Math Skills. Scientific Notation. Uncertainty in Measurements. Appendix A5 SKILLS HANDBOOK

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1 ppendix 5 Scientific Notation It i difficult to work with very large or very mall number when they are written in common decimal notation. Uually it i poible to accommodate uch number by changing the SI prefix o that the number fall between 0.1 and 1000; for example, mm can be expreed a 37 km, and kg can be expreed a mg. However, thi prefix change i not alway poible, either becaue an appropriate prefix doe not exit or becaue it i eential to ue a particular unit of meaurement. In thee cae, the bet method of dealing with very large and very mall number i to write them uing cientific notation. Scientific notation expree a number by writing it in the form a 10 n,where 1 < a < 10 and the digit in the coefficient a are all ignificant. Table 1 how ituation where cientific notation would be ued. Table 1 of Scientific Notation Expreion ommon decimal notation Scientific notation 14.5 million kilometre km km 154 thouand picometre pm pm 60 extillion/mol /mol /mol To multiply number in cientific notation, multiply the coefficient and add the exponent; the anwer i expreed in cientific notation. Note that when writing a number in cientific notation, the coefficient hould be between 1 and 10 and hould be rounded to the ame certainty (number of ignificant digit) a the meaurement with the leat certainty (fewet number of ignificant digit). Look at the following example: ( m)( m) = m = m ( N) ( m) = N/m = N/m On many calculator, cientific notation i entered uing a pecial key, labelled EXP or EE. Thi key include 10 from the cientific notation; you need to enter only the exponent. For example, to enter pre 7.5 EXP pre 3.6 EXP +/ 3 Math Skill Uncertainty in Meaurement Two type of quantitie are ued in cience: exact value and meaurement. Exact value include defined quantitie (1 m = 100 cm) and counted value (5 car in a parking lot). Meaurement, however, are not exact becaue there i alway ome uncertainty or error. There are two type of meaurement error. Random error reult when an etimate i made to obtain the lat ignificant digit for any meaurement. The ize of the random error i determined by the preciion of the meauring intrument. For example, when meauring length, it i neceary to etimate between the mark on the meauring tape. If thee mark are 1 cm apart, the random error will be greater and the preciion will be le than if the mark are 1 mm apart. Sytematic error i aociated with an inherent problem with the meauring ytem, uch a the preence of an interfering ubtance, incorrect calibration, or room condition. For example, if the balance i not zeroed at the beginning, all meaurement will have a ytematic error; uing a lightly worn metre tick will alo introduce error. The preciion of meaurement depend on the gradation of the meauring device. Preciion i the place value of the lat meaurable digit. For example, a meaurement of 1.74 cm i more precie than one of 17.4 cm becaue the firt value wa meaured to hundredth of a centimetre wherea the latter wa meaured to tenth of a centimetre. When adding or ubtracting meaurement of different preciion, the anwer i rounded to the ame preciion a the leat precie meaurement. For example, uing a calculator, 11.7 cm cm cm = cm The anwer mut be rounded to 15.5 cm becaue the firt meaurement limit the preciion to a tenth of a centimetre. No matter how precie a meaurement i, it till may not be accurate. ccuracy refer to how cloe a value i to it accepted value. The percentage error i the abolute value of the difference between experimental and accepted value expreed a a percentage of the accepted value. experimental value accepted value % error = 100% accepted value The percentage difference i the difference between a value determined by experiment and it predicted value. The percentage difference i calculated a experimental value predicted value % difference = 100% predicted value Math Skill 565

