A POWERS OF TEN AND SCIENTIFIC NOTATION In science, very lrge nd very smll deciml numbers re conveniently expressed in terms of powers of ten, some of wic re listed below: 0 3 0 0 0 000 0 3 0 0 0 0.00 02 0 0 00 0 2 0.0 0 0 0 0 0 0. 0 0 0 Using powers of ten, we cn write te rdius of te ert in te following wy, for exmple: Ert rdius 6 380 000 m 6.38 0 6 m Te fctor of ten rised to te sixt power is ten multiplied by itself six times, or one million, so te ert s rdius is 6.38 million meters. Alterntively, te fctor of ten rised to te sixt power indictes tt te deciml point in te term 6.38 is to be moved six plces to te rigt to obtin te rdius s number witout powers of ten. For numbers less tn one, negtive powers of ten re used. For instnce, te Bor rdius of te ydrogen tom is Bor rdius 0.000 000 000 0529 m 5.29 0 m Te fctor of ten rised to te minus elevent power indictes tt te deciml point in te term 5.29 is to be moved eleven plces to B SIGNIFICANT FIGURES Te number of significnt figures in number is te number of digits wose vlues re known wit certinty. For instnce, person s eigt is mesured to be.78 m, wit te mesurement error being in te tird deciml plce. All tree digits re known wit certinty, so tt te number contins tree significnt figures. If zero is given s te lst digit to te rigt of te deciml point, te zero is presumed to be significnt. Tus, te number.780 m contins four significnt figures. As noter exmple, consider distnce of 500 m. Tis number contins only two significnt figures, te one nd te five. Te zeros immeditely to te left of te unexpressed deciml point re not counted s significnt figures. However, zeros locted between significnt figures re significnt, so distnce of 502 m contins four significnt figures. Scientific nottion is prticulrly convenient from te point of view of significnt figures. Suppose it is known tt certin distnce is fifteen undred meters, to four significnt figures. Writing te number s 500 m presents problem becuse it implies tt only two significnt figures re known. In contrst, te scientific nottion of.500 0 3 m s te dvntge of indicting tt te distnce is known to four significnt figures. Wen two or more numbers re used in clcultion, te number of significnt figures in te nswer is limited by te number of significnt figures in te originl dt. For instnce, rectngulr grden wit sides of 9.8 m nd 7. m s n re of (9.8 m)(7. m). A clcultor gives 67.58 m 2 for tis product. However, one of te originl lengts te left to obtin te rdius s number witout powers of ten. Numbers expressed wit te id of powers of ten re sid to be in scientific nottion. Clcultions tt involve te multipliction nd division of powers of ten re crried out s in te following exmples: (2.0 0 6 )(3.5 0 3 ) (2.0 3.5) 0 6 3 7.0 0 9 9.0 0 7 2.0 0 9.0 2.0 4 07 0 4 Te generl rules for suc clcultions re 0 n n 0 0 n 0 m 0 n m 0 n n m 0 m 0 9.0 2.0 07 4 4.5 0 3 (Exponents dded) (Exponents subtrcted) (A-) (A-2) (A-3) were n nd m re ny positive or negtive number. Scientific nottion is convenient becuse of te ese wit wic it cn be used in clcultions. Moreover, scientific nottion provides convenient wy to express te significnt figures in number, s Appendix B discusses. is known only to two significnt figures, so te finl nswer is limited to only two significnt figures nd sould be rounded off to 70 m 2. In generl, wen numbers re multiplied or divided, te number of significnt figures in te finl nswer equls te smllest number of significnt figures in ny of te originl fctors. Te number of significnt figures in te nswer to n ddition or subtrction is lso limited by te originl dt. Consider te totl distnce long biker s tril tt consists of tree segments wit te distnces sown s follows: 2.5 km km 5.26 km Totl 8.76 km Te distnce of km contins no significnt figures to te rigt of te deciml point. Terefore, neiter does te sum of te tree distnces, nd te totl distnce sould not be reported s 8.76 km. Insted, te nswer is rounded off to 9 km. In generl, wen numbers re dded or subtrcted, te lst significnt figure in te nswer occurs in te lst column (counting from left to rigt) contining number tt results from combintion of digits tt re ll significnt. In te nswer of 8.76 km, te eigt is te sum of 2 5, ec digit being significnt. However, te seven is te sum of 5 0 2, nd te zero is not significnt, since it comes from te -km distnce, wic contins no significnt figures to te rigt of te deciml point. A-
