityismostprecise? (a) 4.81mm,0.81mm,48.1mm,0.081mm (b).54cm,1.64cm,16cm,0.5400cm,0.04cm (a) 0.081mm hasdecimalplaces (b) 0.5400cm has4decimalplaces 0 7

Uncertainty in Measurements 6. Copy and complete the following table. Measurement Precision # of Sig.Dig. Measurement now needed rounded in sci.not. a 6.479 km (example) 5 6.5 6.5 10 1 b 46597. cm 1 6 47 000 4.7 x 10 4 c 0.580 L 4 4 1 d 5 kg 0 e 0.06780 mm 5 4 f 485.000 kw 6 4 0.6 0 0.0678 485.0 6 x 10-1. x 10 6.78 x 10-4.850 x 10 1 8

Accuracy & Significant Figures Accuracy relates to how close a measurement agrees with the accepted value. This is an indication of the quality of the measuring instrument and the technique of the user. The difference between an observed value (or the average of the observed values) and the accepted value is called the error. The degree of accuracy of any instrument depends on two things: the precision of the measuring instrument and the skill and care of the user. The precision of a measuring instrument depends on the size of the unit being used. It depends on the place value of the last digit obtained from a measurement or calculation. For example,.861 is more precise than 581.86. Certainty is determined by how many certain digits are obtained by the measuring instrument. Any measurement that falls between the smallest division on the measuring instrument is an estimate and is therefore uncertain. Example: Determine the length of each line with the appropriate number of significant figures. a) b)

Significant figures record all digits that are certain, plus one uncertain digit. To determine the number of significant figures in a measurement, count the number of digits. All digits in a given measurement are significant except for the leading zeros. Example: State the number of significant figures for each of the following measurements. a).58g b) 6.07cm c) 0.005mL d) 0.180L e) 4.148 x 10 g f).00 x 10-1 m CERTAINTY RULE FOR MULTIPLYING AND DIVIDING: ***The answer has the same number of significant figures as the measurement with the fewest significant figures*** Example: State the answer with the appropriate number of significant figures. 1.8 a).9 b) ( 9.80)(7.06) PRECISION RULE FOR ADDING AND SUBTRACTING: ***The answer has the same number of decimal places as the measured value with the fewest decimal places*** Example: State the answer with the appropriate number of significant figures. a) 104.km + 11km + 0.67km = b) 5.5m + 0.597m + 0.16m =

EXACT NUMBERS: When you directly count the number of something, this is an exact value. Objects that have set values, such as 100cm/m or 60s/min, are defined values. Exact and defined values are said to contain an infinite number of significant figures and therefore don t affect the rules of certainty or precision. Example: State the answer with the appropriate number of significant figures. 1 a) (.5)(16.) 4 b) (.) CONVERTING UNITS The method most commonly used is multiplying by conversion factors, which are either memorized or referenced. Example: Perform the following conversions, stating your final answer with the appropriate number of significant figures a) 8.6 minutes to hours. b) 18 minutes to seconds c) 16 km/h to m/s.

SPH4C COLLEGE PHYSICS REVIEW: MATH SKILLS L Calculations Using Measurements (P.546-547) Rounding 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. Use these rules when making calculations and rounding answers to calculations. August 15, 01 4CR - Calculations Using Measurements 1 Rounding RULES FOR ROUNDING 1. When the first digit to be dropped is 4 or less, the last digit retained should not be changed. For example:.141 6 rounded to 4 digits is.141 August 15, 01 4CR - Calculations Using Measurements 1

Rounding RULES FOR ROUNDING. When the first digit to be dropped is greater than 5, or if it is a 5 followed by at least one digit other than zero, the last digit retained is increased by 1 unit. For example:.1 7 rounded to five digits is.14 4.168 501 rounded to four digits is 4.169 August 15, 01 4CR - Calculations Using Measurements Rounding RULES FOR ROUNDING. When the first digit discarded is five or a five followed by only zeros, the last digit retained is increased by 1 if it is odd, but not changed if it is even. For example:.5 rounded to two digits is.4.45 rounded to two digits is.4-6.5 rounded to two digits is -6.4 NOTE! This is sometimes called the even-odd rule. August 15, 01 4CR - Calculations Using Measurements 4 Adding & Subtracting RULES FOR ADDING& SUBTRACTING When adding and/or subtracting, the answer has the same number of decimal places as the measurement with the fewest decimal places. For example: 6.6 cm + 18.74 cm + 0.766 cm = 6.106 cm = 6.1 cm NOTE! The answer must be rounded to 6.1 cm because the first measurement (6.6 cm) limits the precision to a tenth of a centimetre. August 15, 01 4CR - Calculations Using Measurements 5

