Decimal moved after first digit = 4.6 x Decimal moves five places left SCIENTIFIC > POSITIONAL. a) g) 5.31 x b) 0.

Transcription

1 PHYSICS 20 UNIT 1 SCIENCE MATH WORKSHEET NAME: A. Sandard Noaion Very large and very small numbers are easily wrien using scienific (or sandard) noaion, raher han decimal (or posiional) noaion. Sandard noaion expresses a number as a value beween 1 and 10 muliplied by a power of en. Negaive powers are used for small numbers. EXAMPLES i = 1.5 x 10 6 ii = 4.5 x 10-3 iii = 6.71 x 10-6 iv = x 10 3 v = x 10 0 vi. 120 x 10 2 = 1.20 x 10 4 EXERCISE A 1. Wrie each number in opposie noaion. a) b) c) 3.7 x 10 2 d) 6.1 x 10-4 e) f) Decimal moved afer firs digi = 4.6 x 10 5 Decimal moves five places lef SCIENTIFIC > POSITIONAL Decimal moved afer firs digi = 1.2 x 10-4 Decimal moves four places righ negaive exponen POSITIONAL > SCIENTIFIC 6.4 x 10 2 = x 10-3 = Decimal moves wo places righ g) 5.31 x 10 2 h) 1.95 x 10-3 i) j) 4.2 x 10-5 k) 4 x 10 6 l) Decimal moves hree places lef 2. Use a scienific calculaor or graphing calculaor o deermine answers for each of he following. You should be using he EXP or EE key on your calculaors o do hese calculaions; DON T USE POWERS OF 10! The preferred noaion for answers is he one given for quesions no 1.3E4 o represen 1.3 x a) (3.2 x 10 2 )(9.6 x 10 2 ) f) (4.92 x 10 5 )(3.9 x 10 6 ) b) (5.800 x 10-1 )(1.1 x 10 4 ) g (8.518 x 10 2 ) (4.64 x 10 3 ) c) (7.6 x 10 5 ) (9 x 10 5 ) h) (4.8 x 10-3 ) (6.2 x 10-2 ) d) 5 x x 10 3 i) (8.8 x 10-2 )(5.30 x 10-3 ) Your calculaor will do his; by hand you would need o conver back o posiional noaion. e) ( )( ) j) PHYSICS 20N UNIT 1 REVISED JANUARY 08 PAGE 1

2 B. Significan Digis and Rounding Numbers in science usually arise from measuremens, and measuremens are never exac. The digis wrien in a measured value (including zeros, unless hey are leading zeros) are called significan digis (SDs) o indicae ha hey have been measured as precisely as possible. For example: 20.6 cm has 3 SDs km has 5 SDs (hose zeros are significan) N has 3 SDs (leading zeros aren significan) x 10 3 g has 4 SDs (don coun he powers of en) 24 pencils has infinie SDs (This is a couned number hink fingers and oes. I is considered o have an infinie number of significan digis ha is, i is as precise as i can be, or is exac.) If inexac measuremens in fac, all measuremens are used o calculae answers, he answer should no appear o be more precise han he numbers used in he calculaion. Consider: The dimensions of he cover of he book shown o he righ are 20.7 cm by 26.1 cm. Wha is he area of he book cover? A = l x w A = (20.7)(26.1) = cm 2 This answer has 5 SDs, ye he numbers used o find i had only 3 SDs each. Correcly rounded, he area should be 540 cm cm 26.1 cm When muliplying or dividing measured quaniies*, he answer should be rounded** o have he same number of SDs as he number wih he fewes SDs ha was used in he calculaion. *Noe ha all numbers in science are assumed o be measuremens unless indicaed oherwise. **You may have o use sandard noaion o make his rounding correc. Be careful o round only once; remember ha 5 rounds up. For addiion and subracion, look a he number of decimal places, no SDs. When adding or subracing measured quaniies, he answer should be rounded o have he same number of decimal places as he number wih he fewes decimal places used in he calculaion. Noe ha for mixed operaions involving boh muliplying/dividing and adding/subracing, he rule for muliplying and dividing is used. EXAMPLES i. (31.0)(2.25) = 69.75; rounded is 69.8 (3 SDs) ii = ; rounded is 3.6 x 10 2 (2 SDs) iii = ; rounded is 1.9 (2 SDs) 3.5 iv. 6451= ; rounded is (4 SDs) v = 90.45; rounded using decimal places is 90.5 (neares enh) vi = ; rounded using decimal places is 1992 (neares whole number) vii = ; rounded using decimal places is (neares hundredh) PHYSICS 20N UNIT 1 REVISED JANUARY 08 PAGE 2