A-2 APPENDIXES ALGEBRA C C. PROPORTIONS AND EQUATIONS Pysics dels wit pysicl vribles nd te reltions between tem. Typiclly, vribles re represented by te letters of te Englis nd Greek lpbets. Sometimes, te reltion between vribles is expressed s proportion or inverse proportion. Oter times, owever, it is more convenient or necessry to express te reltion by mens of n eqution, wic is governed by te rules of lgebr. If two vribles re directly proportionl nd one of tem doubles, ten te oter vrible lso doubles. Similrly, if one vrible is reduced to one-lf its originl vlue, ten te oter is lso reduced to one-lf its originl vlue. In generl, if x is directly proportionl to y, ten incresing or decresing one vrible by given fctor cuses te oter vrible to cnge in te sme wy by te sme fctor. Tis kind of reltion is expressed s x y, were te symbol mens is proportionl to. Since te proportionl vribles x nd y lwys increse nd decrese by te sme fctor, te rtio of x to y must ve constnt vlue, or x/y k, were k is constnt, independent of te vlues for x nd y. Consequently, proportionlity suc s x y cn lso be expressed in te form of n eqution: x ky. Te constnt k is referred to s proportionlity constnt. If two vribles re inversely proportionl nd one of tem increses by given fctor, ten te oter decreses by te sme fctor. An inverse proportion is written s x /y. Tis kind of proportionlity is equivlent to te following eqution: xy k, were k is proportionlity constnt, independent of x nd y. C.2 SOLVING EQUATIONS Some of te vribles in n eqution typiclly ve known vlues, nd some do not. It is often necessry to solve te eqution so tt vrible wose vlue is unknown is expressed in terms of te known quntities. In te process of solving n eqution, it is permissible to mnipulte te eqution in ny wy, s long s cnge mde on one side of te equls sign is lso mde on te oter side. For exmple, consider te eqution v v 0 t. Suppose vlues for v, v 0, nd re vilble, nd te vlue of t is required. To solve te eqution for t, we begin by subtrcting v 0 from bot sides: v v 0 t v 0 v 0 v v 0 t Next, we divide bot sides of v v 0 t by te quntity : v v 0 t On te rigt side, te in te numertor divided by te in te denomintor equls one, so tt t v v 0 ()t It is lwys possible to ceck te correctness of te lgebric mnipultions performed in solving n eqution by substituting te nswer bck into te originl eqution. In te previous exmple, we substitute te nswer for t into v v 0 t: v v 0 v v 0 v 0 (v v 0 ) v Te result v v implies tt our lgebric mnipultions were done correctly. Algebric mnipultions oter tn ddition, subtrction, multipliction, nd division my ply role in solving n eqution. Te sme bsic rule pplies, owever: Wtever is done to te left side of n eqution must lso be done to te rigt side. As noter exmple, suppose it is necessry to express v 0 in terms of v,, nd x, were v 2 v 2 0 2x. By subtrcting 2x from bot sides, we isolte v 2 0 on te rigt: v 2 v 2 0 2x 2x 2x v 2 2 2x v 0 To solve for v 0, we tke te positive nd negtive squre root of bot sides of v 2 2x v 2 0 : C.3 v 0 ± vv 2 2x SIMULTANEOUS EQUATIONS Wen more tn one vrible in single eqution is unknown, dditionl equtions re needed if solutions re to be found for ll of te unknown quntities. Tus, te eqution 3x 2y 7 cnnot be solved by itself to give unique vlues for bot x nd y. However, if x nd y lso (i.e., simultneously) obey te eqution x 3y 6, ten bot unknowns cn be found. Tere re number of metods by wic suc simultneous equtions cn be solved. One metod is to solve one eqution for x in terms of y nd substitute te result into te oter eqution to obtin n expression contining only te single unknown vrible y. Te eqution x 3y 6, for instnce, cn be solved for x by dding 3y to ec side, wit te result tt x 6 3y. Te substitution of tis expression for x into te eqution 3x 2y 7 is sown below: 3x 2y 7 3(6 3y) 2y 7 8 9y 2y 7 We find, ten, tt 8 y 7, result tt cn be solved for y: 8 y 7 8 8 y Dividing bot sides of tis result by sows tt y. Te vlue of y cn be substituted in eiter of te originl equtions to obtin vlue for x: x 3y 6 x 3( ) 6 x 3 6 3 3 x 3