Multiplying & Dividing RULES FOR MULTIPLYING& DIVIDING When multiplying and/or dividing, the answer has the same number of significant digits as the measurement with the fewest number of significant digits. For example: 77.8 km/h x 0.8967 h = 69.766 km = 69.8 km NOTE! The certainty of the answer is limited to three significant digits, so the answer is rounded up to 69.8 km. The same applies to scientific notation. For example, (5.5 x 10 4 ) ) (5.675 x 10 - ) = 9.7 x 10 5 August 15, 01 4CR - Calculations Using Measurements 6 Multistep Calculations RULES FOR MULTISTEP CALCULATIONS For multistep calculations, round-off errors occur if you use the rounded-off answer from an earlier calculation in a subsequent calculation. Thus, leave all digits in your calculator until you have finished all your calculations and then round the final answer. For example: 5.1 x 0.45 ) 0.00600 =.445 ) 0.00600 or =. ) 0.00600 = 90.75 = 8. = 90 U = 80 Y NOTE! The certainty of the answer is limited to two significant digits, so the answer is rounded accordingly. In the second example though, rounding occurred during the calculation which introduced a round-off error. August 15, 01 4CR - Calculations Using Measurements 7 Calculations Summary ADDING& SUBTRACTING fewest decimal places MULTIPLYING& DIVIDING fewest number of significant digits MULTISTEP CALCULATIONS leave all digits in the calculator until finished and then round NOTE! If a combination of addition, subtraction, multiplication and division are involved, follow the rules for multiplying and dividing. August 15, 01 4CR - Calculations Using Measurements 8

Calculations Using Measurements 1. Perform the following operations. Round your answers accordingly. (a) 67.8 + 968 +.87 (b) 46.66 + 9. + 0.17 (c) 68.7 -.95 (d) (.6)(4.) (e) (65)(0.041)(5) (f) (0.0060)(6)(55.1) (g) 650 ) 4.0 (h).5 (i) (1.6 10 - )(7. 10-1 ) (j) (5.019 10-4 ) (.1 10-7 ) 109.67 49.0 44.75 109.7 866.15 8.5956 16.5 1.5 0.001186 1619.0... = 1040 = 49.0 = 44.8 = 110 = 870 = 8.6 = 160 = 1 = 0.001 = 1600 August 15, 01 4CR - Calculations Using Measurements 9 Calculations Using Measurements. Solve each of the following. Round your answers accordingly. (a) Find the perimeter of a rectangular carpet that has a width and length of.56 m and 4.5 m. (a) 16.1 m August 15, 01 4CR - Calculations Using Measurements 10 Calculations Using Measurements. Solve each of the following. Round your answers accordingly. (b) Find the area of a rectangle whose sides are 4.5 m and 7.5 m. (b) 4 m August 15, 01 4CR - Calculations Using Measurements 11 4

Calculations Using Measurements. Solve each of the following. Round your answers accordingly. (c) A triangle has a base of 5.75 cm and a height of 1.45 cm. Calculate the area of the triangle. (Recall A = ½bh) (c) 5.8 cm August 15, 01 4CR - Calculations Using Measurements 1 Calculations Using Measurements. Solve each of the following. Round your answers accordingly. (d) 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? (d) 49 zings August 15, 01 4CR - Calculations Using Measurements 1 Calculations Using Measurements. Solve each of the following. Round your answers accordingly. (e) The Earth has a mass of 5.98 10 4 kg while Jupiter has a mass of 1.90 10 7 kg. How many times larger is the mass of Jupiter than the mass of the Earth? (e) 18 times August 15, 01 4CR - Calculations Using Measurements 14 5

15/08/01 SPH4C COLLEGE PHYSICS REVIEW: MATH SKILLS L Error in Measurements (P.548) Error in Measurements 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. For example: August 15, 01 4CR - Error in Measurements 1 Error in Measurements 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. August 15, 01 4CR - Error in Measurements 1

