MSCF"B<C%m@~CW[hwRX<EP ; 4V  Ή $.  K : g<J8XR8ao\rx,EPQD.BIFVY~   " !L!$"0"X# )$w'jG(c/(T96 HICALC~1.000iTj7 EL212F~1.548G[j7 EL7D88~2.5477bj7 EL6D80~1.546pj7 EL6D8A~1.545/vj7 EL8D88~1.544'{j7 EL7D84~1.543j7 ELFEEC~1.5428 j7 ELFEE0~1.541 j7 EL7D88~1.540j7 EL6D84~1.539j7 EL6D8C~1.538[q j7 EL6E99~1.537 &j7 EL6A99~1.5362j7 EL6699~1.535z Gj7 EL6299~1.534#XRj7 ELFEE5~1.533{Yj7 ELE1FF~1.532]j7 ELECTR~4.531ej7 ELECTR~3.530_ hj7 ELD48A~1.529Q rj7 ELB88F~1.528]zj7 ELBE7F~1.527j7 ELBA7F~1.526j7 ELA8BE~1.525bj7 ELECTR~2.5241"j7 ELECTR~1.523#&j7 ELA744~1.522.j7 ELA344~1.521W 7j7 ELA544~1.520Bj7 ELA144~1.519 Jj7 ELB482~1.518 JXj7 ELB195~1.5175cj7 ELB185~1.516kj7 ELA24F~1.515 rj7 ELAE3F~1.514 |j7 ELA94F~1.513 j7 ELA54F~1.512p j7 ELA14F~1.511 Bj7 ELA74F~1.510j7 ELA34F~1.509S'j7 ELA244~1.508n,j7 ELB082~1.507j5j7 ELB872~1.506:j7 ELB472~1.505>j7 ELA4C3~1.504yFj7 ELA0C3~1.503% nLj7 ELA913~1.502 Yj7 ELA513~1.501 fj7 ELA113~1.500 qj7 ELAD03~1.499{ Q~j7 ELA903~1.498 j7 ELA503~2.497} j7 ELA103~1.496 aj7 ELB281~2.495Q#j7 ELB681~2.494'j7 ELB281~1.493,j7 ELBC81~1.492E5j7 ELB881~1.491"<j7 ELB481~1.490Cj7 ELB081~2.489 Hj7 ELBC71~1.488Lj7 ELB681~1.487Pj7 ELAFF2~1.486OWj7 ELA803~1.485]T]j7 ELA403~1.484ej7 ELB081~1.483jj7 ELA003~1.482-pj7 ELACF2~1.481"wj7 ELBA81~1.480v}j7 ELA503~1.4797j7 ELADF2~1.478G7j7 ELB671~1.477 ~ j7 ELA603~1.476l Ij7 ELA203~1.475a j7 ELA180~1.474y +j7 ELAD70~1.473,4j7 ELB871~1.472 <j7 ELB471~1.471Ej7 ELBE90~1.470$Jj7 ELBA90~1.469Nj7 ELB690~1.4683Tj7 ELB290~1.467"Xj7 ELBE80~1.466&\j7 ELB680~1.465dj7 ELB890~1.4646ij7 ELA09E~1.463oj7 ELAC8E~1.462=uj7 ELA88E~2.461{j7 ELA88E~1.460;j7 ELA442~1.459;j7 ELA042~1.458jj7 ELAA8E~1.457kGj7 ELA68E~1.456j7 ELA28E~1.455Pj7 ELAF7E~1.454n!j7 ELBC90~1.453X(j7 ELB490~1.452Pw,j7 ELA742~1.451_1j7 ELA342~1.450b&9j7 ELA942~1.449_>j7 ELA542~1.4485Ej7 ELA142~1.447Kj7 ELAC43~1.446wPj7 ELA843~2.445*Vj7 ELA443~2.444R]j7 ELAA43~1.443'cj7 ELA643~2.442U-ij7 ELA243~2.441goj7 ELA843~1.440sj7 ELA443~1.439]yj7 ELA043~1.438~j7 ELA643~1.437j7 ELA243~1.4365j7 ELAE33~1.435 j7 EL45E7~1.434|j7 ELBE70~1.433@Jj7 ELB480~1.432 j7 ELB080~1.431 &j7 ELA08E~1.430F\1j7 ELAC7E~1.429 97: ELECTR~3.428+C7: ELECTR~2.427o7: ELECTR~1.426k79 ELECTR~4.425 kj7Z DYF6D7~1.424xuj7Z DYF6D6~1.423a|j7Z DY6D5C~1.422 j7Z DYE6D4~1.421~  j7Z DY6D58~1.420X g j7Z DY6D52~1.419F j7Z DY6D5E~1.418# j7Z DY5D52~1.417() j7Z DY5D54~1.416&>/ j7Z DY5D5E~2.415S d5 j7Z DY6D5F~1.414A j7Z DY6D51~1.413g Q j7Z DY6D5B~1.412;] j7Z DY6D57~2.411d j7Z DY6D59~2.410|pm j7Z DYAD3E~1.409s j7Z DYAB3E~1.408z j7Y DYA93E~1.407F j7Y DYA73E~1.406=F j7Y DYA53E~1.405 j7Y DYA33E~1.404 j7Y DYB2EF~1.403$ j7Y DYD944~1.402$= j7Y DY4347~1.401a$ j7Y DYBEDF~1.400iP( j7Y DY024F~1.3990 j7Y DY024B~1.398 7 j7Y DY5D5E~1.397 = j7Y DYF65C~1.3968 "I j7Y DY6D57~1.395ZS j7Y DY6D55~1.394IY j7Y DY6D59~1.393A^ j7Y DYNAMI~4.392Rb j7Y DYNAMI~3.391ig j7Y DYNAMI~2.390 zk j7Y DYNAMI~1.389hw 7͑ DYNAMI~3.388 7͑ DYNAMI~2.387 7̑ DYNAMI~1.386F 7͑ DYNAMI~4.385F j7 TH2BDE~1.384-N j7 THERMO~4.383] j7 THERMO~3.382ll j7 THERMO~2.381 j7 THERMO~1.380 7 THERMO~3.379o 7 THERMO~2.378." 7 THERMO~1.377S)K 7 THERMO~4.376|` j7 OP4DE2~1.375i j7 OP4DEE~1.374Yq j7 OP5DE4~1.373njy j7 OP5DE8~1.372w  j7 OP4DE0~1.371j7 OP4DEF~1.370 j7 OPB492~1.369 %j7 OPB092~1.368 c1j7 OP1A8A~1.367=j7 OP40F0~2.366dOj7 OP1D82~1.365 Uj7 OP40F4~1.364Xaj7 OP40F0~1.363 1gj7 OP60FD~1.362sj7 OP50F1~1.361wj7 OP50FD~1.360{j7 OPTICS~4.359 j7 OPTICS~3.358T j7 OPTICS~2.357 j7 OPTICS~1.3567 OPTICS~3.355 "7 OPTICS~2.354<-7 OPTICS~1.353vj7 OPTICS~4.3527 ASTROP~1.351G 7 ASTROP~4.350XB7 ASTROP~3.349?Q7 ASTROP~2.348mj7 MAFEF7~1.3471tj7 MADEF3~1.346O {j7 MATH_1~1.345%j7 MA1B8C~1.344*%j7 MAFEF9~2.343 O+j7 MAEEFD~2.342 4j7 MAEEF1~2.341 >j7 MAEEF5~2.340k Hj7 MAEEF9~1.339#Tj7 MADEFD~2.338 [j7 MADEF1~1.337dj7 MADEF5~1.336\jj7 MAFEF9~1.335r2sj7 MAEEFD~1.334E{j7 MAEEF1~1.333j7 MAEEF5~1.332xj7 MAB7A5~1.331j7 MAB7A9~1.330j7 MAA7AD~1.329Wj7 MADEFD~1.328{?%j7 MATH_1~4.327~,j7 MATH_1~3.32682j7 MA1781~1.325+ 8j7 MATH_1~2.324\6:i7M MATH_I~1.323Z<i7M MATH_F~2.322>Di7M MATH_F~1.321:*`i7M MATH_F~3.320"9dp3G SOUND1~3.3193G SOUND1~2.318n 3G SOUND1~1.317~$3G SO633A~1.316<3G SO533E~1.315F Z>#4< SO5332~1.3146J#4< SO5034~1.313N#4< SO533A~1.312c#4< SOUND0~4.311Tj#4 SOUND0~3.310*#4< SOUND0~2.309*#4< SOUND0~1.308 , 7ܚ BLACKS~1.307 # 7ܚ 0Default.306~-7{^ _TOOLS~4.305 x.8 _TOOLS~3.304:7{^ _TOOLS~2.303:7{^ _TD20E~1.302mI8 _T97A2~1.3013 AZ7|^ _TOOLS~1.300?td7|^ _SD1A7~1.29928f> _S1777~1.298y27|^ _S1AEE~1.297QP7|^ _S1BFA~1.296C#co8 _SB329~1.2957|^ _S2F44~1.294#>7}^ _S2AAB~1.293:>L8 _S3B71~1.2927}^ _S7B07~1.291G7}^ _S8D2E~1.290/8 _S6670~1.289F7}^ _SA24E~1.288B O7}^ _S8E46~1.287 Io8 _SE68C~1.2867^ _S86CE~1.285L17^ _SB8F9~1.284m0B8 _S82A7~1.2831Gs7^ _SA548~1.282:7^ _SB508~1.281`::8= _S09A7~1.280%u7^ _S9970~1.279T 7|^ _S4F01~1.278!T 8 _SDD2C~1.277u7^ _SMALL~4.276_#7^ _SMALL~3.27558 _SMALL~2.274 1R7^ _SMALL~1.273'"\7^ _SKINP~3.2724~8nA _SKINP~2.2717^ _SKINP~1.270<8 _NUMBE~4.269,N88C _NUMBE~3.268" {8 _NUMBE~2.2677^ _NE26C~1.2668 _N97D1~1.265 77^ _NUMBE~1.264ZD7^ _NORMA~3.263yU8U _NORMA~2.262g7^ _NORMA~1.261!m7^ _MIMES~3.2601 p8 _MIMES~2.259~7^ _MIMES~1.2587^ _LOGO_~2.257(8 _LOGO_~1.256 ;7^ 000_Logo.255F7^ _LISTB~3.254K8 _LISTB~2.253PZ7^ _LISTB~1.252k']7^ _L72CD~1.251~r8 _LISTB~4.2507^ _ListBox.24947^ _LE511~1.248Z*u77^ _L7469~1.247*a8=A _L6858~1.2467^ _L94F5~1.245[7^ _L2039~1.244[-7^ _L7AB4~1.243xF7^ _LARGE~3.242 X8C _LF3F3~1.241e v7^ _LARGE~2.240* 7^ _LARGE~4.239Z * 7^ _LARGE~1.238-6 7^ _KEYBO~2.237'd 83 _KEYBO~1.236!7^ _KEYBO~3.235@!7^ _G7C61~1.234 G!8' _GD510~1.233"!7^ _G5E3B~1.232#!7^ _GDDE2~1.231 >!8, _G9C93~1.230 1_!7^ _G4618~1.229"j!7^ _G8377~1.2289p!84 _GRAPH~4.227g"7^ _GD131~1.226g"7^ _GRAPH~3.225^ "89 _GRAPH~2.224T"7^ _GRAPH~1.2236"7^ _FE4D5~1.222 "8@ _F909E~1.221T("7^ _F66BF~1.220a-*" 7% _FC4D6~1.219{."8I _F0ACE~1.218z =" 7$ _F7FD0~1.217?" 7$ _F7CE5~1.216 N"8 _F6C96~1.215a" 7$ _FA410~1.214g" 7$ _FF1E9~1.2135k"8t _F9056~1.2129{" 7$ _F341D~1.211~" 7$ _F30C1~1.210 #8V _FCF3A~1.209Q # 7$ _F5D84~1.208X #7J` _F04E8~1.207F#8k _FORMU~4.206/0#7N` _FORMU~3.2055#7v _FORMU~2.204 7#8c _FORMU~1.20320C#7v _FC0F8~1.202$bD#7v _F2E33~1.201x K#8{ _F454D~1.200V#7v _F8B5D~1.199 X#7M` _FF5B9~1.198&d#8~ _F9A36~1.197x#7L` _F3464~1.196#7v _FE1AE~1.195 $8 _F27FA~1.194" $7v _F9AEA~1.193G $7L` _FF85A~1.192$8l' _FA2D8~1.191.$7K` _FA6C9~1.1904$7^ _E094A~1.189r -7$8 _EAAC0~1.188`C$7^ _EE689~1.187D$7^ _E1F88~1.186N G$8 _EBDE7~1.185eS$7^ _E1F92~1.184bU$7^ _E06D0~1.183i W$8 _EAAC7~1.182kb$7^ _E94E8~1.181Qc$7^ _ED5A0~1.180 f$8 _E82BF~1.179~s$7^ _EQUAT~4.178! @u$7t _EQUAT~3.177a$8 _EQUAT~2.176-%7t _EQUAT~1.175-%7^ _DATET~3.174%8r _DATET~2.173 9%7^ _DATET~1.172XbE%7^ _CHANG~2.171NZ%8 _CHANG~3.170p%7^ _CHANG~1.169>v% 7" _CB8DA~1.168>&8 _C0297~1.167$>& 7! _CA2CA~1.166W&7^ _CATEG~4.165f&8 _CATEG~3.164x&7^ _CATEG~2.163J|~& 7Z _C4152~1.162xN'8 _C5E4B~1.161xN' 7[ _CATEG~1.160m'7^ _BODER~1.159n'7^ _BODER~3.158#o'7^ _BODER~2.157h3p'7g^ TOOLSP~4.156l q'8( TOOLSP~3.155}'7g^ TOOLSP~2.154}'7g^ TOC802~1.153(8& TO0462~1.152(7h^ TOOLSP~1.151+=(7i^ SM4492~1.1503V(8( SMC9A0~1.149()7i^ SMADDF~1.148()7i^ SMBA04~1.147S%;)8( SM056B~1.146Ir`)7j^ SM67B5~1.145:n)7j^ SM5EC5~1.144=*8( SMC640~1.143=*7j^ SMC5C4~1.142}W*7j^ SM5E3C~1.1410j*8z SM55C9~1.140+7j^ SM2C51~1.139+7j^ SM35B5~1.138N#(+8$& SMC086~1.137L+7k^ SMA73F~1.136z.,[+7k^ SMD57E~1.135/,8( SM9A9A~1.134/,7k^ SMD36F~1.133V7E,7k^ SMALLF~4.1328|,8( SMALLF~3.131-7l^ SMALLF~2.130 -7i^ SM3B33~1.129$-8( SM4DF8~1.1286-7l^ SMALLF~1.127<-7l^ SMALLB~3.1265X-8( SMALLB~2.125.7m^ SMALLB~1.124Ĉ.7m^ SKINPR~3.123=/8c@ SKINPR~2.1228=/7m^ SKINPR~1.121:9Z/8E NUMBER~4.120 +085D NUMBER~3.119<" +08 NUMBER~2.118kGM07m^ NU7B65~1.117f08( NU5516~1.116~ 17n^ NUMBER~1.115~ 17n^ NORMAL~3.114+18BB NORMAL~2.113.117n^ NORMAL~1.112617o^ MIMESC~3.1111 :18( MIMESC~2.110G17o^ MIMESC~1.109I17o^ Logo_VGA.108&gb18 LOGO_S~1.107 27p^ 0000Logo.1063 27p^ LISTBO~3.105 28 LISTBO~2.104[27p^ LISTBO~1.103 [!27p^ LIB3C6~1.102.28|( LISTBO~4.101H27p^ 0ListBox.1004yO27q^ LA2BD2~1.099+37q^ LARGEF~4.0981+38@ LARGEF~3.097]37q^ LARGEF~2.096Epp37q^ LA3ACE~1.095E47q^ LARGEF~1.094E47r^ LARGEB~4.09361486 LARGEB~3.092g47r^ LARGEB~2.091&vx47r^ LAD6FF~1.09057r^ LARGEB~1.089G,57s^ KEYBOA~2.088(A58 KEYBOA~1.087j57s^ Keyboard.086:~57s^ GRAPHT~3.085 58[( GRAPHT~2.084 67s^ GRAPHT~1.083Q 67s^ GRAPHO~3.082E ^68T( GRAPHO~2.081 ;67s^ GRAPHO~1.080F67t^ GRAPHN~3.079kM68M( GRAPHN~2.078g_67t^ GRAPHN~1.077b67t^ GRAPHC~3.0766h68E( GRAPHC~2.075w67t^ GRAPHC~1.074y67w^ FODC67~1.073 {68;( FO061F~1.072T77w^ FOBF33~1.071T7 7 FOB6A9~1.070c 783( FO8978~1.069wn7 7 FO9DFE~1.068 77`` FOA462~1.067q$78.' FOF190~1.066 <77`` FORMUL~4.065&I78 FORMUL~3.064b783 FORMUL~2.063^~77u^ FORMUL~1.06289` FODD8D~1.061 88%( FO1D62~1.060- 8 7Ú FO4A96~1.059X87`` FOAAFA~1.058<88g FOFCDC~1.057%187]` FO0771~1.056687u FO6FA1~1.055 v888( FO5732~1.054 D87u FO1A44~1.053E87u FO5486~1.052K88 ( FOC1A6~1.051[87u FOBD9F~1.050< h^87]` FO4F8D~1.049&k88~ FODD6A~1.048?87\` FO00A6~1.0479 7 FOD077~1.046 98( FO081F~1.045% 97u FOBF43~1.044G+97\` FO9FD1~1.043r98l' FO7F4A~1.042Q097a` FODA0D~1.041 697w^ EQ4660~1.040r 898L' EQC297~1.039` E97w^ EQ9527~1.038F97w^ EQ776C~1.037N 0I98C' EQ5395~1.036e~U97x^ EQ4A27~1.035V97x^ EQ05B6~1.034j X987' EQA001~1.033c97x^ EQE633~1.032:d97x^ EQ0B39~1.031Z g98/' EQA340~1.030 gs97x^ EQUATI~4.029_ pu97t EQUATI~3.028:8(' EQUATI~2.027:7t EQUATI~1.026}7:7y^ DATETI~2.0259:8v DATETI~1.024 5Y:7y^ DateTime.023\ f:7y^ CHANGE~3.022Vs:8' CHANGE~2.021;7y^ CHANGE~1.020;8` CAD317~1.019P<8& CA08E5~1.018 P< 7 CATEGO~4.017#{q<7z^ CATEGO~3.016=8# CATEGO~2.015 =7z^ CATEGO~1.014Lr'= 7 CA51AD~1.013#Rs=8 CACE2F~1.012 > 7 Category.0115 >7{^ BODERC~3.010!