2 ppendix 5 (a) (b) (c) Figure 1 The poition of the dart in each of thee figure are analogou to meaured or calculated reult in a laboratory etting. The reult in (a) are precie and accurate, in (b) they are precie but not accurate, and in (c) they are neither precie nor accurate. Figure 1 how an analogy between preciion and accuracy, and the poition of dart thrown at a dartboard. How certain you are about a meaurement depend on two factor: the preciion of the intrument ued and the ize of the meaured quantity. More precie intrument give more certain value. For example, a ma meaurement of 13 g i le precie than a meaurement of 1.76 g; you are more certain about the econd meaurement than the firt. ertainty alo depend on the meaurement. For example, conider the meaurement 0.4 cm and 15.9 cm; both have the ame preciion. However, if the meauring intrument i precie to ± 0.1 cm, the firt meaurement i 0.4 ± 0.1 cm (0.3 cm or 0.5 cm) or an error of ± 5%, wherea the econd meaurement could be 15.9 ± 0.1 cm (15.8 cm or 16.0 cm) for an error of ± 0.6%. For both factor the preciion of the intrument ued and the value of the meaured quantity the more digit there are in a meaurement, the more certain you are about the meaurement. Significant Digit The certainty of any meaurement i communicated by the number of ignificant digit in the meaurement. In a meaured or calculated value, ignificant digit are the digit that are certain plu one etimated (uncertain) digit. Significant digit include all digit correctly reported from a meaurement. Follow thee rule to decide whether a digit i ignificant: 1. If a decimal point i preent, zero to the left of the firt non-zero digit (leading zero) are not ignificant.. If a decimal point i not preent, zero to the right of the lat non-zero digit (trailing zero) are not ignificant. 3. ll other digit are ignificant. 4. When a meaurement i written in cientific notation, all digit in the coefficient are ignificant. 5. ounted and defined value have infinite ignificant digit. Table how ome example of ignificant digit. Table ertainty in Significant Digit Meaurement Number of ignificant digit 3.07 m g kg cm kj people (counted) infinite n anwer obtained by multiplying and/or dividing meaurement i rounded to the ame number of ignificant digit a the meaurement with the fewet number of ignificant digit. For example, if we ue a calculator to olve the following equation: (77.8 km/h)( h) = km However, the certainty of the anwer i limited to three ignificant digit, o the anwer i rounded up to 69.8 km. Rounding Off Ue thee rule when rounding anwer to calculation: 1. When the firt digit dicarded i le than five, the lat digit retained hould not be changed rounded to 4 digit i When the firt digit dicarded i greater than five, or if it i a five followed by at leat one digit other than zero, the lat digit retained i increaed by 1 unit rounded to 5 digit i rounded to 4 digit i When the firt digit dicarded i five followed by only zero, the lat digit retained i increaed by 1 if it i odd, but not changed if it i even..35 rounded to digit i.4.45 rounded to digit i rounded to digit i ppendix 5

3 Meauring and Etimating Many people believe that all meaurement are reliable (conitent over many trial), precie (to a many decimal place a poible), and accurate (repreenting the actual value). ut many thing can go wrong when meauring. There may be limitation that make the intrument or it ue unreliable (inconitent). The invetigator may make a mitake or fail to follow the correct technique when reading the meaurement to the available preciion (number of decimal place). The intrument may be faulty or inaccurate; a imilar intrument may give different reading. For example, when meauring the temperature of a liquid, it i important to keep the thermometer at the correct depth and the bulb of the thermometer away from the bottom and ide of the container. If you et a thermometer with it bulb on the bottom of a liquid-filled container, you will be meauring the temperature of the bottom of the container, and not the temperature of the liquid. There are imilar concern with other meaurement. To be ure that you have meaured correctly, you hould repeat your meaurement at leat three time. If your meaurement appear to be reliable, calculate the mean and ue that value. To be more certain about the accuracy, repeat the meaurement with a different intrument. Trigonometry The word trigonometry come from the Greek word trigonon and metria, meaning triangle meaurement. The earliet ue of trigonometry wa for urveying. Today, trigonometry i ued in navigation, electronic, muic, and meteorology, to mention jut a few. The firt application of trigonometry wa to olve right triangle. Trigonometry derive from the fact that for imilar triangle, the ratio of correponding ide will be equal. Q P.3 cm 54 We can apply the ine ratio to determine the length of the two unknown ide of the triangle hown in Figure 3. in R = o pp = P Q hyp PR PQ in 54 =. 3 cm Figure 3 Right triangle To ue your calculator to find in 54, make ure it i in degree mode, and enter in 54. Thi hould produce the anwer To find QR, we can ue the Pythagorean theorem or apply the ine ratio to angle P. in P = Q R PR QR in 36 =. 3 cm R QR = (0.5878)(.3 cm) QR = 1.4 cm (to two ignificant digit) Two other trigonometric ratio that are frequently ued when working with right triangle are the coine and tangent ratio, abbreviated co and tan repectively. They are defined a adj coine v = tangent v = o pp h yp adj For the triangle hown in Figure 4 co = tan = co = tan = Figure Right triangle In the right triangle in Figure, conider the ratio. In relation to the angle, i the oppoite ide and i the hypotenue. Thi ratio of o ppoite ide i called the ine hypotenue ratio (abbreviated in). For the given triangle in = and in = Figure 4 Right triangle Trigonometry can alo be ued for triangle other than right triangle. The ine law and the coine law can be ueful when dealing with problem involving vector. Math Skill 567