3 EXERCISE B 1. Assuming hey are all measuremens, sae he number of significan digis in each of he following numbers. a) 3200 b) c) d) e) f) Sae he number of significan digis in each of he following. a) 30.5 seconds b) 100 people c) km d) 10 pencils e) ml f) 1200 kg 3. Selec he number wih he fewes decimal places; round he oher numbers o he same number of decimal places. a) 0.231, 4.6, 5.02 b) 1200, 654.1, c) , , 0.7 d) 35.91, 3.451, Round each of he following numbers o wo significan digis; use sandard form as needed. a) 169 b) 4.02 c) 6459 d) e) f) Add/subrac as indicaed; round answers o he correc number of decimal places. Where unis are given, be sure o use hem correcly. a) = b) = c) 20 g + 4 kg 656 g = d) = e) 3.30 km m 625 m = f) = 6. Muliply/divide as indicaed; round answers o he correc number of significan digis. Use sandard form where needed. a) 3.40 x 3 b) x 12.4 x 0.21 c) 22.3 x 39.1 x d) 60 candies spli evenly among 12 children e) ( 5.73) ( ) f) ( 174) ( ) g) half of a 40 m rope h) (5.5 x 10 3 )(4.1 x 10 2 ) i) x 1506 j) PHYSICS 20N UNIT 1 REVISED JANUARY 08 PAGE 3

4 Symbol C. SI Conversions Sysème Inernaional d'uniés, or SI for shor, is he meric sysem of unis almos exclusively used in science and nano n x 10-9 giga G x 10 9 engineering, and he official sysem of unis in use in many micro µ x 10-6 mega M x 10 6 counries, including Canada. The base unis of SI are he milli m x 10-3 kilo k x 10 3 kilogram (kg), he meer (m), and he second (s). Derived unis include he newon (N), he joule (J), and he wa (W). Unis for larger or smaller measured and calculaed values are ceni deci c d x 10-2 x 10-1 heco deka h da x 10 2 x 10 1 obained using prefixes, as given a righ (and on your daa shee.) To conver SI unis, use he seps below wih he char given (he char is also provided on your daa shee.) M k h da No prefix d c m µ x 10 6 x 10 3 x 10 2 x 10 1 x 10 0 x 10-1 x 10-2 x 10-3 x 10-6 I. Locae he given prefix and he needed prefix on he char. II. Deermine he change in powers of en beween hem. (Eg. From kilo o ceni is a change of 10 5 ) III. If he needed prefix is righ of he given one, muliply* by he change in he power of en. If he needed prefix is lef of he given one, divide* by he change in he power of en. EXAMPLES *If you are moving o he righ in he char, you are convering a given value ino smaller unis here will be more of hem, and you should muliply by he change in he power of en. If you are moving o he lef in he char, you are convering a given value ino larger unis here will be fewer of hem, and you should divide by he change in he power of en. i. 120 hm = cm From heco o ceni is a change of 10 4 ; ceni is he smaller uni, so 120 hm = 120 x 10 4 cm or 1.20 x 10 6 cm. ii mg = kg From milli o kilo is a change of 10 6 ; kilo is he larger uni, so 2.40 mg = kg or 2.40 x 10-6 kg. iii. 35 MN = dn From mega o deci is a change of 10 7 (noe he jump of 10 3 from mega o kilo); deci is he smaller uni, so 35 MN = 35 x 10 7 dn or 3.5 x 10 8 dn. Noe ha you can mainain he same precision (number of significan digis) when convering SI values by using sandard noaion as needed. Oher Conversions Remember he kilomeers per hour < > meers per second conversion based on 3600 sec/h and 1000 m/km: m s 3.6 = km h km h 3.6 = m s PHYSICS 20N UNIT 1 REVISED JANUARY 08 PAGE 4