>8P BODERC~1.009|#>8P BODERC~2.008 %>8Ȗ 00Hicalc.007PP?7 Biorythm.006PP?9jd 000units.005R@9" currency.004cR@7 000const.003A7d 0000mtex.002B9d 00Hicalc.001S8vMSCET c$=TMTkTd HiCalc Your Trusted CalculatorPPCLINKHPCJUPITER %CE1%\HiCalc%CE2%%CE11%Help interfacesDefault Black SkinSounds Formulas Math Image Astrophysic OpticsThermodynamicsDynamics and mechanicsElectric Engineering SoftwarePPCLINKHiCalc HiCalc.lnk%LinkFilename2%.lnk                     # Hicalc.exe$ mtex.dll const.dat currency.dat units.dat Biorythm.dat Hicalc.htmBoderColores.dat BoderColores_SQVGA.dat BoderColores_VGA.dat  Category.png Category_SQVGA.png Category_VGA.pngCategoryButtons.pngCategoryButtons_SQVGA.pngCategoryButtons_VGA.pngCategoryLabels.pngCategoryLabels_SQVGA.pngCategoryLabels_VGA.pngChangePageButtons.pngChangePageButtons_SQVGA.pngChangePageButtons_VGA.png DateTime.PNGDateTime_SQVGA.pngDateTime_VGA.pngEquationsSolver_Back.PNGEquationsSolver_Back_SQVGA.pngEquationsSolver_Back_VGA.PNG"EquationsSolver_DeleteButtons.PNG(EquationsSolver_DeleteButtons_SQVGA.png&EquationsSolver_DeleteButtons_VGA.PNG EquationsSolver_Operators.PNG!$EquationsSolver_Operators_SQVGA.png""EquationsSolver_Operators_VGA.PNG#%EquationsSolver_ScrollDownButton.PNG$ +EquationsSolver_ScrollDownButton_SQVGA.png% )EquationsSolver_ScrollDownButton_VGA.PNG& #EquationsSolver_ScrollUpButton.PNG' )EquationsSolver_ScrollUpButton_SQVGA.png( 'EquationsSolver_ScrollUpButton_VGA.PNG)Formula_Back.PNG*Formula_Back_SQVGA.png+Formula_Back_VGA.PNG,Formula_List_DownArrow.PNG-!Formula_List_DownArrow_SQVGA.png.Formula_List_DownArrow_VGA.PNG/Formula_List_IconItem.PNG0 Formula_List_IconItem_SQVGA.png1Formula_List_IconItem_VGA.PNG2Formula_List_Thumb.PNG3Formula_List_Thumb_SQVGA.png4Formula_List_Thumb_VGA.PNG5Formula_List_UpArrow.PNG6Formula_List_UpArrow_SQVGA.png7Formula_List_UpArrow_VGA.PNG8Formula_Next.PNG9Formula_Next_SQVGA.png:Formula_Next_VGA.PNG; Formula_Operators.PNG<!Formula_Operators_SQVGA.png="Formula_Operators_VGA.PNG>#Formula_Panel.PNG?$Formula_Panel_SQVGA.png@%Formula_Panel_VGA.PNGA&Formula_Play.PNGB'Formula_Play_SQVGA.pngC(Formula_Play_VGA.PNGD)Formula_Switch_ClearAll.PNGE*"Formula_Switch_ClearAll_SQVGA.pngF+ Formula_Switch_ClearAll_VGA.PNGG,FormulaButtons.PNGH-FormulaButtons_SQVGA.pngI.FormulaButtons_VGA.PNGJ/GraphCloseIcon.PNGK0GraphCloseIcon_SQVGA.pngL1GraphCloseIcon_VGA.PNGM2GraphNewIcon.pngN3GraphNewIcon_SQVGA.pngO4GraphNewIcon_VGA.PNGP5GraphOperators.PNGQ6GraphOperators_SQVGA.pngR7GraphOperators_VGA.pngS8GraphTextXY.PNGT9GraphTextXY_SQVGA.pngU:GraphTextXY_VGA.PNGV; 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B" FA# .$ &% +% C% & ' ( ) 0* + , ]- . / {0 +1 2 B3 4<5(H5T55o 6"7#8_%u9&T:&:'<'/='=g(>(>K)A?)?F*@*A+B+SCK, D,Do-E.\F.G.G~/GH;0H0I<1WJ1J2cL}2AM2Nj3O3Q3QH4GS4'T4,UY5V5XO6iX7uX8Yj9Yj:^Zv; [<[=\b>]?T^1@._ A_ cA^` A` Bxa dBbBb=CbCcDd%E+eMFfFfFJgLGhGh HhsHhHi`IjI`kI X_C= 1 .4.2.5.2 =V_L and V_C are anti-phase. X_L < X_C= 1 .4.2.6 =Series resonance= 1 .4.2.7 =Q-factor= 1 .4.2.8 =Bandwidth= 1 .4.2.9 =Power in a.c. circuits(R-L, R-C or R-L-C series circuit)= 2 .4.2.9.1 =Based on resistance and current= 1 .4.2.9.2 =Based on the value of angle= 1 .4.3 =Single-phase parallel a.c. circuits= 2 .4.3.1 =Parallel Resonance and Q-factor= 2 .4.3.1.1 =Resonant frequency= 1 .4.3.1.2 =Dynamic resistance= 1 .4.3.1.3 =Q-factor= 1 .4.4 =Series resonance and Q-factor= 2 .4.4.1 =Series Resonance= 1 .4.4.2 =Q factor= 1 .4.4.3 =Voltage magnification= 2 .4.4.3.1 =Resonant frequency= 1 .4.4.3.2 =Maximum p.d.= 1 .4.4.4 =Overall Q-factor for series components= 1 .4.4.5 =Bandwidth= 0 .4.4.6 =Small deviations from the resonant frequency= 2 .4.4.6.1 =frequency w1= 0 .4.4.6.1 =frequency w2= 0 .4.5 =Parallel resonance and Q-factor= 2 .4.5.1 =Parallel resonance= 1 .4.5.2 =The LRC parallel network= 1 .4.5.3 =Dynamic resistance= 1 .4.5.4 =Q-factor in a parallel network= 1 .4.5.5 =Natural and forced resonant frequency= 1 .4.5.6 =The LR-CR parallel network= 1 .4.5.7 =Q-factor of parallel components= 1 .4.6 =Three-phase systems= 2 .4.6.1 =Star connection= 2 .4.6.1.1 =Line currents= 0 .4.6.1.2 =Line voltages= 1 .4.6.2 =Delta connection= 2 .4.6.2.1 =Line voltages= 0 .4.6.2.2 =Line currents= 1 .4.6.3 =Power in three-phase systems= 2 .4.6.3.1 =Power in three-phase systems(Based on resistance and phase current)= 1 .4.6.3.2 =Power in three-phase systems(Based on line voltage and line current)= 1 .4.6.3.3 =Total volt-amperes= 1 .4.6.4 =Measurement of power in three-phase systems= 2 .4.6.4.1 =One-wattmeter method for a balanced load= 1 .4.6.4.2 =Two-wattmeter method for balanced or unbalanced loads(Total power)= 1 .4.6.4.3 =Two-wattmeter method for balanced or unbalanced loads(The power factor)= 1 .4.6.4.4 =Three-wattmeter method for a three-phase, 4-wire system for balanced and unbalanced loads= 1 .4.7 =A.c. motors= 2 .4.7.1 =Synchronous speed= 1 .4.7.2 =Slip= 1 .4.7.3 =Rotor e.m.f. and frequency= 2 .4.7.3.1 =Rotor e.m.f.= 1 .4.7.3.2 =Rotor frequency= 1 .4.7.4 =Rotor impedance and current= 2 .4.7.4.1 =Rotor impedance= 1 .4.7.4.2 =Rotor current - Starting current= 1 .4.7.4.3 =Rotor current - Running current= 1 .4.7.5 =Rotor copper loss= 1 .4.7.6 =Efficiency of induction motor= 1 .4.7.7 =Torque equation for an induction motor= 1 .4.8 =A.c. bridges= 2 .4.8.1 =Balance conditions for an a.c. bridge= 1 .4.8.2 =Types of a.c. bridge circuit= 2 .4.8.2.1 =The simple Maxwell bridge(Rx)= 1 .4.8.2.10 =The Schering bridge(Rx)= 1 .4.8.2.11 =The Schering bridge(Cx)= 1 .4.8.2.12 =The Schering bridge(Loss angle)= 1 .4.8.2.13 =The Wien bridge(The balance bridge)= 1 .4.8.2.14 =The Wien bridge(Frequency)= 1 .4.8.2.2 =The simple Maxwell bridge(Lx)= 1 .4.8.2.3 =The Hay bridge(R_x)= 1 .4.8.2.4 =The Hay bridge(L_x)= 1 .4.8.2.5 =The Owen bridge(Rx)= 1 .4.8.2.6 =The Owen bridge(Lx)= 1 .4.8.2.7 =The Maxwell-Wien bridge(Rx)= 1 .4.8.2.8 =The Maxwell-Wien bridge(Lx)= 1 .4.8.2.9 =The de Sauty bridge= 1 .4.9 =Delta-star and star-delta transformations= 2 .4.9.1 =Delta-Star to Star-Delta= 1 .4.9.2 =Delta-Star to Star-Delta= 1 .4.9.3 =Delta-Star to Star-Delta= 1 .4.9.4 =Star-Delta to Delta-Star= 1 .4.9.5 =Star-Delta to Delta-Star= 1 .4.9.6 =Star-Delta to Delta-Star= 1 Q!The quantity of electrical chargeaC0ICurrentbA0t'The time during which the current flowscs0b*ca/ca/bRThe resistancea\Omega0VPotential differencebV0I The currentcA0b/ca*cb/ar&The internal resistance of the batterya\Omega0E"The battery voltage without a loadbV0VThe battery voltage with a loadcV0I)The total current supplied by the batterydA0(b-c)/da*d+cb-a*d(b-c)/aRThe resistancea\Omega0\rhoThe resistivityb\Omega m0lThe length of the conductorcm0a#Cross-sectional area of a conductordm^20b*c/da*d/ca*d/bb*c/aR_tResistance at t^oCa\Omega0R_0Resistance at 0^oCb\Omega0\alpha_0-Temperature coefficient of resistance at 0^oCc/^oC0t Temperatured^oC0 b*(1+c*d) a/(1+c*d) (a/b-1)/d (a/b-1)/cR_tResistance at t^oCa\Omega0R_{20}Resistance at 20^oCb\Omega0 \alpha_{20}.Temperature coefficient of resistance at 20^oCc/^oC0t Temperatured^oC0 b*(1+c*d) a/(1+c*d) (a/b-1)/d (a/b-1)/cR_1Resistance at t1^oCa\Omega0R_2Resistance at t2^oCb\Omega0\alpha_0-Temperature coefficient of resistance at 0^oCc/^oC0t_1 Temperatured^oC0t_2 Temperaturee^oC0b*(1+c*d)/(1+c*e)a(1+c*e)/(1+c*d)I The currentaA0VPotential differencebV0RThe resistancec\Omega0b/ca*cb/aPElectrical poweraW0VPotential differencebV0I The currentcA0b*ca/ca/bPElectrical poweraW0I The currentbA0RThe resistancec\Omega0b^2*c sqrt(a/c)a/b^2PElectrical poweraW0VPotential differencebV0RThe