FSK 116 Semester 1 Mathematics and Other Essentials. Priorities

Transcription

1 FSK 6 Semeste Mthemtics nd Othe Essentils Pioities Know how YOUR clculto woks nd lwys hve YOUR clculto with you. Alwys hve pencil (nd n ese) t hnd when doing Physics.

2 Geek Alphbet Alph Et Nu Tu Bet Thet Xi Upsilon Gmm Iot Omicon Phi Delt Kpp Pi Chi Epsilon Lmbd Rho Psi Zet Mu Sigm Omeg 3 Geek Alphbet cont. Alph Not Bet Not B Epsilon Not E Et Not n Nu Not v Kpp Not k, K Rho Not p Mu Not u t Tu Not T,t Upsilo n Not u Chi Not X Omeg Not w 4

3 Symbols nd Constnts = Equl to Not equl to Sme ode of mgnitude Appox. equl to Identicl Popotionl to Much less thn Much gete thn Less thn Gete Thn Gete thn o equl to Less thn o equl to Becuse Theefoe Implies, fom this follows x, x Chnge in x Absolute vlue of n! = n(n-)(n-) (3)()() n i x i Tends to Sum of ll the vlues of x =3.459 e = Pefixes SI Units te T 0 centi c 0 - gig G 0 9 milli m 0-3 meg M 0 6 mico 0-6 kilo k 0 3 nno n 0-9 hecto h 0 pico p 0 - dek d 0 femto f 0-5 deci d 0 - tto

4 Exmples Wite the following using the coect pefixes nd o 3 significnt figues: mm bytes m s 7 89 N J 7 Exponents When n is el nd positive: n... n times n n... n times n n 3 p q p q p 3 p q pq q If >0 nd n =b, then b>0 is the bse, n is the exponent nd b is the n th powe of. When using bse e: e =

5 Exmples Detemine the following in exponent nottion: Exmple Exponents nd gphs y x Fo = Fo >, eg. = Fo = ½ Fo = e x y = x ( = ) y = x ( = ) y = x ( = ½) y = x ( = e) Y Y=^x Y=^x x 0 5

6 x x y Fo = Fo >, eg. = Fo = ½ Fo = e y = x ( = ) y = x ( = ) -3 /8 - ¼ - ½ y = x ( = ½) Exmple cont y = x ( = e) Y Y=^x Y=^x Y=^x x x x y Fo = Fo >, eg. = Fo = ½ Fo = e y = x ( = ) y = x ( = ) y = x ( = ½) -3 /8 8 - ¼ 4 - ½ 0 ½ 4 ¼ 3 8 /8 4 6 /6 Exmple cont y = x ( = e) Y Y=^x Y=^x Y=½^x Y=^x x 6

7 x x y Fo = Fo >, eg. = Fo = ½ Fo = e y = x ( = ) y = x ( = ) y = x ( = ½) Exmple cont3 y = x ( = e) -3 / ¼ ½ ½.7 4 ¼ / / Y Y=^x Y=^x Y=½^x Y=e^x Y=^x x 3 Logithms & Ntul Logs n If b, then n log b Also b n, then logb n log b log b If =0 then bse is usully left out - if log s, then s 0 Bse e ntul logithm e = so if then ln s log e s(ponounced lin s ), s e 4 7

8 Rules fo Logs If, P, Q e positive, then log n P nlog P n log P log n log 0 log log P P log PQ log Plog Q log P log PlogQ Q log P log P log b b P 5 Exmples Detemine the following 3 0 Log 000 = 0 Log 00 = Log = 0 0 Log = Log = 6 8

9 Exmples Detemine n in the following: n log3.4 n log.34 n log.34 n log 3.4 n log3.4 n ln3.4 n ln e 7 Appliction of Logs: Ethqukes Richte Scle Not ccute fo lge qukes A ML log Alog A0 log A0 A is mximum excusion of the Wood-Andeson seismogph. A 0 depends only on the epicentl distnce of the sttion Moment Mgnitude Scle Moe ccute fo lge qukes. MW log M system_in_ll_its_jpnese_loctions.php 8 9

10 Appliction of Logs: Sound Fequency of 000Hz is just udible when: Intensity I 0 =0-6 W/cm Noise level is given in decibel by the following: L 0log I I0 Heing dmge cn occu fom 85dB with long tem exposue db t 0 Hz 300 feet wy Whispe: L 0dB Tlking: L 65 db Detemine the the eltionship of the intensities between tlk to whispe? Iw 0 0 log I0 Iw 0 I0 It 65 0log I0 6.5 It 0 I0 It I w Poblems to solve: Re-wite these using the coect SI unit pefixes nd 3 significnt figues: J W km s m m Using you own clculto detemine the following:.5 e 0.5 e 0.5 e.5 e.5 e Wht tend do you obseve? Using you own clculto detemine the following: /