4 ppendix 5 The Sine Law Thi law tate that for any given triangle, the ratio of the ine of an angle to the length of the oppoite ide i contant. Thu, for Figure 5 c a b Figure 5 Scalene triangle in = in = in a b c The oine Law Thi law tate that for any given triangle, the quare of the length of any ide i equal to the um of the quare of the length of the other two ide, minu twice the product of the length of thee two ide and the coine of the angle between them (the included angle). Thu, for Figure 5 a = b + c bc co b = a + c ac co c = b + a ab co Note that the firt part of the coine law i the Pythagorean theorem; the lat factor imply adjut for the fact that the angle i not a right angle. If the angle i a right angle, the coine i zero and the term diappear, leaving the Pythagorean theorem. Look at Figure 6 and calculate the length of ide. 53. cm y the coine law we have = + ()() co = (47.3 cm) + (53. cm) (47.3 cm)(53. cm) co 115 = 37 cm cm (5033 cm ) (co 115 ) = 5067 cm (5033 cm )( 0.46) = 7194 cm = 84.8 cm Equation and Graph Linear Equation ny equation that can be written in the form x + y = i called a linear, or firt-degree equation in two variable. However, mot often the equation i rearranged a y = mx + b, where y i the dependent variable (on the y-axi) and x i the independent variable (on the x-axi). Thi equation i known a the lope-intercept form becaue m i the lope of the line on the graph and b i the y-intercept. Linear equation are encountered in many area of cience. For example, the equation for the velocity (v) of an object at a given time (t) i given by the linear equation v = v i + at, where v i i the initial velocity and a i the acceleration. If we change thi equation to the lope-intercept form (y = mx + b), it read v = at + v i,where v repreent the y variable and t repreent the x variable. Plot the graph of the equation for the velocity of an object with initial velocity 0.0 m/ [E] and acceleration 5.0 m/ [E]. v = 0.0 m/ + (5.0 m/ )t The lope of the line can be calculated uing the following equation: rie ( y m = = y1) r un ( x x ) 1 where y 1 and x 1, and y and x are any two point on the line. Figure cm ppendix 5

5 From Figure 7, we can chooe two point (1, 5) and (5.5, 47.5); note that one i a data point and one i not. We can now calculate the lope. Velocity (m/ [E]) Time () Figure 7 Velocity-time graph v v1 m = t t 1 ( ) m/ [E] = ( ) =.5 m/ [E] 4.5 m = 5.0 m/ [E] The lope of the line i 5.0 m/ [E]. Thi i a poitive value, indicating a poitive lope. negative lope (a line loped the other way) would have a negative value. If the value of one of the variable i known, the other value can be read from the graph or obtained by olving the equation uing algebraic kill. For example, if t =.0 you can ee on the graph that the correponding y coordinate i 30.0 m/. Solving algebraically we get the ame reult: In many ituation there i a combination of direct and invere variation, commonly referred to a joint variation. Problem dealing with joint variation are olved by ubtituting the value of the variable from a known experiment to calculate k and then uing the value of k to determine the miing variable in another experiment. The electrical reitance of a wire (R) varie directly a it length and inverely a the quare of it diameter. n invetigation determine that 50.0 m of wire of diameter 3.0 mm ha a reitance of 8.0. Determine the contant of variation for thi type of wire. Without doing another invetigation, determine the reitance of 40.0 m of the ame type of wire if the diameter i 4.0 mm. The variation equation i R = k d l m 8.0 = k (3.0 mm ) k = (8.0 )( 3.0 mm) m = 1.44 mm m 1 m 1000 mm k = mm Ue thi value of k to find R: m R = mm (4.0 mm) mm 1 m = mm mm = mm 16 mm R = 3.6 v = v i + at = 0.0 m/ m/ (.0 ) = 0.0 m/ m/ v = 30.0 m/ Variation Equation When y varie directly a x, written a y x, it mean that y = kx, where k i the contant of variation. When y varie inverely a x, written a y 1 x, it mean that y = x k or xy = k. Logarithm ny poitive number N can be expreed a a power of ome bae b where b > 1. Some obviou example are 16 = 4 bae, exponent 4 5 = 5 bae 5, exponent 7 = 3 3 bae 3, exponent = 10 3 bae 10, exponent 3 In each example, the exponent i an integer. However, exponent may be any real number, not jut an integer. If you ue the x y button on your calculator, you can experiment to get a better undertanding of thi concept. Math Skill 569