5 EXERCISE C mm= cm L= ml cn = N ml= dl km= m 6. 9 x 10 5 µl= L km = cm dal= ml hm= dam cl= L m= cm kg= g dam= dm cg= mg m= km s= min h= s Gm= cm min= hours x 10 2 days= hours cm= mm x µm= m a (years)= hours N= µn g = kg kj= J µm= cm J= kj x 10 9 J= MJ ns= s D. Solving Equaions A single equaion can be solved for one unknown value. As in Science 10, you may eiher subsiue given numbers firs, hen rearrange as needed o isolae he unknown variable, or rearrange algebraically before subsiuing values. If you have compleed a leas Pure Mah 10, algebraic subsiuion is recommended. If your las compleed mah course was Applied Mah 10, you may have more success subsiuing values firs. Whichever echnique you adop, you mus be able o efficienly and reliably solve equaions. Rearranging an equaion o solve for an unknown variable, algebraically or wih numbers, can be done by: -muliplying or dividing all erms (boh sides ) of he equaion by he same number or variable -adding or subracing he same number or variable o boh sides of he equaion When you arrive a a value for he unknown variable, remember ha his answer is a resul of manipulaing measured values significan digis and unis mus be considered for your final answer. Generally, you should: -wrie an unrounded answer wih unis (unrounded means wo or hree more significan digis han will be needed for your final answer) -wrie a correcly rounded answer wih correc unis*. Your rounding should be done o give your answer he same number of significan digis as he value you subsiued ha had he fewes significan digis. Noe ha numbers ha are already presen in he equaion as i comes from your daa shee are no considered when you round. *I is perfecly correc o use uni analysis when solving equaions. By including unis wih all values as you subsiue hem ino equaions, you can double-check ha dimensions mach (no using meres combined wih cenimeers, for example), and generally be correc in aaching he correc unis o your answer. An equally good pracice (followed by your eacher, and shown in he examples below) is o carefully check all values before subsiuing o ensure unis are consisen hese should be generally in SI base unis of meres, seconds, kilograms, Newons, Joules, ec. You should hen be able o aach correc unis o your answer, wihou using uni analysis. Wach ou paricularly for speed conversions don subsiue km/h values. Well-prepared sudens should be able o use eiher mehod o ge correc unis! PHYSICS 20N UNIT 1 REVISED JANUARY 08 PAGE 5

6 EXAMPLES Consisen SI v = d base unis =? = i. v = 65 cm/s = 0.65 m/s v = d = s d = 200 cm = 2.00 m = d v = 3.1 s Unrounded answer wih unis Correcly rounded answer wih unis ii. = 23 s v f =? v f = 2d v i v i = 12 m/s d = v i + v f v f = 2(625) d d = 625 m = v i + v f v f = m/s 2 2d = v i + v f v f = 42 m/s Alernaely: d = v i + v f = 12 + v f v f = = 12 + v f 23 (2)( ) = 12 + v f v f = m/s = 12 + v f v f = v f = 42 m/s STARTING BY REARRANGING ALGEBRAICALLY PREFERRED STARTING WITH NUMERICAL SUBSTITUTION EXERCISE D Remember ha consans included wih he equaion - like he values 5/9 and 32 in quesion #1, and he ½ in quesion #2 - are NOT included in he rounding process. 1. C = 5 (F 32) C = 20 C F ( F) = 9 2. E k = 1 2 mv2 E k = 125 J m = 10 kg v (m/s) = 3. v = d v = 10 m/s d = 20 m (s) = 4. a = v f v i v i = 1.2 m/s v f = 0.80 m/s = 15 s a (m/s 2 ) = PHYSICS 20N UNIT 1 REVISED JANUARY 08 PAGE 6