11 Poblems to solve: Using you own clculto detemine the following: Log (549 3) Log (549/0) Log (549)/Log(0) Log (549/0) 3 Log 0 Log (-549) Do you obseve ny tends? ln (549 3) ln (549/0) ln (549)/ln(0) ln (549/0) 3 ln 0 Do you obseve ny similities/diffeences between the Log nd ln? Exmples Wite the following using the coect pefixes nd o 3 significnt figues: mm bytes m s 7 89 N J. km 46 Gbytes 3.9 m ps 7.9 kn 0.65 MJ

12 Exmples Detemine the following in exponent nottion: ( 3) 0 ( 3) ( 6) 8 8/ Rules fo Logs If, P, Q e positive, then log n P nlog P n log P log n log 0 log log P P log PQ log Plog Q log P log PlogQ Q log P log P log b b P 4

13 Exmples Detemine the following Log 000 = Log 00 = Log = Log = Log = Exmples Detemine n in the following: n log n log n log n log n log n ln n ln e 6 3

14 Poblems to solve: Re-wite these using the coect SI unit pefixes nd 3 significnt figues: J W km s m m Using you own clculto detemine the following:.5 e 0.5 e 0.5 e.5 e.5 e Wht tend do you obseve? Using you own clculto detemine the following: / Poblems to solve: Re-wite these using the coect SI unit pefixes nd 3 significnt figues: 54.9 MJ 3 MW Tm (987 Pm (Pet m) s 6.37 mm nm Detemine the following: = = = = 0.5 Using you own clculto detemine the following:.5 = 0.86 e 0.5 e = e =.8.5 e = = 9.49 e Do you obseve ny tends? 8 4

15 Poblems to solve: Using you own clculto detemine the following: Log (549 3) Log (549/0) Log (549)/Log(0) Log (549/0) 3 Log 0 Log (-549) Do you obseve ny tends? ln (549 3) ln (549/0) ln (549)/ln(0) ln (549/0) 3 ln 0 Do you obseve ny similities/diffeences between the Log nd ln? Eo / undefined Eo / undefined Eo / undefined 9 Angles nd Tig Functions 30 5

16 Angles O B + - A Angle is positive fo nticlockwise ottion Angle is negtive fo clockwise ottion Rottion = 360 ½ Rottion = 80 ¼ Rottion = 90 s s s 3 3 s s s s Rdins (d) Angles - s s s s s 3 3 s 3 3 s constnt Rottion = cic of cicle = \ Rottion = dins (360 ) ½ Rottion = ½cic of cicle = ½ = dins (80 ) ¼ Rottion = ¼cic of cicle = ¼ = / dins (90 ) \ = d d =

17 Exmples Convet to Rdins = 3 = 75 = 70 = 35 = A cicle hs dius of 3 cm wht is the c length subtended by n ngle of 50 Convet to degees ½ d = ½ d = 4 / 3 d = ⅔ d = 7 d = 6 d = d = 33 Clculto Skills Chnge ngle setting between degees din gdient: DGR Convet ngle between degees din gdient: DGR Set deciml nd exponents: FSE Fixed Scientific Engineeing Deciml 0 x 0 3n Using 0 x 3 0 pess 3 then Exp then Exp

18 Tigonomety Angle cn be in o d. Mke sue clculto is in ppopite mode. sin y ' x ' cos y ' tn x ' Mke sue you know how you clculto equies the input. As you see it witten - Function then ngle (DAL) O Angle then Function O y θ x P(x,y ) y x 35 Tigonomety 0.5 Vlue Sin Vlue Tn Angle - degees Vlue Cos

19 Tigonomety If the eltionship between the tingle sides e known, then the ngles cn be detemined. The c function: y' Sin y' Sin x' Cos x' Cos O y θ x P(x,y ) y x y' Tn x' y' Tn x' Sin Sin 37 Exmples Give the following nswes fo in : Sin = Tn = Cos - (-0.5) = Cos = Sin - (-.000) = Tn - 0 = Cos -.34 = Give the following nswes fo in dins (d). Confim you nswe by conveting to nd compe with pevious nswe. Sin = Tn = Cos - (-0.5) = Cos = Sin - (-.000) = Tn - 0 = 38 9