resistancec\Omega0b^2/c sqrt(a*c)b^2/aWElectrical energyaJ0PElectrical powerbW0tThe timecs0b*ca/ca/bEPotential gradientaV/m0VThe vontage between the platesbV0dThe distance between the platescm0b/ca*cb/aC CapacitanceaF0Q The chargebC0VThe vontage between the platescV0b/ca*cb/aQ The charge stored in a capacitoraC0I The currentbA0tThe timecs0b*ca/ca/bDThe electric flux densityaC/m^20Q The charge stored in a capacitorbC0A6The area is perpendicular to the direction of the fluxcm^20b/ca*cb/a \epsilon_0The permittivity of free spaceaF/m 8.85*10^(-12)DThe electric flux densitybC/m^20EThe potential gradientcV/m0b/ca*cb/aDThe electric flux densityaC/m^20EThe potential gradientbV/m0 \epsilon_0The permittivity of free spacecF/m 8.85*10^(-12) \epsilon_rThe relative permittivitydF/m0b*c*da/(c*d)a/(b*c)\epsilonThe absolute permittivityaF/m0 \epsilon_0The permittivity of free spacebF/m 8.85*10^(-12) \epsilon_rThe relative permittivitycF/m0b*ca/(c)CThe capacitanceaF/m0 \epsilon_0The permittivity of free spacebF/m 8.85*10^(-12) \epsilon_rThe relative permittivitycF/m0AThe area of one of the platesdm^20dThickness of dielectricem0b*c*d/e a*e/(b*d) a*e/(b*c)b*c*d/aC CapacitanceaF/m0 \epsilon_0The permittivity of free spacebF/m 8.85*10^(-12) \epsilon_rThe relative permittivitycF/m0AThe area of one of the platesdm^20dThickness of dielectricem0NThe count platesfm0 b*c*d*(f-1)/ea*e/(b*d*(f-1))a*e/(b*c*(f-1)) b*c*d*(f-1)/aC(The total equivalent circuit capacitanceaF0C_iThe capacitance of capacitor ibF0C(The total equivalent circuit capacitanceaF0C_iThe capacitance of capacitor ibF0 E_mDielectric strengthaV/m0V_mThe voltage maxbV0d The distancecm0b/ca*cb/a W The energy stored by a capacitoraJ0CThe capacitance of capacitorbF0VThe Voltage of capacitorcV0b*c^2/22*a/c^2 sqrt(2*a/b)BThe magnetic flux densityaT0\PhiThe magnetic fluxbWb0A6The area is perpendicular to the direction of the fluxcm^20b/ca*cb/aF_mThe magnetomotive forceaA0N.The number of conductors or turns of the coil.b0I The current in conductor or coilcA0b*ca/ca/bHThe magnetic field strengthaA/m0NThe number of conductorsb0I The currentcA0lThe length of the flux pathdm0b*c/da*d/ca*d/bb*c/aHThe magnetic field strengthaA/m0\mu _0The permeability of free spacebH/m 4*pi*10^(-7)BThe magnetic flux densitycT0c/ba*b\muThe absolute permeabilityaH/m0\mu _0The permeability of free spacebH/m 4*pi*10^(-7)\mu_rThe relative permeabilityc0b*ca/ca/bSThe reluctancea1/H0lThe length of the flux pathbm0\mu _0The permeability of free spacecH/m 4*pi*10^(-7)\mu_rThe relative permeabilitydH/m0AThe cross-sectional areaem^20 b/(c*d*e)a*c*d*e b/(a*d*e) b/(a*c*e) b/(a*c*d)SThe total reluctanca1/H0S_iThe reluctanc of part ib1/H0F The forceaN0BThe flux density of the fieldbT0iThe strength of the currentcA0lThe length of the conductordm0\thetaAnglee^o0 b*c*d*sin(e)a/(c*d*sin(e))a/(b*d*sin(e))a/(c*b*sin(e))FThe force experienced by a wireaN0QThe electric charge of wirebC0vThe velocity of wirecm/s0BThe flux density of the fielddT0b*c*da/(c*d)a/(b*d)a/(b*c) EThe electromotive forceaV0BThe flux density of the fieldbT0lThe length of the conductorcm0vThe velocity of thingdm/s0b*c*da/(c*d)a/(b*d)a/(b*c) LThe inductanceaH0\PhiThe change in fluxbWb0i The currentcA0b/ca*cb/a EThe electromotive forceaV0LThe self inductancebH0dIThe change in currentcWb0dt(The time taken for the current to changeds0 LThe inductance of coilaH0NThe count of turnb0\PhiThe fluxcWb0I The currentdA0b*c/da*d/ca*d/bb*c/a WThe energy storedaJ0LThe self inductancebH0I The currentcA0b*c^2/22*a/c^2 sqrt(2*a/b) V_0 output signalaV0GOpen-loop voltage gainbV0V_2non-inverting inputcV0V_1inverting inputdV0b*(c-d)a/(c-d)a/b + dc - a/b A_scommon-mode gaina0V_0 output signalbV0V_{cm}common input signalcV0b/ca*cb/a CMRRCommon-mode rejection ratioadB0A_ddifferential voltage gainbV0A_scommon-mode gainc020*log(b/abs(c)) V_{out}output voltageaV0V_{in} input voltagebV0R_fFeedback Resistorc\Omega0R_{in}Resistord\Omega0 - b*(c/d)-(a*d)/c-(a*d)/b-(b*c)/a V_{out}output voltageaV0V_{in} input voltagebV0R_2Feedback Resistorc\Omega0R_1Resistord\Omega0 b*(1 + c/d) a*d/(c+d) (a-b)*(d/b) b*c/(a-b)   V_{initial}2the output voltage of the integrator at time t = 0  V_{out}aV0V_2bV0V_1cV0R_1d\Omega0R_2e\Omega0R_3f\Omega0R_fg\Omega0#b*(((g + d)*f)/((e + f)*d)) - c*g/d FLThe force of attraction or repulsion between two electrically charged bodiesaN0kConstantb0q_1Electric charge of body 1cC0q_2Electric charge of body 2dC0dThe distance of two bodiesem0 b*c*d/e^2 DDissipation factora0R_S Resistanceb\Omega0\omega FrequencycRad0C_S CapacitancedF0b*c*da/(c*d)a/(b*d)a/(b*c) DDissipation factora0R_P Resistanceb\Omega0\omega FrequencycRad0C_P CapacitancedF0 1/(b*c*d) 1/(a*c*d) 1/(a*b*d) 1/(a*b*c) W_dDielectric power lossaW0VVoltagebV0\omega FrequencycRad0C CapacitancedF0DDissipation factore0 b^2*c*d*esqrt(a/(c*d*e)) a/(b^2*d*e) a/(b^2*c*e) a/(b^2*c*d) C CapacitanceaF/m0 \epsilon _0Permittivity of free spacebF/m 8.85*10^(-12) \epsilon _rRelative permittivityc0bOuter conductor radiusdm0aInner conductor radiusem0(2*pi*b*c)/ln(d/e) EDielectric stressaV/m0VCore potentialbV0rConductor radiuscm0aInner conductor radiusdm0bOuter conductor radiusem0b/(c*(ln(e/d))) a*c*ln(e/d)b/(a*(ln(e/d))) E_{max}Dielectric stress maxaV/m0VCore potentialbV0aInner conductor radiuscm0bOuter conductor radiusdm0b/(c*(ln(d/c))) a*c*ln(d/c) E_{min}Dielectric stress minaV/m0VCore potentialbV0aInner conductor radiuscm0bOuter conductor radiusdm0b/(d*(ln(d/c))) a*d*ln(d/c) CCapacitance per metreaF/m0 \epsilon _0Permittivity of free spaceeF/m 8.85*10^(-12) \epsilon _rRelative permittivityb0D2Distance between the centres of the two conductorscm0aRadius of each conductordm0pi*e*b/ln(c/d) W_f Energy storedaJ0C CapacitancebF0VCharge VoltagecV0 0.5*b*c^2 a/(0.5*c^2)sqrt(a/(0.5*b))  \omega _fEnergy stored per unit volumeaJ/m^30DElectric flux densitybC/m^20 \epsilon _0Permittivity of free spacecF/m 8.85*10^(-12) \epsilon _rRelative permittivityd0 b^2/(2*c*d) sqrt(2*a*c*d) LInductance per metreaH/m0\mu _rPermeability relativec0bOuter conductor radiusdm0aInner conductor radiusem0(c/(2*pi))*(0.25+ln(d/e)) LInductance per metreaH/m0\mu _0Permeability of free spacebH/m 4*pi*10^(-7)\mu _rPermeability relativec0D2Distance between the centers of the two conductorsdm0aRadius of each conductorem0(b*c/(pi))*(0.25+ln(d/e))  \omega _fMagnetic energyaJ/m^30B Flux densitybT0\mu _0Permeability of free spacecH/m 4*pi*10^(-7) b^2/(2*c) sqrt(2*a*c) W_fMagnetic energyaJ0L InductancebH0ICurrentcA0 (1/2)*b*c^22*a/c^2 sqrt(2*a/b)I The currentaA0VPotential differencebV0RThe resistancec\Omega0b/ca*cb/aV_{AB} Voltage OuputaV0R_2b\Omega0R_3c\Omega0R_4d\Omega0V_1original voltage eV0(b + c)/(b + c + d)*eR_{AB}Equivalent resistor a\Omega0R_1b\Omega0R_2c\Omega0R_3d\Omega0R_4e\Omega0b + e*(c +d)/(e + c +d)IEquivalent current aA0R_1b\Omega0R_2c\Omega0R_3d\Omega0R_4e\Omega0V_1fV0,(c+d)/(b +c+d)*f/(e + b*(c + d)/(b + c +d)) R_{AB}Equivalent resistor a\Omega0R_1b\Omega0R_2c\Omega0R_3d\Omega0R_4e\Omega0b + e*(c +d)/(e + c +d)R=|z|Maximum power tranfera\Omega0R"Real (Resistance) part of the loadb\Omega0r0Real (Resistance) part of the internal impedancec\Omega0 sqrt(b^2+c^2)RMaximum power tranfera\Omega0r0Real (Resistance) part of the internal impedanceb\Omega0R"Real (Resistance) part of the loada\Omega0r0Real (Resistance) part of the internal impedanceb\Omega0X"Image (Reactance) part of the loadc\Omega0x0Image (Reactance) part of the internal impedanced\Omega0RMaximum power tranfera\Omega0r0Real (Resistance) part of the internal impedanceb\Omega0X"Image (Reactance) part of the loadc\Omega0x0Image (Reactance) part of the internal impedanced\Omega0sqrt(b^2+(c+d)^2)VThe sum of voltageaV0V_iApplied voltagebV0 R_{total}The total circuit resistancea\Omega0R_iThe separate resistanceb\Omega0ICurrentaA0VThe sum of voltagebV0 R_{total}The total circuit resistancec\Omega0b/ca*cb/aIThe total circuit currentaA0I_i.The currents through the individual componentsbA0RThe total resistancea\Omega0R_i!The resistances of each componentb\Omega0RThe total resistancea\Omega0R_1The resistances of component 1b\Omega0R_2The resistances of component 2c\Omega0 b*c/(b+c)I_1$The currents through the component 1aA0IThe total circuit currentbA0R_1The resistance of component 1c\Omega0R_2The resistance of component 2d\Omega0 b*d/(c+d) a*(c+d)/dI_1$The currents through the component 1aA0IThe total circuit currentbA0R_1The resistance of component 1c\Omega0R_2The resistance of component 2d\Omega0 b*c/(c+d) a*(c+d)/c\tau Time Constantas0C