15/08/01 Error in Measurements 1. What three things can you do during an experiment to help eliminate errors? 1. To be sure that 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 precise about the accuracy, repeat the measurements with a different instrument. August 15, 01 4CR - Error in Measurements Error in Measurements. There are two types of measurement error. What are they? random error and systematic error August 15, 01 4CR - Error in Measurements 4 Random Error Random error results when an estimate is made to obtain the last digit for any measurement. The size of the random error is determined by the precision of the measuring instrument. For example, when measuring length with a measuring tape, it is necessary to estimate between the marks on the measuring tape. If these marks are 1 cm apart, the random error will be greater and the precision will be less than if the marks are 1 mm apart. Such errors can be reduced by taking the average of several readings. RANDOM ERROR results when the last digit is estimated reduced by taking the average of several readings August 15, 01 4CR - Error in Measurements 5

15/08/01 Systematic Error Systematic error is associated with an inherent problem with the measuring system, such as the presence of an interfering substance, incorrect calibration, or room conditions. For example, if a balance is not zeroed at the beginning, all measurements will have a systematic error; using a slightly worn metre stick will also introduce error. Such errors are reduced by adding or subtracting the known error or calibrating the instrument. SYSTEMATIC ERROR due to a problem with the measuring device reduced by adding/subtracting the error or calibrating the device August 15, 01 4CR - Error in Measurements 6 Accuracy & Precision In everyday usage, "accuracy" and "precision" are used interchangeably to describe how close a measurement is to a true value, but in science it is important to make a distinction between them. Accuracy refers to how close a value is to its accepted value. Precision is the place value of the last measureable digit. ACCURACY how close a value is to its accepted value PRECISION place value of last measureable digit August 15, 01 4CR - Error in Measurements 7 Accuracy & Precision For example, the position of the darts in each of the figures are analogous to measured or calculated results in a laboratory setting. The results in (a) are precise and accurate, in (b) they are precise but not accurate, and in (c) they are neither precise nor accurate. August 15, 01 4CR - Error in Measurements 8

15/08/01 Percentage Error No matter how precise a measurement is, it still may not be accurate. The percentage error is the absolute value of the difference between experimental and accepted values expressed as a percentage of the accepted value. % error = experimental value accepted value accepted value x 100 NOTE! The bars ( ) in the equation above represent absolute value. means that, mathematically, if a = and b = - then a = b =. This August 15, 01 4CR - Error in Measurements 9 Percentage Difference 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. measurement 1 measurement % difference = x 100 measurement 1 + measurement NOTE! Magnitude is a term frequently used by physicists. The magnitude of a quantity is the same as its absolute value. August 15, 01 4CR - Error in Measurements 10 Error in Measurements. 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] (b) 9.95 m/s [down] (a) 8.96 (b) 1. August 15, 01 4CR - Error in Measurements 11 4

15/08/01 Error in Measurements 4. Calculate the percentage difference between the two experimental values (8.94 m/s and 9.95 m/s ) used in question #. 10.7 August 15, 01 4CR - Error in Measurements 1 U Check Your Learning WIKI (REVIEW) O... 4CR - WS1 (Math Skills) O... 4CR - QUIZ1 (Math Skills - Part 1) August 15, 01 4CR - Error in Measurements 1 5

SPH4C COLLEGE PHYSICS REVIEW: MATH SKILLS L Measuring & Estimating (P.549) Making Precise 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. vernier scale L main scale L August 15, 01 4CR - Measuring & Estimating 1 Making Precise Measurements If you look at the diagram carefully you will see there are 10 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. vernier scale L main scale L August 15, 01 4CR - Measuring & Estimating 1

Making Precise Measurements 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, the inside diameter of a hollow cylinder, or the depth of a hole. August 15, 01 4CR - Measuring & Estimating Making Precise Measurements Another instrument with a double scale is the outside micrometer caliper. August 15, 01 4CR - Measuring & Estimating 4 Using a Vernier Caliper 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. Digital models are also available. August 15, 01 4CR - Measuring & Estimating 5