20 Fo ight ngled tingle Tig nd Tingles c Pythgos Theoem: c = + b Fo n equiltel tingle b Exmple Complete the following tble using the specil tingles on pevious pge d 0 /6 /4 /3 Sin Cos Tn 40 0

21 Tig. Identities Not used vey often in this couse but good to know them when needed. sin tn cos sin cos sin( A B) sin Acos Bcos Asin B sin( A B) sin Acos Bcos Asin B cos( A B) cos Acos Bsin Asin B cos( A B) cos Acos Bsin Asin B cos cos sin sin cos sin cos 4 Tig. Identities Cosine Rule c = + b -bcos C If C = 90 A c B C???? b Sine Rule b c sin A sin B sin C A c b B C 4

22 Exmples If A = B deive, using tig identities, n eqution fo: Sin A Cos A Wite the following in tems of cos A sin A = cos A = 43 Geometic Fomule Rectngul shpes Ae = b Volume = Ae depth = bc b c Tingul shpes Ae = ½bse height =½h h 44

23 Cicle Geometic Fomule- Cicumfeence = Ae = Sphee Ae = 4 Volume = 4 / 3 3 Cylinde Ae = ( ) + h Volume = coss-sectionl e height Volume = h h 45 Geometic Fomule-3 Tpesium Ae = ½(+b)h h b 46 3

24 Exmples A flt metl plte (dimensions in cm) is bent in the fom of gutte (see figue below). Find fomul fo the volume of the gutte in tems of the ngle Exmples - Conside n equiltel tingle with side lengths of m Using the ngles of the tingle, detemine the e of the tingle. Without using the ngles of the tingle, detemine its e. 48 4

25 Exmples 3 A cicul cetck hs n inne dius of km nd tck width of 6 m. Detemine the e of the odwy in sque metes. 49 Exmples 4 An empty cylindicl pipe hs n oute dimete of 30 mm nd n inne dimete of 5 mm. Detemine the volume of mteil needed to poduce m length of this pipe. Detemine the e of the pipe exposed to the tmosphee. Wht is the volume of wte which m of this pipe would contin if it wee closed t one end? 50 5

26 Exmples Convet to Rdins = d 3 = d 75 = d 70 = 4.74 d 35 = d A cicle hs dius of 3 cm wht is the c length subtended by n ngle of d s s cm.6cm Convet to degees ½ d = 8.65 ½ d = 90 4 / 3 d = 40 ⅔ d = 0 7 d = 6 d = d = Exmples Give the following nswes fo in : Sin = 60 Tn = 45 Cos - (-0.5) = 0 Cos = 60 Sin - (-.000) = -90 Tn - 0 = 0 Cos -.34 = Undefined/Eo Give the following nswes fo in dins (d). Confim you nswe by conveting to nd compe with pevious nswe. Sin =.047 d Tn = d Cos - (-0.5) =.094 d Cos =.047 d Sin - (-.000) =.57 d Tn - 0 = 0 d 5 6

27 Exmples A flt metl plte (dimensions in cm) is bent in the fom of gutte (see figue below). Find fomul fo the volume of the gutte in tems of the ngle. bse Sin bse sin Volume cente bse height length cos 0 40cos height cos height cos Volume totl 0sin cos 40cos sin 40cos 0 Volume side ½bse height length cos cos ½ sin 0sin 0 53 Exmples - Conside n equiltel tingle with side lengths of m Using the ngles of the tingle, detemine the e of the tingle. Without using the ngles of the tingle, detemine its e. Angles in equiltel tingle: ll 60 Height of tingle detemined fom Sin = h/ h = sin 60 Ae = ½ bse x h Ans:.73 m Use Pythgos theoem to detemine height Ae = ½ bse x h 54 7

28 Exmples 3 A cicul cetck hs n inne dius of km nd tck width of 6 m. Detemine the e of the odwy in sque metes. Decide which units you wnt to use. Detemine inne e of tck. Detemine oute e of tck. Find diffeence between two es. Ae Ae inne oute Ensue tht you finl nswe is in metes Ae od out in in out Ans: m 55 Exmples 4 An empty cylindicl pipe hs n oute dimete of 30 mm nd n inne dimete of 5 mm. Detemine the volume of mteil needed to poduce m length of this pipe. Detemine the e of the pipe exposed to the tmosphee. Wht is the volume of wte which m of this pipe would contin if it wee closed t one end? Ans: m m m 3 wte 56 8