CapacitancebF0R Resistancec\Omega0b*ca/ca/bv_CCapacitor voltageaV0VThe battery voltagebV0tTimescs0\tau Time constantds0b*(1-e^(-c/d))v_RResistor voltageaV0VThe battery voltagebV0tTimescs0\tau Time constantds0 b*(e^(-c/d))iCurrent flowingaA0ICurrent resistorbA0tTimescs0C CapacitordF0RResistorf\Omega0b*(e^(-c/(d*f)))v_CCapacitor voltageaV0VThe battery voltagebV0tTimescs0\tau Time constantds0 b*(e^(-c/d)) d*ln(b/a)iCurrentaA0ICurrent RegisterbA0tTimescs0\tau Time constantds0 b*(e^(-c/d))\tau Time Constantas0L InductancebH0R Resistancec\Omega0b/ca*cb/av_LInduced voltageaV0VBattery voltagebV0tTimescs0\tau Time constantds0 b*(e^(-c/d))v_RResistor voltageaV0VBattery voltagebV0tTimescs0\tau Time constantds0b*(1 - e^(-c/d))iCurrentaA0I Current flowsbA0tTimescs0\tau Time constantds0b*(1 - e^(-c/d))v_LInduced voltageaV0VBattery voltagebV0tTimescs0\tau Time constantds0 b*(e^(-c/d))iCurrentaA0I Current flowsbA0tTimescs0\tau Time constantds0 b*(e^(-c/d))EE.m.f. Generatedavolts0pnumber of pairs of polesb0narmature speedcrev/s0Znumber of armature conductorsd0cRnumber of parallel paths through the winding between positive and negative brushese0\Phi flux per polefWb (2*b*f*c*d)/eEGenerated e.m.f.aV0VTerminal voltagebV0I_a Load currentcA0R_aArmature resistanced\Omega0b+c*da - c*d(a-b)/d(a-b)/cEGenerated e.m.f.aV0VTerminal voltagebV0I_a Load currentcA0R_aArmature resistanced\Omega0b+c*da - c*d(a-b)/d(a-b)/c\eta Efficiencya0VTerminal voltagebV0IOutput currentcA0I_a#The current in the armature circuitdA0RTotal resistancee\Omega0I_f The current in the shunt circuitfA0C,Sum of the iron, friction and windage lossesgW0b*c/(b*c + d^2*e + f*b + g)*100VTerminal voltageaV0I_a#The current in the armature circuitbA0RTotal resistancec\Omega0I_f The current in the shunt circuitdA0C,Sum of the iron, friction and windage losseseW0 (b^2*c - e)/dsqrt((a*d + e)/c)(a*d + e)/(b^2) (b^2*c - e)/a b^2*c - a*dEInduced e.m.f.aV0VSupply voltagebV0I_aCurrentcA0R_aArmature resistanced\Omega0b-c*da + c*d(b-a)/d(b-a)/cTtorqueaNm0pnumber of pairs of polesb0Znumber of armature conductorsc0I_acurrentdA0cRnumber of parallel paths through the winding between positive and negative brushese0\Phi flux per polefWb0(b*f*c*d)/(pi*e)EGenerated e.m.f.aV0VSupply voltagebV0I_aArmature currentcA0R_aArmature resistanced\Omega0b-c*da + c*d(b-a)/d(b-a)/cEGenerated e.m.f.aV0VSupply voltagebV0ISupply currentcA0R_aArmature resistanced\Omega0R_fField winding resistancee\Omega0 b-c*(d+e) a + c*(d+e)\eta Efficiencya0VSupply voltagebV0ISupply currentcA0I_aArmature currentdA0R_aArmature resistancee\Omega0I_fField winding currentfA0C,Sum of the iron, friction and windage lossesgW0!(b*c - d^2*e - f*b - g)/(b*c)*100VTerminal voltageaV0I_a#The current in the armature circuitbA0R_aTotal resistancec\Omega0I_fField winding currentdA0C,Sum of the iron, friction and windage losseseW0 (b^2*c - e)/dsqrt((a*d + e)/c)(a*d + e)/(b^2) (b^2*c - e)/a b^2*c - a*dT Periodic timeas0f FrequencybHz01/b1/aK_f Form factora0RMSRoot Mean Squareb0Averagec0b/ca*cb/aX_Linductive reactancea\Omega0L InductancebH0f FrequencycHz02*pi*b*c a/(2*pi*c) a/(2*pi*b)X_Ccapacitive reactancea\Omega0C CapacitancebF0f FrequencycHz0 1/(2*pi*b*c) 1/(2*pi*a*c) 1/(2*pi*a*b)Z impedancea\Omega0X_Linductive reactanceb\Omega0R Resistancec\Omega0 sqrt(b^2+c^2)sqrt(a^2 - c^2)sqrt(a^2 - b^2)Z impedancea\Omega0X_Ccapacitive reactanceb\Omega0R Resistancec\Omega0 sqrt(b^2+c^2)sqrt(a^2 - c^2)sqrt(a^2 - b^2)Z impedancea\Omega0X_Linductive reactanceb\Omega0X_Ccapacitive reactancec\Omega0R Resistanced\Omega0sqrt(d^2+(b -c)^2)Z impedancea\Omega0X_Linductive reactanceb\Omega0X_Ccapacitive reactancec\Omega0R Resistanced\Omega0sqrt(d^2+(c - b)^2)f_rresonant frequencyaHz0L InductancebH0C CapacitancecF01/(2*pi*sqrt(b*c))1/(((2*pi)^2)*(a^2)*c)1/(((2*pi)^2)*(a^2)*b)Q-factor3This ratio is a measure of the quality of a circuita0L InductancebH0C CapacitancecF0R Resistanced\Omega0 1/d*sqrt(b/c) f_2 - f_1half-power points.aHz0f_rSeries resonancebHz0QQ-factorc0b/ca*cb/a PPoweraW0ICurrentbA0R Resistancec\Omega0(b^2)*c sqrt(a/c)a/(b^2) PPoweraW0ICurrentbA0VVoltagecV0\PhiAngled0 b*c*cos(d) SApparent poweraVA0ICurrentbA0VvoltagecV0b*ca/ca/b QReactive poweraVA0ICurrentbA0VvoltagecV0\Phid0 b*c*sin(d) f_RResonant frequencyaHz0L InductancebH0C CapacitancecF0R Resistanced\Omega0"1/(2*pi)*sqrt(1/(b*c) - d^2/(b^2))R_DDynamic resistancea\Omega0L InductancebH0C CapacitancecF0R Resistanced\Omega0b/(d*c)a*c*db/(a*d)b/(a*c)Q-factora0L InductancebH0f_RResonant frequencycHz0R Resistanced\Omega0 2*pi*c*b/d a*d/(2*pi*c) a*d/(2*pi*b) 2*pi*b*c/af_rresonant frequencyaHz0L InductancebH0C CapacitancecF01/(2*pi*sqrt(b*c))Q_r The Q factora0RResistorb\Omega0L Inductancec\H0C CapacitancedF0 1/b*sqrt(c/d) 1/a*sqrt(c/d) a^2*b^2*d c/(a^2*b^2)f FrequencyaHz0f_rResonant frequencybHz0Q The Q factorc0b*(sqrt(1 - 1/(2*(c^2))))V_C_m!Maximum p.d. across the capacitoraV0V Input voltagebV0QQ factorc0b*c/(sqrt(1 - 1/(4*c^2)))Q_TOverall Q-factora0Q_LThe Q-factor of the inductorb0Q_CThe Q-factor of the capacitorc0 (b*c)/(b+c) a*c/(c -a) a*b/(b-a)Q_rQ-factor f_rResonant frequency f_2The upper half-power frequency f_1The lower half-power frequency f_rResonant frequencyaHz0LPure inductancebH0CPure capacitancecF01/(2*pi*sqrt(b*c))1/(4*pi^2*a^2*c)1/(4*pi^2*a^2*b)f_rResonant frequencyaHz0LPure inductancebH0R Resistancec\Omega0CPure capacitancedF0sqrt(1/(b*d)-c^2/b^2)/(2*pi)R_DDynamic resistancea\Omega0LPure inductancebH0CPure capacitancecF0R Resistanced\Omega0b/(c*d)a*c*db/(a*d)b/(a*c)Q_rQ-factora0w_r2*pi*Resonant frequencybRad/s0LPure inductancecH0R Resistanced\Omega0(b*c)/d(a*d)/c(a*d)/b(b*c)/af_rForced resonant frequencyaHz0f_nNatural frequencybHz0Q3Q-factor of in a parallel network (refer to 3.30.4)c0b*sqrt(1-1/c^2)a/sqrt(1-1/c^2)f_rResonant frequencyaHz0LPure inductancebH0CPure capacitancecF0R_L L Resistanced\Omega0R_C C Resistancee\Omega0,1/(2*pi*sqrt(d*e))*sqrt((d^2-b/c)/(e^2-b/c))Q_TOverall Q-factora0Q_LQ-factor of coilb0Q_CQ-factor of capacitorc0 (b*c)/(b+c) -a*c/(a-c) -a*b/(a-b)V_L Line voltageaV0V_P Phase voltagebV0 sqrt(3)*b a/sqrt(3)I_L Line currentaA0I_P Phase currentbA0 sqrt(3)*b a/sqrt(3)PPoweraW0R_P Resistanceb\Omega0I_P Phase currentcA03*c^2*b a/(3*c^2) sqrt(a\(3*b))PPoweraW0V_L Line voltageb\Omega0I_L Line currentcA0\PhiAngled0 sqrt(3)*b*c*dSPoweraVA0V_L Line voltagebV0I_L Line currentcA0 sqrt(3)*b*c a/(sqrt(3)*c) a/(sqrt(3)*b)P Total PoweraW0P_mwattmeter readingbW03*ba/3P Total PoweraW0P_1Power 1bW0P_2Power 2cW0b+ca - ca - btan\PhiAnglea0P_1Power 1bW0P_2Power 2cW0sqrt(3)*(b-c)/(b+c)P Total PoweraW0P_1Power 1bW0P_2Power 2cW0P_3Power 3dW0b+c + d a - c - d a - b - d a - b - cn_ssynchronous speedarev/s0f4the frequency of the currents in the stator windingsbHz0ppairs of polesc0b/ca*cb/asSlipa0n_sSynchronous speedbrev/s0n_r Rotor speedcrev/s0 (b-c)/b*100E_2The rotor e.m.f. at standstillaV0E_1*The supply voltage per phase to the statorbV0N_1Stator windingsc0N_2Rotor windingsd0b*d/ca*c/dd*b/aa*c/bf_rFrequency of the rotoraHz0sSlipb0fSupply frequencycHz0b*ca/ca/bZ_rRotor impedance per phasea\Omega0R_2 Resistanceb\Omega0sSlipc0X_2!Reactance per phase at standstilld\Omega0sqrt(b^2 + (c*d)^2)I_2Starting currentaA0E_1*The supply voltage per phase to the statorbV0N_1Stator windingsc0N_2Rotor windingsd0R_2 Resistancee\Omega0X_2!Reactance per phase at standstillf\Omega0(d/c)*b/sqrt(e^2 + f^2)I_rRunning currentaA0E_1*The supply voltage per phase to the statorbV0N_1Stator windingsc0N_2Rotor windingsd0R_2 Resistancee\Omega0X_2!Reactance per phase at standstillf\Omega0sSlipg0(d/c)*b/sqrt(e^2 + (s*f)^2)P_2power input to the rotoraW0sSlipb0I_rcA0R_2 Resistanced\Omega0c^2*d/bc^2*d/a\etaEfficiency of induction motora0P_m Output powerbW0P_1 Input powercW0 (b/c)*100c*a/100 (b/a)*100 T The Torquea0E_1*The supply voltage per phase to the statorbV0N_1Stator windingsc0N_2Rotor windingsd0R_2 Resistancee\Omega0X_2!Reactance per phase at standstillf\Omega0sSlipg0mPhasesh0n_sSynchronous speedirev/s0.