APPENDIX D EXPONENTS AND LOGARITHMS A-3 C.4 THE QUADRATIC FORMULA Equtions occur in pysics tt include te squre of vrible. Suc equtions re sid to be qudrtic in tt vrible, nd often cn be put into te following form: x 2 bx c 0 (C-) were, b, nd c re constnts independent of x. Tis eqution cn be solved to give te qudrtic formul, wic is x b ± vb2 4c 2 (C-2) Te ± in te qudrtic formul indictes tt tere re two solutions. For instnce, if 2x 2 5x 3 0, ten 2, b 5, nd c 3. Te qudrtic formul gives te two solutions s follows: Solution : Plus sign Solution 2: Minus sign x b vb2 4c 2 ( 5) v( 5)2 4(2)(3) 2(2) 5 v 3 4 2 x b vb2 4c 2 ( 5) v( 5)2 4(2)(3) 2(2) 5 v 4 D EXPONENTS AND LOGARITHMS Appendix A discusses powers of ten, suc s 0 3, wic mens ten multiplied by itself tree times, or 0 0 0. Te tree is referred to s n exponent. Te use of exponents extends beyond powers of ten. In generl, te term y n mens te fctor y is multiplied by itself n times. For exmple, y 2, or y squred, is fmilir nd mens y y. Similrly, y 5 mens y y y y y. Te rules tt govern lgebric mnipultions of exponents re te sme s tose given in Appendix A (see Equtions A-, A-2, nd A-3) for powers of ten: y n y n y m y n y n y m y n m yn m (Exponents dded) (Exponents subtrcted) (D-) (D-2) (D-3) To te tree rules bove we dd two more tt re useful. One of tese is y n z n (yz) n (D-4) Te following exmple elps to clrify te resoning beind tis rule: 3 2 5 2 (3 3)(5 5) (3 5)(3 5) (3 5) 2 Te oter dditionl rule is (y n ) m y nm (Exponents multiplied) To see wy tis rule pplies, consider te following exmple: (5 2 ) 3 (5 2 )(5 2 )(5 2 ) 5 2 2 2 5 2 3 (D-5) Roots, suc s squre root or cube root, cn be represented wit frctionl exponents. For instnce, vy y /2 nd 3 vy y /3 In generl, te nt root of y is given by n vy y /n (D-6) Te rtionle for Eqution D-6 cn be explined using te fct tt (y n ) m y nm. For instnce, te fift root of y is te number tt, wen multiplied by itself five times, gives bck y. As sown below, te term y /5 stisfies tis definition: (y /5 )(y /5 )(y /5 )(y /5 )(y /5 ) (y /5 ) 5 y (/5) 5 y Logritms re closely relted to exponents. To see te connection between te two, note tt it is possible to express ny number y s noter number B rised to te exponent x. In oter words, y B x (D-7) Te exponent x is clled te logritm of te number y. Te number B is clled te bse number. One of two coices for te bse number is usully used. If B 0, te logritm is known s te common logritm, for wic te nottion log pplies: Common logritm y 0 x or x log y (D-8) If B e 2.78..., te logritm is referred to s te nturl logritm, nd te nottion ln is used: Nturl logritm y e z or z ln y (D-9) Te two kinds of logritms re relted by ln y 2.3026 log y (D-0) Bot kinds of logritms re often given on clcultors. Te logritm of te product or quotient of two numbers A nd C cn be obtined from te logritms of te individul numbers ccording to te rules below. Tese rules re illustrted ere for nturl logritms, but tey re te sme for ny kind of logritm. ln (AC) ln A ln C ln A ln A ln C C (D-) (D-2) Tus, te logritm of te product of two numbers is te sum of te individul logritms, nd te logritm of te quotient of two numbers is te difference between te individul logritms. Anoter useful rule concerns te logritm of number A rised to n exponent n: ln A n n ln A (D-3) Rules D-, D-2, nd D-3 cn be derived from te definition of te logritm nd te rules governing exponents.