6 ppendix 5 The mot common bae i bae 10. Some example for bae 10 are = = = y definition, the exponent to which a bae b mut be raied to produce a given number N i called the logarithm of N to bae b (abbreviated a log b ). When the value of the bae i not written it i aumed to be bae 10. Logarithm to bae 10 are called common logarithm. We can expre the previou example a logarithm: log 3.16 = 0.5 log = 1.3 log =.7 nother bae that i ued extenively for logarithm i the bae e (approximately.7183). Logarithm to bae e are called natural logarithm (abbreviated a ln). Mot meaurement cale are linear in nature. For example, a peed of 80 km/h i twice a fat a a peed of 40 km/h and four time a fat a a peed of 0 km/h. However, there are everal example in cience where the range of value of the variable being meaured i o great that it i more convenient to ue a logarithmic cale to bae 10. One example of thi i the cale for meauring the intenity level of ound. For example, a ound with an intenity level of 0 d i 100 time (10 ) a loud a a ound with an intenity level of 0 d, and 40 d i (10 4 ) time more intene than a ound of 0 d. Other ituation that ue logarithmic cale are the acidity of a olution (the ph cale) and the intenity of earthquake (the Richter cale). Logarithmic Graph Quite often, graphing the reult from experiment how a logarithmic progreion. For example, the erie 1,, 3, 4, 5, 6 i a linear progreion, wherea the erie 10, 100, 1000, , i a logarithmic progreion. Thi mean that the value increae exponentially. Figure 8 i a graph of type y = log x to illutrate the ound intenity level cale, where ound intenity level in decibel (d) i equal to the logarithm of intenity in watt per metre quared (W/m ). Intenity Level (d) Intenity (W/m ) Figure 8 Graph of ound intenity level veru intenity Notice that the cale on both axe are linear, o that we can ee very little of the detail on the x-axi. Where the data range on one axi i extremely large and/or doe not follow a linear progreion, it i more convenient to change the cale (uually on the x-axi) o that we can ee more detail of the entire range of value. Semi-log graph paper can be ued to contruct uch graph. If the cale on the x-axi i changed to a logarithmic cale, the graph of ound intenity level veru ound intenity on emi-log graph paper i hown in Figure 9. Intenity Level (d) E-1 1E-10 1E-08 1E Intenity (W/m ) Figure 9 Semi-log graph of ound intenity level veru intenity 570 ppendix 5

7 Dimenional nalyi Dimenional analyi i a ueful tool to determine whether an equation ha been written correctly and to convert unit. i the cae with many topic in phyic, dimenional analyi can be eay or hard depending on the treatment we give it. Dimenion i a term that refer to quantitie that we can meaure in our univere. Three common dimenion are ma (m), length (l), and time (t). Note that the unit of thee dimenion are all bae unit kilogram (kg), metre (m), and econd (). In dimenional analyi, all unit are expreed a bae unit. fter a while, dimenional analyi become econd nature. Suppoe, for example, that after you olve an equation in which time t i the unknown, the final line in your olution i t =.1 kg. You know that omething ha gone eriouly wrong on the right-hand ide of the equation. It might be that care wa not taken in cancelling certain unit or that the equation wa written incorrectly. For example, we can ue dimenional analyi to determine if the following expreion i valid: d = v i + 1 at One way to check i to inert the appropriate unit. The uual technique when working with unit i to put them in quare bracket and to ignore number like the 1 in the expreion. The quare bracket indicate that we are dealing with unit only. The expreion become [m] = m + m [ ] You can alo ue dimenional analyi to change from one unit to another. For example, to convert 95 km/hr to m/, kilometre mut be changed to metre and hour to econd. It help to realize that 1 km = 1000 m and 1 hr = Thee two equivalencie allow the following two term to be written: km 1 hr = 1 and = 1 1 km Of coure, the numerator and denominator could be witched (i.e., 1 km and 3600 could be in the numerator) and the ratio would till be 1. However, a you will ee, it i convenient to keep the ratio a they are for cancelling purpoe. ecaue multiplying by 1 doe not change the value of anything, we can write the following expreion and cancel the unit: 95 k h m r = 95 km h m 1 r h r 1 km Therefore, 9 5 km = m 1 hr 3600 = 6.4 m/, or 6 m/ (to two ignificant digit) What will be the magnitude of the acceleration of a 100-g object that experience a net force of magnitude 38. N? Firt convert gram to kilogram: ma = (100 g) 1 kg 1000 g ma =.1 kg [m] = m + [m] The expreion i not valid becaue the unit on the right-hand ide of the equation do not equal the unit on the left-hand ide. The correct expreion i d = v i t + 1 at You can check it out yourelf by inerting the unit in quare bracket. If you wih, you can ue the actual dimenion of length [l] and time [t] intead of ubtituting unit. The dimenional analyi of the equation i l [l] = t [t] + tl [t ] [l] = [l] + [l] [l] = [l] Remember that becaue we are dealing only with dimenion, there i no need to ay l on the right-hand ide. From Newton econd law, F net = ma a = F net m = 3 8. N.1 kg a = 18 N/kg It i omewhat botherome to leave acceleration with the unit N/kg, o we will ue dimenional analyi to change the unit: F net = ma [N] = [kg][m/ ] a = 18 N = 18 [k g] [ m/ ] kg [ kg] a = 18 m/ Math Skill 571