((d/c)^2*h/(2*pi*i))*(g*b^2*e/(e^2 + (g*f)^2))Z_1 Impedance 1a\Omega0Z_2 Impedance 2b\Omega0Z_3 Impedance 3c\Omega0Z_4 Impedance 4d\Omega0c*d/bc*d/aa*b/da*b/cR_xUnknown resistancea\Omega0R_2 Resistance 2b\Omega0R_3 Resistance 3c\Omega0R_4 Resistance 4d\Omega0b*d/cL_xUnknown inductanceaH0R_2 Resistance 2b\Omega0R_3 Resistance 3c\Omega0L_4 Inductance 4dH0b*d/cR_xUnknown resistancea\Omega0R_2 Resistance 2b\Omega0R_3 Resistance 3c\Omega0R_4 Resistance 4d\Omega0C_3 Capacitor 3e\Omega0\omegaAngular frequencyf\Omega0f^2*(e^2)*b*c*d/(1 + (f*e*b)^2)L_xUnknown inductancea\Omega0R_2 Resistance 2b\Omega0R_3 Resistance 3c\Omega0R_4 Resistance 4d\Omega0C_3 Capacitor 3e\Omega0\omegaAngular frequencyf\Omega0e*b*d/(1 + (f*e*b)^2)R_xUnknown resistancea\Omega0R_4 Resistance 4b\Omega0C_2 Capacitor 2cF0C_3 Capacitor 3dF0b*d/cL_xUnknown inductanceaH0R_4 Resistance 4b\Omega0R_2 Resistance 2c\Omega0C_3 Capacitor 3dF0b*d*cR_xUnknown resistanceaH0R_2 Resistance 2b\Omega0R_3 Resistance 3c\Omega0R_4 Resistance 4d\Omega0b*d/cL_xUnknown inductanceaH0R_4 Resistance 4b\Omega0R_2 Resistance 2c\Omega0C_3 Capacitor 3dF0b*d*c C_xUnknow capacitoraF0R_2 Resistance 2b\Omega0C_4 Capacitor 4cF0R_3 Resistance 3d\Omega0c*d/bc*d/aa*b/da*b/c R_xUnknown resistancea\Omega0R_4 Resistance 4b\Omega0C_2 Capacitor 2cF0C_3 Capacitor 3dF0b*d/c C_xUnknow capacitoraF0R_4 Resistance 4b\Omega0R_3 Resistance 3c\Omega0C_2 Capacitor 2dF0c*d/b \deltaAnglea0R_3 Resistance 3b\Omega0C_3 Capacitor 3cF0\omegaAngular frequencyd0 atan(d*b*c) C_2 Capacitor 2aF0C_3 Capacitor 3bF0R_2 Resistance 2c\Omega0R_3 Resistance 3d\Omega0R_4 Resistance 4e\Omega0R_1 Resistance 1f\Omega0 b/(e/f - d/c) a*(e/f - d/c) d/(e/f - b/a) c*(e/f - b/a) f*(b/a + d/c) e/(b/a + d/c)f FrequencyaHz0C_2 Capacitor 2bF0C_3 Capacitor 3cF0R_2 Resistance 2d\Omega0R_3 Resistance 3e\Omega01/(2*pi*sqrt(b*c*d*e)) Z_1 Impedance 1a\Omega0Z_A Impedance Ab\Omega0Z_B Impedance Bc\Omega0Z_C Impedance Cd\Omega0 (b*c)/(b+c+d)(a*c+a*d)/(c-a)(a*b+a*d)/(b-a)(b*c-a*b-c*a)/a Z_2 Impedance 2a\Omega0Z_A Impedance Ab\Omega0Z_B Impedance Bc\Omega0Z_C Impedance Cd\Omega0 (c*d)/(b+c+d)(c*d-a*c-a*d)/a(a*b+a*d)/(d-a)(a*b+a*c)/(c-a) Z_3 Impedance 3a\Omega0Z_A Impedance Ab\Omega0Z_B Impedance Bc\Omega0Z_C Impedance Cd\Omega0 (b*d)/(b+c+d)(a*c+a*d)/(d-a)(b*d-a*b-a*d)/a(a*b+a*c)/(b-a) Z_A Impedance Aa\Omega0Z_1 Impedance 1b\Omega0Z_2 Impedance 2c\Omega0Z_3 Impedance 3d\Omega0(b*c+c*d+d*b)/(c)(a*c-c*d)/(c+d) d*b/(a-b-d)(a*c-b*c)/(b+c) Z_B Impedance Ba\Omega0Z_1 Impedance 1b\Omega0Z_2 Impedance 2c\Omega0Z_3 Impedance 3d\Omega0(b*c+c*d+d*b)/(d)(a*d-c*d)/(c+d)(a*d-d*b)/(b+d) b*c/(a-c-b) Z_C Impedance Ca\Omega0Z_1 Impedance 1b\Omega0Z_2 Impedance 2c\Omega0Z_3 Impedance 3d\Okmega0(b*c+c*d+d*b)/(b) c*d/(a-c-d)(a*b-d*b)/(b+d)(a*b-b*c)/(b+c) P_{dB} Power ratioadB0P_1Power 1bW0P_2Power 2cW0 10*log(c/b) P_{dB} Power ratioadB0V_1 Voltage 1bV0V_2 Voltage 2cV0 20*log(c/b) P_d_B Power ratioadB0I_1 Current 1bA0I_2 Current 2cA0 20*log(c/b) A Power ratioadB0V_1 Voltage 1bV0V_2 Voltage 2cV0 20*log(b/c) Z_0Characteristic impedancea\Omega0Z_1 Impedance 1b\Omega0Z_2 Impedance 2c\Omega0sqrt(b^2+2*b*c) Z_1 Impedance 1a\Omega0Z_0Characteristic impedanceb\Omega0A AttenuationcdB0b*((10^(c/20)-1)/(10^(c/20)+1)) Z_2 Impedance 1a\Omega0Z_0Characteristic impedanceb\Omega0A AttenuationcdB0b*((2*10^(c/20))/(10^(c/10)-1)) Z_0Characteristic impedancea\Omega0Z_1 Impedance 1b\Omega0Z_2 Impedance 2c\Omega0sqrt((b*c^2)/(b+2*c)) Z_1 Impedance 1a\Omega0Z_0Characteristic impedanceb\Omega0A AttenuationcdB0b*((10^(c/10)-1)/(2*10^(c/20))) Z_2 Impedance 1a\Omega0Z_0Characteristic impedanceb\Omega0A AttenuationcdB0b*((10^(c/20)+1)/(10^(c/20)-1)) Z_1 Impedance 1a\Omega0Z_0Characteristic impedanceb\Omega0A AttenuationcdB0b*(10^(c/20)-1)A= \frac {V_1} {V_2} Z_2 Impedance 2a\Omega0Z_0Characteristic impedanceb\Omega0A AttenuationcdB0(b*10^(c/20))/(10^(c/20)-1) \beta Phase delayaRad/m0\omega FrequencybRad0LInductance per metrecH0CCapacitance per metredF0 b*sqrt(c*d) a/sqrt(c*d) a^2/(b^2*d) a^2/(b^2*c) \lambda Wavelengtham0\beta Phase delaybRad/m02*pi/b2*pi/a uVelocityam/s0\lambda Wavelengthbm0f FrequencycHz0b*ca/ca/b  E_1E.m.f. induced in the primaryaV0f frequencybHz0\Phi_m Maximum fluxcWb0N_1Primary windingsdV0 4.44*b*c*d a/(4.44*c*d) a/(4.44*b*d) a/(4.44*b*c) E_2E.m.f. induced in the secondaryaV0f frequencybHz0\Phi_m Maximum fluxcWb0N_1Primary windingsdV0 4.44*b*c*d a/(4.44*c*d) a/(4.44*b*d) a/(4.44*b*c) R_eTotal equivalent resistancea\Omega0R_1#Resistances of the primary windingsb\Omega0R_2%Resistances of the secondary windingsc\Omega0V_1Primary voltagedV0V_2Secondary voltageeV0b + c*((d/e)^2) X_eEquivalent reactancea\Omega0X_1"Reactances of the primary windingsb\Omega0X_2$Reactances of the secondary windingsc\Omega0V_1Primary voltagedV0V_2Secondary voltageeV0b + c*((d/e)^2)  Regulationa0E_2Noload secondary voltagebV0V_2Terminal voltage on loadcV0 ((b-c)/b)*100 \etaTransformer efficiencya0P_{lst}Total Power lossesbW0P_{in} Input powercW01 - b/c R_1The equivalent input resistancea\Omega0N_1Primary windingb0N_2 Secondaryc0R_Lload of resistanced\Omega0 (b/c)^2*d Z Impedance X_LInductive reactance  Z Impedance X_CCapacitive reactance  Z Impedance X_LInductive reactance R Resistance  Z Impedance X_CCapacitive reactance R Resistance  Z Impedance X_CCapacitive reactance X_LInductive reactance R Resistance  |Z|Modulusa\Omega0R Resistanceb\Omega0X_LInductive reactancec\Omega0X_CCapacitive reactanced\Omega0sqrt(b^2 + (c - d)^2) \PhiAngulara0R Resistanceb\Omega0X_LInductive reactancec\Omega0X_CCapacitive reactanced\Omega0atan((c -d)/b)Y AdmittanceS0G ConductanceS0B SusceptanceS0Z Impedance Y Admittance I_1 Current 1 I_2 Current 2 I Total Current Z_1 Impedance 1 Z_2 Impedance 2 a,Electric current and quantity of electricity \text Q = It Resistance\text R = \frac{V}{I}!Internal resistance of a battery.\text r = \frac{E-V}{I}Electric engineering_3_3_1a.PNGElectric engineering_3_3_1b.PNGResistance and resistivity\text R = \frac{\rho l}{a},Temperature coefficient of resistance at 0C)\text R_t = R_0\left(1+\alpha_0 t \right)-Temperature coefficient of resistance at 20C/\text R_t = R_{20}\left(1+\alpha_{20} t \right)%Temperature coefficient of resistance;\text \frac{R_1}{R_2} = \frac{1+\alpha_0t_1}{1+\alpha_0t_2} Ohm's lawThe current passing through a conductor is directly proportional to the potential difference across the two terminal points, and inversely proportional to the resistance of the conductor.\text I = \frac{V}{R}5Calculation based on potential difference and current \text P = VI+Calculation based on resistance and current\text P = I^2R8Calculation based on potential difference and resistance\text P = V^2/RElectrical energy \text W = PtPotential gradient\text E = \frac{V}{d}Electric engineering_3_5_2.PNG Capacitance\text C = \frac{Q}{V}Electric engineering_3_5_2.PNG The charge stored in a capacitor \text Q = ItElectric engineering_3_5_4.PNGElectric flux density\text D = \frac{Q}{A}Electric engineering_3_5_5.PNGVacuum permittivity\text \epsilon_0 = \frac{D}{E}The relative permittivity(\text \frac{D}{E} = \epsilon_0\epsilon_rThe absolute permittivity%\text \epsilon = \epsilon_0\epsilon_r Capacitance*\text C = \frac{\epsilon_0\epsilon_r A}{d} Electric engineering_3_5_7_1.PNG)Capacitance (N parallel-plates capacitor)/\text C = \frac{\epsilon_0\epsilon_r A(N-1)}{d} Electric engineering_3_5_7_2.PNG Capacitors connected in parallel\text C = \sum_{i=1}^n C_iElectric engineering_3_5_8.PNGCapacitors connected in series.\text \frac{1}{C} = \sum_{i=1}^n \frac{1}{C_i}Electric engineering_3_5_9.PNG Dielectric strength\text E_m = \frac{V_m}{d}3Electric engineering_Electric engineering_3_5_2.PNG Energy stored in capacitors\text W = \frac{CV^2}{2}Magnetic flux density\text B = \frac{\Phi}{A}Magnetomotive force\text F_m = NIMagnetic field strength\text H = \frac{NI}{l}Permeability of free space\text \mu_0 = \frac{B}{H}The absolute permeability\text \mu = \mu_0\mu_r Reluctance \text S = \frac{l}{\mu_0\mu_r A}%Magnetic circuits connected in series\text S = \sum_{i=1}^n S_i%Force on a current-carrying conductor\text F = B\,i\,l sin\thetaElectric engineering_3_8_1.PNGForce on a charge \text F = QvB The electromotive force \text E = BlvElectric engineering_3_9_1.PNG 3Calculation based on the change in flux and current\text L = \frac{\Phi}{i} 5Calculation based on the e.m.f and the change current\text E = -L\frac{dI}{dt} Inductance of a coil\text L = \frac{N\Phi}{I}Electric engineering_3_9_3.PNG  Energy stored\text W = \frac{LI^2}{2}Electric engineering_3_9_3.PNG Open-loop voltage gainV_0 = G(V_2 - V_1)4Electric engineering_Electric engineering_3_19_1.png Common-mode gainkThe common mode gain, A_s, is the ratio of the output voltage, V_0, to the common mode input signal, V_{cm}A_s = \frac{V_0}{V_{cm}} Common-mode rejection ratioThe common-mode rejection ratio (CMRR) of a differential amplifier (or other device) measures the tendency of the device to reject input signals common to both input leads&CMRR = 20log_{10}{(\frac{A_d}{|A_s|})} Inverting amplifier$V_{out} = -V_{in}\frac{R_f}{R_{in}}Electric engineering_3_19_3.png Non-inverting amplifier$V_{out}= V_{in}(1 + \frac{R_2}{R_1})Electric engineering_3_19_4.png Summing amplifier\V_{out} = -R_f{(\frac{V_1}{R_1} + \frac{V_2}{R_2} + \frac{V_3}{R_3} + .. + \frac{V_n}{R_n})}Electric engineering_3_19_5.png  Integrator8V_{out}= \int_0^t - {\frac{V_{in}} {RC}dt} + V_{initial}Electric engineering_3_19_6.png Differential amplifieraV_{out} = V_2\left(\frac{(R_f + R_1)R_3}{(R_3 + R_2)R_1}\right) - V_1\left(\frac{R_f}{R_1}\right)Electric engineering_3_19_7.png Coulomb's law.The magnitude of the electrostatic force between two point charges is directly proportional to the product of the magnitudes of each charge and inversely proportional to the square of the distance between the charges.