A-4 APPENDIXES E GEOMETRY AND TRIGONOMETRY E. ANGLES GEOMETRY b c Two ngles re equl if. Tey re verticl ngles (see Figure E). 2. Teir sides re prllel (see Figure E2). b 2 2 Figure E Figure E2 3. Teir sides re mutully perpendiculr (see Figure E3). 6. Two similr tringles re congruent if tey cn be plced on top of one noter to mke n exct fit. CIRCUMFERENCES, AREAS, AND VOLUMES OF SOME COMMON SHAPES. Tringle of bse b nd ltitude (see Figure E6): c 2 Figure E5 Are 2 b b Figure E6 Figure E3 TRIANGLES. Te sum of te ngles of ny tringle is 80 (see Figure E4). + + γ = 80 Figure E4 2. A rigt tringle s one ngle tt is. 3. An isosceles tringle s two sides tt re equl. 4. An equilterl tringle s tree sides tt re equl. Ec ngle of n equilterl tringle is 60. 5. Two tringles re similr if two of teir ngles re equl (see Figure E5). Te corresponding sides of similr tringles re proportionl to ec oter: b c 2 b 2 c 2 γ 2. Circle of rdius r: Circumference 2 r Are r 2 3. Spere of rdius r: Surfce re 4 r 2 4 3 Volume r 3 4. Rigt circulr cylinder of rdius r nd eigt (see Figure E7): Surfce re 2 r 2 2 r Volume r 2 r Figure E7
APPENDIX F SELECTED ISOTOPES A-5 E.2 TRIGONOMETRY BASIC TRIGONOMETRIC FUNCTIONS. For rigt tringle, te sine, cosine, nd tngent of n ngle re defined s follows (see Figure E8): sin cos tn Side opposite Hypotenuse Side djcent to Hypotenuse Side opposite Side djcent to Figure E8 o o 2. Te secnt (sec ), cosecnt (csc ), nd cotngent (cot ) of n ngle re defined s follows: sec cos csc cot tn TRIANGLES AND TRIGONOMETRY. Te Pytgoren teorem sttes tt te squre of te ypotenuse of rigt tringle is equl to te sum of te squres of te oter two sides (see Figure E8): 2 2 2 o o 2. Te lw of cosines nd te lw of sines pply to ny tringle, not just rigt tringle, nd tey relte te ngles nd te lengts of te sides (see Figure E9): Lw of cosines Lw of sines c b Figure E9 c 2 2 b 2 2b cos OTHER TRIGONOMETRIC IDENTITIES. sin ( ) 2. cos ( ) cos 3. tn ( ) tn 4. () / (cos ) tn 5. sin 2 cos 2 6. sin ( ± ) cos ± cos If, sin ( ± ) cos If, sin 2 2 cos 7. cos ( ± ) cos cos ± If, cos ( ± ) If, cos 2 cos 2 sin 2 2 sin 2 b γ c ± F SELECTED ISOTOPES Atomic % Abundnce, Atomic Mss Atomic or Decy Mode Hlf-Life (if No. Z Element Symbol No. A Mss u If Rdioctive Rdioctive) 0 (Neutron) n.008 665 0.37 min Hydrogen H.007 825 99.985 Deuterium D 2 2.04 02 0.05 Tritium T 3 3.06 050 2.33 yr 2 Helium He 3 3.06 030 0.000 38 4 4.002 603 00 3 Litium Li 6 6.05 2 7.5 7 7.06 003 92.5 4 Beryllium Be 7 7.06 928 EC, 53.29 dys 9 9.02 82 00 5 Boron B 0 0.02 937 9.9.009 305 80. Dt for tomic msses re tken from Hndbook of Cemistry nd Pysics, 66t ed., CRC Press, Boc Rton, FL. Te msses re tose for te neutrl tom, including te Z electrons. Dt for percent bundnce, decy mode, nd lf-life re tken from E. Browne nd R. Firestone, Tble of Rdioctive Isotopes, V. Sirley, Ed., Wiley, New York, 986. lp prticle emission, negtive bet emission, positron emission, -ry emission, EC electron cpture.