Using a Vernier Caliper NOTE! When 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 0.000 cm but instead reads 0.00 cm, the error must be taken into consideration with each reading. August 15, 01 4CR - Measuring & Estimating 6 Using a Vernier Caliper HOWTOREADAMETRIC VERNIERCALIPER 1. Find the first line (the ZERO line) on the vernier (sliding) scale. Look on the main (stationary) scale and record the number you just passed (or are currently on) as #.# cm. 5.0 cm August 15, 01 4CR - Measuring & Estimating 7 Using a Vernier Caliper HOWTOREADAMETRIC VERNIERCALIPER. Find the FIRST pair of lines that match up perfectly. Read the line number off the vernier (sliding) scale and add this to the measurement. 5.08 cm August 15, 01 4CR - Measuring & Estimating 8

Using a Vernier Caliper HOWTOREADAMETRIC VERNIERCALIPER. Determine the error half of the smallest measurement possible. In our case the smallest measurement possible is 1 mm so the error is 0.5 mm or 0.05 cm. Add this to your measurement. 5.08 " 0.05 cm August 15, 01 4CR - Measuring & Estimating 9 Using a Vernier Caliper 1. What is the reading (including the error) of the following metric vernier calipers? (a) 0 1 0.69 " 0.05 cm 6 8 0 4 0 (b) 4.18 " 0.05 cm 6 8 0 4 0 August 15, 01 4CR - Measuring & Estimating 10 Using a Vernier Caliper 1. What is the reading (including the error) of the following metric vernier calipers? (c) 0 1 0.87 " 0.05 cm 6 8 0 4 0 (d) 4.6 " 0.05 cm 6 8 0 4 0 August 15, 01 4CR - Measuring & Estimating 11 4

Using a Vernier Caliper 1. What is the reading (including the error) of the following metric vernier calipers? (e) 1 1.1 " 0.05 cm 6 8 0 4 0 (f) 6 7 8 7.05 " 0.05 cm 6 8 0 4 0 August 15, 01 4CR - Measuring & Estimating 1 5

SPH4C COLLEGE PHYSICS REVIEW: MATH SKILLS L Trigonometry (P.550-551) Trigonometry The first application of trigonometry was to solve right-angle triangles. Trigonometry derives from the fact that for similar triangles, the ratio of corresponding sides will be equal. For a given angle " in a right triangle, there are three important ratios: sine, cosine, and tangent. These are called the primary trigonometric ratios and they can be used to find the measures of unknown sides and angles in right triangles. August 15, 01 4CR - Trigonometry 1 Trigonometry NOTE! DEG, RAD, and GRAD are different units/modes used for measuring angles. For this course make sure that your calculator is always in DEG mode. (Hint: if you are not getting the correct answers for a trigonometry problem start by checking this!) August 15, 01 4CR - Trigonometry 1

Trigonometry 1. Determine the value of each ratio rounded to four decimal places. (a) sin 5 (b) cos 60 (c) tan 45 (d) cos 75 (e) sin 18 0.576 0.5000 1.0000 0.588 0.090 August 15, 01 4CR - Trigonometry Trigonometry RECALL! If the value of a trigonometric ratio is known, its corresponding angle can be found using the inverse of that ratio. For example, if cos = 0.50 then = cos -1 (0.50). for sin -1 use: nd sin for cos -1 use: nd cos for tan -1 use: nd tan August 15, 01 4CR - Trigonometry 4 Trigonometry. Determine the size of A rounded to the nearest degree. (a) sin A = 0.599 (b) cos A = 0.46 (c) tan A = 4.15 (d) cos A = 0.5000 (e) sin A = 0.419 E 65E 77E 60E 14E August 15, 01 4CR - Trigonometry 5

Trigonometry. Solve for x rounded to one decimal place. (a) sin 5 = x/8 (b) cos 70 = x/15 (c) tan 0 = /x (d) sin 85 = 6/x (e) cos 5 = 5/x 4.6 5.1 8. 6.0 5.5 August 15, 01 4CR - Trigonometry 6 Trigonometry 4. Use two different methods to find the value of the unknown(s) in each triangle. Round your answers to one decimal place. X 7 mm 60 cm 6 cm X = 1. mm Y = 54.1 cm = 64. E 10 mm 5E Y August 15, 01 4CR - Trigonometry 7 U Check Your Learning WIKI(REVIEW) O... 4CR - QUIZ (Math Skills - Part ) August 15, 01 4CR - Trigonometry 8