\text F = k\frac{q_1q_2}{d^2}Electric engineering_3_5_1a.PNGElectric engineering_3_5_1b.PNGElectric engineering_3_5_1c.PNGElectric engineering_3_5_1d.PNG Dissipation factor(Series)[In physics, the dissipation factor is a measure of loss-rate of power of a mechanical mode.D = tan\delta = R_S \omega C_S!Electric engineering_3_39_1_a.png!Electric engineering_3_39_1_b.png/loss \; angle, \delta = \left(90^o-\phi \right)D= \frac {1} {Q} = tan\delta Dissipation factor(Parallel)[In physics, the dissipation factor is a measure of loss-rate of power of a mechanical mode.*D = tan\delta = \frac {1} {R_P \omega C_P}!Electric engineering_3_39_2_a.png!Electric engineering_3_39_2_b.pngcos\phi \approx tan\delta Dielectric power lossW_d = V^2\omega CD (Capacitance between concentric cylinders:C = \frac {2\pi \epsilon _0 \epsilon _r} {ln\frac {b} {a}}Electric engineering_3_40_1.png Dielectric stress E = \frac {V} {rln\frac {b} {a}} Dielectric stress max&E_{max} = \frac {V} {aln\frac {b} {a}} Dielectric stress min&E_{min} = \frac {V} {bln\frac {b} {a}} $Capacitance of an isolated twin line9C = \frac {\pi \epsilon _0 \epsilon _r} {ln\frac {D} {a}} 2Energy stored in the electric field of a capacitorW_f = \frac {1} {2} CV^2 +Energy stored per unit volume of dielectricq\omega _f = \frac {D^2} {2\epsilon _0 \epsilon _r} = \frac {1} {2} DE = \frac {1} {2} \epsilon _0 \epsilon _r E^2 6Inductance of a concentric cylinder (or coaxial cable)HL = \frac {\mu _r} {2\pi} \left(\frac {1} {4} + ln \frac {b} {a} \right) #Inductance of an isolated twin lineNL = \frac {\mu _0 \mu _r} {\pi} \left(\frac {1} {4} + ln \frac {D} {a} \right) 'Magnetic energy in a nonmagnetic mediumO\omega _f = \frac {B^2} {2 \mu _0}= \frac {1} {2} HB = \frac {1} {2} \mu _0 H^2 %Magnetic energy stored in an inductorW_f = \frac {1} {2} LI^2 Current lawAt any point in an electrical circuit where charge density is not changing in time, the sum of currents flowing towards that point is equal to the sum of currents flowing away from that point\sum I=0!Electric engineering_3_14_1_1.png Voltage lawVThe directed sum of the electrical potential differences around a circuit must be zero&\oint_C \mathbf{E} \cdot d\mathbf{I}=0!Electric engineering_3_14_1_2.png Ohm's lawThe current passing through a conductor is directly proportional to the potential difference across the two terminal points, and inversely proportional to the resistance of the conductor.\text I = \frac{V}{R} Equivalent single voltage sourceAny combination of voltage sources, current sources and resistors with two terminals is electrically equivalent to a single voltage source V and a single series resistor R. For single frequency AC systems the theorem can also be applied to general impedances, not just resistors-V_{AB}=\frac{R_2 + R_3}{(R_2 + R_3) + R_4}V_1Electric engineering_3_14_2.png!Equivalent single series resistorAny combination of voltage sources, current sources and resistors with two terminals is electrically equivalent to a single voltage source V and a single series resistor R. For single frequency AC systems the theorem can also be applied to general impedances, not just resistors5R_{AB} = R_1 + \frac{R_4(R_2 + R_3)}{R_2 + R_3 + R_4}Electric engineering_3_14_2.pngEquivalent current source Any collection of voltage sources and resistors with two terminals is electrically equivalent to an ideal current source, I, in parallel with a single resistor, R. For single-frequency AC systems the theorem can also be applied to general impedances, not just resistors]I = \frac{R_3 + R_2}{R_1 + R_2 + R_3}\frac{V_1}{R_4 + \frac{R_1(R_2 + R_3)}{R_1 + R_2 + R_3}}Electric engineering_3_14_3.pngEquivalent single resistor Any collection of voltage sources and resistors with two terminals is electrically equivalent to an ideal current source, I, in parallel with a single resistor, R. For single-frequency AC systems the theorem can also be applied to general impedances, not just resistors5R_{AB} = R_1 + \frac{R_4(R_2 + R_3)}{R_2 + R_3 + R_4}Electric engineering_3_14_3.pngYMaximum power theorems when the load is purely resistive (i.e. x equals 0) and adjustableR=|z|=\sqrt{r^2+R^2}Electric engineering_3_36_1.pngWMaximum power theorems when both the load and the source impedance are purely resistiveR=rElectric engineering_3_36_1.png*Maximum power theorems with adjustable R,XR=r \; and \; X=-xElectric engineering_3_36_1.png4Maximum power theorems with fixed X and adjustable RR=\sqrt{r^2+(x+X)^2}Electric engineering_3_36_1.pngVoltage of series circuits\text V = \sum_{i=1}^n V_iElectric engineering_3_4_1.PNGResistance of series circuits"\text R_{total} = \sum_{i=1}^n R_iElectric engineering_3_4_1.PNGOhm's law for series circuits\text I = \frac{V}{R_{total}}Electric engineering_3_4_1.PNG The current of parallel circuits\text I = \sum_{i=1}^n I_i Electric engineering_3_4_2_1.PNG#The resistance of parallel circuits.\text \frac{1}{R} = \sum_{i=1}^n \frac{1}{R_i} Electric engineering_3_4_2_1.PNGTwo resistors in parallel. \text R = \frac{R_1R_2}{R_1+R_2} Electric engineering_3_4_2_1.PNGCurrent division \text I_1 = I\frac{R_2}{R_1+R_2} Electric engineering_3_4_3_1.PNGCurrent division \text I_2 = I\frac{R_1}{R_1+R_2} Electric engineering_3_4_3_1.PNGRC time constantEIt is the time required to charge the capacitor, through the resistor \tau = CRCapacitor voltage?The voltage across the capacitor tends towards V as time passes:v_C = V(1 - e^{\frac{-t}{CR}}) = V(1-e^{\frac{-t}{\tau}})!Electric engineering_3_18_2_1.pngResistor voltage/The voltage across the resistor tends towards 0/v_R = Ve^{\frac{-t}{CR}} = Ve^{\frac{-t}{\tau}}!Electric engineering_3_18_2_2.pngDecay of current flowing,i = Ie^{\frac{-t}{CR}}= Ie^{\frac{-t}{\tau}}!Electric engineering_3_18_2_3.pngDecay of voltage4v_C = v_R = Ve^{\frac{-t}{CR}}= Ve^{\frac{-t}{\tau}}#Electric engineering_3_18_3_1_a.png#Electric engineering_3_18_3_1_b.pngDecay of current,i = Ie^{\frac{-t}{CR}}= Ie^{\frac{-t}{\tau}}!Electric engineering_3_18_3_2.pngRL time constant\tau = \frac{L}{R}Decay of induced voltage>The voltage across the inductor tends towards 0 as time passes/v_L = Ve^{\frac{-Rt}{L}} = Ve^{\frac{-t}{\tau}}!Electric engineering_3_18_5_1.pngGrowth of resistor voltage/The voltage across the resistor tends towards V;v_R = V(1 - e^{\frac{-Rt}{L}}) = V(1 - e^{\frac{-t}{\tau}})!Electric engineering_3_18_5_2.pngGrowth of current flow9i = I(1 - e^{\frac{-Rt}{L}}) = I(1 - e^{\frac{-t}{\tau}})!Electric engineering_3_18_5_3.pngDecay of voltages1v_L=v_R=Ve^{\frac{-Rt}{L}} = Ve^{\frac{-t}{\tau}}Electric engineering_3_18_6.pngDecay of current.i = Ie^{\frac{-Rt}{L}} = I e^{\frac{-t}{\tau}}Electric engineering_3_18_6.png'E.m.f. generated in an armature winding.generated \, e.m.f. \, E = \frac{2p\Phi nZ}{c}Separately-excited generatorE = V + I_aR_a!Electric engineering_3_22_2_1.pngShunt wound generatorE = V + I_aR_a!Electric engineering_3_22_2_2.pngThe efficiencyL\eta= \frac{output}{input}= {(\frac{VI}{VI + I_a^2R + I_fV + C})}\times 100%Electric engineering_3_22_3.png"The efficiency maximum (condition)I_a^2R = I_fV + CD.c. motors(Back e.m.f.) E = V- I_aR_aTorque of a d.c. machine2T = \frac{EI_a}{2\pi n} = \frac{p\Phi ZI_a}{\pi c}Shunt wound motorE = V - I_aR_a!Electric engineering_3_22_6_1.pngSeries-wound motorE = V - I(R_a + R_f)!Electric engineering_3_22_6_2.pngThe efficiency8\eta={(\frac{VI - I_a^2R_a - I_fV - C}{VI })}\times 100%!The efficiency maximum(condition)I_a^2R_a = I_fV + CPeriodic time of the waveformMAn interval of time between the recurrence of phases in a vibration, waveform T=\frac{1}{f}Electric engineering_3_15_1.pngRoot mean squareRoot mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g. waves.5x_{rms} = \sqrt{\frac{x_1^2 + x_2^2 + .. + x_n^n}{n}} Form factorThe form factor of an alternating current