7 5. a = v f v i a = 9.80 m/s 2 v f = 22 m/s v i = 10 m/s (s) = 6. v = d v = m/s = 120 s d (m) = 7. F = ma a = m/s 2 m = 150 kg F (N) = 8. v f 2 = v i 2 + 2ad v i = 0 m/s a = m/s 2 d = -120 m v f (m/s) = 9. C = 5 (F 32) F = -45 F C ( C) = Q = mc T m = 2.40 kg c = 4.19 kj/kg C T = 20 C Q (kj) = 11. Q = mc T m = kg c = 4.19 kj/kg C Q = 3.10 kj T ( C) = 12. E p = mgh m = 1200 kg g = 9.80 N/kg h = 3.50 m E p (J) = 13. E p = mgh E p = 200 J g = 9.80 N/kg m = 7.9 kg h (m) = PHYSICS 20N UNIT 1 REVISED JANUARY 08 PAGE 7

8 14. F = ma F = 380 N m = 250 kg a (m/s 2 ) = 15. E k = 1 2 mv2 v = 4.2 m/s m = 2.40 kg E k (J) = E. Experimens and Graphing In a ypical physics experimen, a relaionship beween wo quaniies is being esed. Each ime a rial of he experimen is performed, one of he quaniies is changed or manipulaed his is he manipulaed (or independen) variable. On each rial, he second quaniy s value changes as a resul of manipulaing he independen variable. The value of his second quaniy is measured his is he responding (or dependen) variable. The experimener also ofen has he choice o change oher quaniies in an experimen; if hese quaniies are inenionally lef unchanged, hey are conrolled variables. (Noe ha a conrolled variable is no he same as an experimenal conrol.) One way of showing relaionships in physics is o draw a graph. When drawing a graph, be sure o: Plo he independen variable as x (horizonal axis) Plo he dependen variable as y (verical axis) Creae linear scales on boh axes firs (wih a linear scale, he same number change is represened by he same number of divisions hroughou he lengh of he scale), hen plo he poins Sar each scale a zero unless you are old oherwise Label he quaniies ploed on each axis, ogeher wih heir unis Carefully plo poins afer choosing each scale Draw a smooh curve or sraigh line hrough he ploed poins a bes-fi line Give he graph a ile Graphs provide informaion in a number of ways. The mos obvious is hrough reading he graph noing wha value for he independen variable corresponds o a given value for he dependen variable, for example. Informaion can also be obained from he shape of he graph a sraigh horizonal line on a velociy-ime graph, for example means he objec whose moion is graphed is raveling a a consan velociy during his ime. Finally, calculaions such as slope ( slope = rise ) or area under he graph line can provide addiional informaion he slope of a run posiion-ime graph, for example, gives velociy. EXERCISE E 1. An objec is acceleraed from res by applying various forces; as he force is increased, he acceleraion changes. Use he given daa o draw a graph of force versus acceleraion on grid (A) below. F (N) a (m/s 2 ) PHYSICS 20N UNIT 1 REVISED JANUARY 08 PAGE 8

9 2. A oy car rolls from res down a driveway; is posiion a differen imes is measured by a moion sensor. Use he given daa o plo a graph of posiion agains ime on he grid (B) below. (Remember ha for moion graphs, ime is always he independen variable.) (s) d (m) Grid C below shows he force ha acs on an objec over differen disances. Recall from Science 10 ha he area under a force-displacemen graph gives work done. Use he graph o find he oal work done by he force. (Hin: use riangles and recangles.) GRID C GRID A F (N) Work done = d (m) GRID B Use he posiion-ime graph below o find he velociy of each moving objec. A 12 C 8 B d (m) (s) v A = v B = v C = PHYSICS 20N UNIT 1 REVISED JANUARY 08 PAGE 9