waveform (signal) is the ratio of the RMS (Root Mean Square) value to the average value (mathematical mean of absolute values of all points on the waveform)K_f = \frac{RMS}{Average}Inductive reactance8Proportional to the signal frequency and the inductanceX_L=\frac{V_L}{I_L}= 2\pi fL!Electric engineering_3_16_1_a.png!Electric engineering_3_16_1_b.png!Electric engineering_3_16_1_c.pngCapacitive reactanceCInversely proportional to the signal frequency and the capacitance&X_C=\frac{V_C}{I_C}= \frac{1}{2\pi fC}!Electric engineering_3_16_2_a.png!Electric engineering_3_16_2_b.png!Electric engineering_3_16_2_c.pngRL series circuitZ=\sqrt{R^2 + X_L^2}`Electric engineering_ tan\Phi = \frac{X_L}{R} \\ sin\Phi = \frac{X_L}{Z}\\ cos\Phi = \frac{R}{Z}!Electric engineering_3_16_3_a.png!Electric engineering_3_16_3_b.png!Electric engineering_3_16_3_c.pngRC series circuitZ=\sqrt{R^2 + X_C^2}eElectric engineering_tan\alpha = \frac{X_C}{R} \\ sin\alpha = \frac{X_C}{Z}\\ cos\alpha = \frac{R}{Z}!Electric engineering_3_16_4_a.png!Electric engineering_3_16_4_b.png!Electric engineering_3_16_4_c.png%V_L and V_C are anti-phase. X_L > X_CZ=\sqrt{R^2 + (X_L-X_C)^2}4Electric engineering_tan(\Phi) = \frac{X_L - X_C}{R}#Electric engineering_3_16_5_1_a.png#Electric engineering_3_16_5_1_b.png#Electric engineering_3_16_5_1_c.png%V_L and V_C are anti-phase. X_L < X_CZ=\sqrt{R^2 + (X_C-X_L)^2}6Electric engineering_tan(\alpha) = \frac{X_C - X_L}{R}#Electric engineering_3_16_5_2_a.png#Electric engineering_3_16_5_2_b.pngSeries resonanceXR-L-C series circuit, when XL = XC, the applied voltage V and the current I are in phasef_r = \frac{1}{2\pi\sqrt{LC}}Electric engineering_3_16_6.pngQ-factorRepresents the effect of electrical resistance and, for electromechanical resonators such as quartz crystals, mechanical friction.(Q-factor = \frac{1}{R}\sqrt{\frac{L}{C}}IElectric engineering_Q-factor = \frac{2\Pi f_r}{R} = \frac{1}{2\Pi f_rCR} BandwidthThe difference between the upper and lower cutoff frequencies of for example a filter, a communication channel or a signal spectrum(f_2 - f_1) = \frac{f_r}{Q}Electric engineering_3_16_8.png Based on resistance and currentP = I^2RElectric engineering_ Based on the value of angle P = VIcos\Phi"Electric engineering_P = VIcos\Phi Apparent powerS = VI Electric engineering_3_16_10.png Reactive power Q = VIsin\Phi Electric engineering_3_16_10.png  Power factorp.f. = \frac{P}{S} = cos\PhiElectric engineering_Resonant frequency9f_R = \frac{1}{2\pi}\sqrt{\frac{1}{LC} - \frac{R^2}{L^2}}!Electric engineering_3_17_1_1.pngDynamic resistanceZThe ratio of the change in voltage to the change in current is known as dynamic resistanceR_D = \frac{L}{RC}Q-factorThe Q-factor of a parallel resonant circuit is the ratio of the current circulating in the parallel branches of the circuit to the supply currentQ-factor = \frac{2\pi f_RL}{R}Series Resonancef_r = \frac{1}{2\pi\sqrt{LC}}!Electric engineering_3_28_1_a.png!Electric engineering_3_28_1_b.pngQ factorThe Q factor or quality factor compares the time constant for decay of an oscillating physical system's amplitude to its oscillation period#Q_r = \frac{1}{R}\sqrt{\frac{L}{C}}(Electric engineering_\frac{\omega_rL}{R} \frac{1}{\omega_rCR}\frac{V_C(or V_L)}{V}Resonant frequency f = f_r\sqrt{1 - \frac{1}{2Q^2}} Maximum p.d.AV_C_m = V_L_m = \frac{QV}{\sqrt{1 - \left(\frac{1}{2Q}\right)^2}}&Overall Q-factor for series componentsQ_T = \frac{Q_LQ_C}{Q_L + Q_C} Bandwidth8Q_r = \frac{f_r}{f_2 - f_1} \,and \, f_r = \sqrt{f_1f_2} frequency w2I\frac{I}{I_r} = \frac{1}{1 + j\delta Q} \\ \frac{Z}{Z_r} = 1 + j2\delta Q frequency w1I\frac{I}{I_r} = \frac{1}{1 + j\delta Q} \\ \frac{Z}{Z_r} = 1 - j2\delta QParallel resonance f_r=\frac{1}{2\pi\sqrt{LC}} (Hz)Electric engineering_3_30_1.pngThe LRC parallel network5f_r=\frac{1}{2\pi}\sqrt{\frac{1}{LC}-\frac{R^2}{L^2}}Electric engineering_3_30_2.pngDynamic resistanceR_D=\frac{L}{CR}Q-factor in a parallel networkQ_r=\frac{w_rL}{R}Electric engineering_3_30_4.png%Natural and forced resonant frequencyf_r=f_n\sqrt{1-\frac{1}{Q^2}}The LR-CR parallel networkMf_r=\frac{1}{2\pi\sqrt{LC}}\sqrt{\frac{R_L^2-\frac{L}{C}}{R_C^2-\frac{L}{C}}}!Electric engineering_3_30_6_a.pngQ-factor of parallel componentsQ_T=\frac{Q_LQ_C}{Q_L+Q_C} Line currents I_L = I_PElectric engineering_3_20_1.png Line voltagesV_L = \sqrt{3}V_PElectric engineering_3_20_1.png Line voltages V_L = V_PElectric engineering_3_20_2.png Line currentsI_L = \sqrt{3}I_PElectric engineering_3_20_2.pngCPower in three-phase systems(Based on resistance and phase current) P = 3I_P^2R_PDPower in three-phase systems(Based on line voltage and line current)P = \sqrt{3}V_LI_Lcos\PhiTotal volt-amperesS = \sqrt{3}V_LI_L(One-wattmeter method for a balanced loadP = 3\times P_m#Electric engineering_3_20_4_1_a.png#Electric engineering_3_20_4_1_b.pngBTwo-wattmeter method for balanced or unbalanced loads(Total power) P = P1 + P2!Electric engineering_3_20_4_2.pngGTwo-wattmeter method for balanced or unbalanced loads(The power factor)1tan\Phi = \sqrt{3}{(\frac{P_1 - P_2}{P_1 + P_2})}!Electric engineering_3_20_4_2.pngYThree-wattmeter method for a three-phase, 4-wire system for balanced and unbalanced loadsP = P_1 + P_2 + P_3!Electric engineering_3_20_4_3.pngSynchronous speedn_s = \frac{f}{p}Slip(s = {(\frac{n_s - n_r}{n_s})}\times 100% Rotor e.m.f.E_2 = {(\frac{N_2}{N_1})}E_1!Electric engineering_3_23_3_1.pngRotor frequencyf_r = sfRotor impedanceZ_r = \sqrt{R_2^2 + (sX_2)^2}Electric engineering_3_23_4.png Rotor current - Starting current8I_2 ={\frac{(\frac{N_2}{N_1})E_1}{\sqrt{R_2^2 + X_2^2}}}!Electric engineering_3_23_3_1.pngRotor current - Running current=I_r = {\frac{s(\frac{N_2}{N_1})E_1}{\sqrt{R_2^2 + (sX_2)^2}}}Electric engineering_3_23_4.pngRotor copper lossIs = \frac{rotor \, copper \, loss}{rotor \, input} = \frac{I_r^2R_2}{P_2}Efficiency of induction motorJ\eta = \frac{output \, power}{input \, power} = \frac{P_m}{P_1}\times 100%&Torque equation for an induction motortT = \left({\frac{m{\left(\frac{N_2}{N_1}\right)^2}}{2\pi n_s}}\right)\left(\frac{sE_1^2R_2}{R_2^2 + (sX_2)^2}\right)%Balance conditions for an a.c. bridgeZ_1Z_3 = Z_2Z_4Electric engineering_3_27_1.pngThe simple Maxwell bridge(Rx)R_x=\frac{R_2R_4}{R_3}!Electric engineering_3_27_2_1.pngThe simple Maxwell bridge(Lx)L_x = \frac{R_2L_4}{R_3}!Electric engineering_3_27_2_1.pngThe Hay bridge(R_x);R_x=\frac{\omega^2C_3^2R_2R_3R_4}{(1 + \omega^2C_3^2R_3^2)}!Electric engineering_3_27_2_2.pngThe Hay bridge(L_x)0L_x = \frac{C_3R_2R_4}{(1 + \omega^2C_3^2R_3^2)}!Electric engineering_3_27_2_2.pngThe Owen bridge(Rx)R_x=\frac{R_4C_3}{C_2}!Electric engineering_3_27_2_3.pngThe Owen bridge(Lx)L_x =R_2R_4C_3!Electric engineering_3_27_2_3.pngThe Maxwell-Wien bridge(Rx)R_x=\frac{R_2R_4}{R_3}!Electric engineering_3_27_2_4.pngThe Maxwell-Wien bridge(Lx)L_x =C_3R_2R_4!Electric engineering_3_27_2_4.png The de Sauty bridgeC_x=\frac{R_3C_4}{R_2}!Electric engineering_3_27_2_5.png The Schering bridge(Rx)R_x=\frac{C_3R_4}{C_2}!Electric engineering_3_27_2_6.png The Schering bridge(Cx)C_x=\frac{C_2R_3}{R_4}!Electric engineering_3_27_2_6.png The Schering bridge(Loss angle) \delta = tan^{-1}(\omega R_3C_3)!Electric engineering_3_27_2_6.png #The Wien bridge(The balance bridge)3\frac{R_3}{R_2} + \frac{C_3}{C_2} = \frac{R_4}{R_1}!Electric engineering_3_27_2_7.pngThe Wien bridge(Frequency)#f=\frac{1}{2\pi\sqrt{C_2C_3R_2R_3}}!Electric engineering_3_27_2_7.png Delta-Star to Star-DeltaZ_1=\frac{Z_AZ_B}{Z_A+Z_B+Z_C}!Electric engineering_3_35_1_a.png!Electric engineering_3_35_1_b.png Delta-Star to Star-DeltaZ_2=\frac{Z_BZ_C}{Z_A+Z_B+Z_C}!Electric engineering_3_35_1_a.png!Electric engineering_3_35_1_b.png Delta-Star to Star-DeltaZ_3=\frac{Z_AZ_C}{Z_A+Z_B+Z_C}!Electric engineering_3_35_1_a.png!Electric engineering_3_35_1_b.png Star-Delta to Delta-Star$Z_A=\frac{Z_1Z_2+Z_2Z_3+Z_3Z_1}{Z_2}!Electric engineering_3_35_1_a.png!Electric engineering_3_35_1_b.png Star-Delta to Delta-Star$Z_B=\frac{Z_1Z_2+Z_2Z_3+Z_3Z_1}{Z_3}!Electric engineering_3_35_1_a.png!Electric engineering_3_35_1_b.png Star-Delta to Delta-Star$Z_C=\frac{Z_1Z_2+Z_2Z_3+Z_3Z_1}{Z_1}!Electric engineering_3_35_1_a.png!Electric engineering_3_35_1_b.png Power ratio(Power)P_{dB} = 10lg\frac {P_2} {P_1} Power ratio(Voltage)P_{dB} = 20lg\frac {V_2} {V_1} Power ratio(Current)P_{dB} = 20lg\frac {I_2} {I_1}  AttenuationA = 20lg\frac {V_1} {V_2} Characteristic impedanceZ_0 = \sqrt {Z_1^2+2Z_1Z_2}Electric engineering_3_41_2.png  Impedance 1SZ_1 = Z_0\left( \frac {10^{\frac {A} {20}} - 1} {10^{\frac {A} {20}} + 1} \right)Electric engineering_3_41_2.png  Impedance 2LZ_2 = Z_0\left(\frac {2*10^{\frac {A} {20}}} {10^{\frac {A} {10}}-1} \right)Electric engineering_3_41_2.png Characteristic impedance6Z_0 = \sqrt {\left(\frac {Z_1Z_2^2} {Z_1+2Z_2}\right)}Electric engineering_3_41_3.png  Impedance 1QZ_1 = Z_0\left( \frac {10^{\frac {A} {10}} - 1} {2*10^{\frac {A} {20}}} \right)Electric engineering_3_41_3.png  Impedance 2NZ_2 = Z_0\left(\frac {10^{\frac {A} {20}} + 1} {10^{\frac {A} {20}}-1} \right)Electric engineering_3_41_3.png  Impedance 1/Z_1 = Z_0 \left ( 10^{\frac {A} {20}-1} \right)Electric engineering_41_4.png  Impedance 2<Z_2= \frac {Z_0*10^{\frac {A} {20}}} {10^{\frac {A} {20}}-1}Electric engineering_41_4.png  Phase delay\beta = \omega \sqrt {LC}  Wavelength\lambda = \frac {2\pi} {\beta} Velocity u = f\lambda "Transformer principle of operation3\frac{V_1}{V_2} = \frac{N_1}{N_2} = \frac{I_2}{I_1}Electric engineering_3_21_1.png 1The r.m.s. value of e.m.f. induced in the primaryE_1 = 4.44f\Phi_mN_1 3the r.m.s. value of e.m.f. induced in the secondaryE_2 = 4.44f\Phi_mN_2 2total equivalent resistance in the primary circuit-R_e = R_1 + R_2\left(\frac{V_1}{V_2}\right)^2!Electric engineering_3_21_3_1.png /the equivalent reactance in the primary circuit-X_e = X_1 + X_2\left(\frac{V_1}{V_2}\right)^2!Electric engineering_3_21_3_1.png Regulation of a transformer8Regulation=\left(\frac{E_2 - V_2}{E_2}\right)\times 100%Electric engineering_ Transformer efficiency!\eta = 1 - \frac{P_{lst}}{P_{in}}Electric engineering_ Resistance matching'R_1 = \left(\frac{N_1}{N_2}\right)^2R_LElectric engineering_3_21_6.png Purely inductive a.c. circuitZ=jX_LElectric engineering_3_24_1.png Purely capacitive a.c. circuitZ=-jX_CElectric engineering_3_24_2.png RL series circuit Z=R + jX_LElectric engineering_3_24_3.png RC series circuit Z=R - jX_CElectric engineering_3_24_4.png  ImpedanceZ=R - j(X_L - X_C)Electric engineering_3_24_5.png Modulus |Z| = \sqrt{R^2 + (X_L - X_C)^2}Electric engineering_3_24_5.png Argular&\Phi = tan^{-1}({\frac{X_L - X_C}{R}})Electric engineering_3_24_5.png'Admittance, conductance and susceptance'Y = \frac{I}{V} = \frac{1}{Z}= G \pm jBElectric engineering_Parallel a.c. networks\\frac{1}{Z_T} = \frac{1}{Z_1} + \frac{1}{Z_2} + ..+ \frac{1}{Z_n}\\ Y_T = Y_1 + Y_2 + ..+Y_nElectric engineering_3_25_2.png!Current division in a.c. circuitsGI_1=I{(\frac{Z_2}{Z_1 + Z_2})} \, and \, I_2=I{(\frac{Z_1}{Z_1 + Z_2})}Electric engineering_3_25_3.pngPNG  IHDR 0PLTE)))111999JJJRRRcccsssHr pHYsgRtIME 2)tEXtAuthorH tEXtDescription !# tEXtCopyright:tEXtCreation time5 tEXtSoftware]p: tEXtDisclaimertEXtWarningtEXtSourcetEXtComment̖tEXtTitle'%IDAThݚklE-V{@+-6Ac5>@#!5A-?QZD$޵Ԩ5A4-/D$4";ugn3{?9F>7K#89a`=wGƦw} 1|H#c^Z?Fg/gW#AaFV_ቱ"!?B(c{ Ji!!V a B0Ʌ|p1Fc]q2b0rf 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" $r%&4')*]T+ ,4,nu-,../r.011u2Db3{6455h67EK8B9h:::*~;@<*X=3>N?<@'A4,BrB@D:'EE0GlWH`IrJJYtKLOLM N!O!P5"FQ"Q3#R #@S "$S $T %U %V  &bWh&MX&XR'Y'ZQ(x[(w\;)] )^ '*J` *Ga >+b +d ,e [--g.h[.)i.i/@j/)k/kZ0l08m0m81nc11o1o1p 025q{2q2r 3s{3t3u4vh4w4x5y5z5{5|K6}6]~6 f770888`)9Ll9B9:C:_::q;;Ȋ;8<z<Ս=z=|=[m>V>!,???}@"@=AAB\BBћ C C CpSDD1DŠE|]E'E¢EZ*F& jF.1 =Frames of reference= 2 .1.1 =Galilean relativity= 2 .1.1.1 =Position= 1 .1.1.2 =Time= 0 .1.1.3 =Velocity= 1 .1.1.zQJJ4 =Momentum= 1 .1.1.5 =Angular momentum= 1 .1.1.6 =Kinetic energy.= 1 .1.2 =Lorentz transformations= 2 .1.2.1 =Lorentz Factor= 1 .1.2.2 =Position x= 1 .1.2.3 =Position x'= 1 .1.2.4 =Time t= 1 .1.2.5 =Time t'= 1 .1.3 =Velocity transformations= 2 .1.3.1 =Velocity x-axis in frame O= 1 .1.3.2 =Velocity x'-axis in frame O'= 1 .1.3.3 =Velocity y-axis in frame O= 1 .1.3.4 =Velocity y'-axis in frame O'= 1 .1.3.5 =Velocity z-axis in frame O= 1 .1.3.6 =Velocity z'-axis in frame O'= 1 .1.4 =Momentum and energy transformations= 2 .1.4.1 =Momentum x= 1 .1.4.2 =Momentum x'= 1 .1.4.3 =Energy x= 1 .1.4.4 =Energy x'= 1 .1.4.5 =Energy equality.= 0 .1.5 =Propagation of light= 2 .1.5.1 =Doppler= 1 .1.5.2 =Aberration x= 1 .1.5.3 =Aberration x'= 1 .1.5.4 =Relativistic beaming= 1 .1.6 =Rotating frames= 2 .1.6.1 =Vector transformations= 0 .1.6.2 =Acceleration= 0 .1.6.3 =Coriolis force= 0 .1.6.4 =Centrifugal force= 0 .1.6.5 =Motion relative to Earth= 2 .1.6.5.1 =x-axis= 1 .1.6.5.2 =y-axis= 1 .1.6.5.3 =z-axis= 1 .2 =Gravity= 2 .2.1 =Newton's law of gravitation= 1 .2.2 =Gravitational field strength= 2 .2.2.1 =If r > a= 1 .2.2.2 =If r < a= 1 .2.3 =Gravitational potential= 2 .2.3.1 =If r > a= 1 .2.3.2 =If r < a= 1 .3 =Particle motion= 2 .3.1 =Dynamic definitions= 2 .3.1.1 =Newtonian force= 1 .3.1.2 =Momentum= 1 .3.1.3 =Kinetic energy= 1 .3.1.4 =Angular momentum= 1 .3.1.5 =Couple= 1 .3.1.6 =Centre of mass= 0 .3.10 =Oblique elastic collision= 2 .3.10.1 =Direction of motion= 1 .3.10.2 =Final velocity of first object= 1 .3.10.3 =Final velocity of second object= 1 .3.2 =Relativistic dynamics= 2 .3.2.1 =Lorentz Factor= 1 .3.2.2 =Momentum= 1 .3.2.3 =Force= 1 .3.2.4 =Rest energy= 1 .3.2.5 =Kinetic energy= 1 .3.2.6 =Total energy= 1 .3.3 =Constant acceleration= 2 .3.3.1 =Initial velocity= 1 .3.3.2 =Square initial velocity= 1 .3.3.3 =Distance travelled calculates based on initial velocity and acceleration= 1 .3.3.4 =Distance travelled calculates based on initial velocity and final velocity= 1 .3.4 =Reduced mass= 2 .3.4.1 =Reduced mass= 1 .3.4.2 =Distances from centre of mass to centre of first object= 1 .3.4.3 =Distances from centre of mass to centre of second object= 1 .3.4.4 =Moment of inertia= 1 .3.4.5 =Total angular momentum.= 0 .3.4.6 =Lagrangian= 0 .3.5 =Projectile motion= 2 .3.5.1 =Velocity (Based on elevation angle)= 1 .3.5.2 =Velocity (Based on y coordinate)= 1 .3.5.3 =Trajectory= 1 .3.5.4 =Maximum height= 1 .3.5.5 =Horizontal range= 1 .3.6 =Rocketry= 2 .3.6.1 =Escape velocity= 1 .3.6.2 =Specific impluse= 1 .3.6.3 =Effective exhaust velocity (into a vacuum)= 1 .3.6.4 =Rocket equation= 1 .3.6.5 =Multistage rocket= 0 .3.6.6 =In a constant gravitational field= 1 .3.7 =Gravitationally bound orbital motion= 2 .3.7.1 =Potential energy of interaction= 1 .3.7.10 =Phase= 1 .3.7.11 =Period= 2 .3.7.11.1 =Calculation m and E= 1 .3.7.11.2 =Calculation m and a= 1 .3.7.12 =Positive constant= 1 .3.7.2 =Total energy= 2 .3.7.2.1 =Calculation based on distance of m from M= 1 .3.7.2.2 =Calculation based on semi-major axis= 1 .3.7.3 =Orbital equation= 1 .3.7.4 =Semi-major axis= 2 .3.7.4.1 =Calculation based on semi-latus rectum= 1 .3.7.4.2 =Calculation based on total energy= 1 .3.7.5 =Semi-minor axis= 2 .3.7.5.1 =Calculation based on semi-latus rectum= 1 .3.7.5.2 =Calculation based on total energy and total angular momentum= 1 .3.7.6 =Eccentricity= 2 .3.7.6.1 =Calculation based on total energy and total angular momentum= 1 .3.7.6.2 =Calculation based on semi-major axis and semi-minor axis= 1 .3.7.7 =Semi-latus-rectum= 2 .3.7.7.1 =Calculation based on total angular momentum and orbiting mass= 1 .3.7.7.2 =Calculation based on semi-major axis and semi-minor axis= 1 .3.7.7.3 =Calculation based on semi-major axis and eccentricity= 1 .3.7.8 =Pericentre= 2 .3.7.8.1 =Calculation based on semi-latus-rectum= 1 .3.7.8.2 =Calculation based on semi-major axis= 1 .3.7.9 =Apocentre= 2 .3.7.9.1 =Calculation based on semi-latus-rectum= 1 .3.7.9.2 =Calculation based on semi-major axis= 1 .3.8 =Rutherford scattering= 2 .3.8.1 =Scattering potential energy= 1 .3.8.2 =Scattering angle= 1 .3.8.3 =Closest approach= 1 .3.8.4 =Semi axis= 1 .3.8.5 =Eccentricity= 1 .3.8.6 =Motion trajectory= 1 .3.8.7 =Scattering centre= 1 .3.9 =Inelastic collision= 2 .3.9.1 =Coefficien of restitution= 1 .3.9.2 =Loss of kinetic energy= 1 .3.9.3 =Final velocity of object 1= 1 .3.9.4 =Final velocity of object 2= 1 .4 =Elasticity= 2 .4.1 =Elasticity definitions (simple)= 2 .4.1.1 =Stress= 1 .4.1.2 =Strain= 1 .4.1.3 =Young modulus(Hooke's law)= 1 .4.1.4 =Poisson ratio= 1 .4.2 =Isotropic elastic solids= 2 .4.2.1 =Lame coefficients= 2 .4.2.1.1 =Lame coefficients 1= 1 .4.2.1.2 =Lame coefficients 2= 1 .4.2.2 =Longitudinal modulus= 1 .4.2.3 =Stress tensor= 0 .4.2.4 =Bulk modulus= 1 .4.2.5 =Pressure= 1 .4.2.6 =Shear modulus= 1 .4.2.7 =Transverse stress= 1 .4.2.8 =Young modulus= 1 .4.2.9 =Poisson ratio= 1 .4.3 =Torsion= 2 .4.3.1 =Torsional rigidity= 1 .4.3.2 =Thin circular cylinder= 1 .4.3.3 =Thick circular cylinder= 1 .4.3.4 =Arbitrary thin-walled tube= 1 .4.3.5 =Long flat ribbon= 1 .4.4 =Bending beams= 2 .4.4.1 =Bending moment= 1 .4.4.2 =Light beam= 1 .4.4.3 =Heavy beam= 0 .4.4.4 =Euler strut failure= 2 .4.4.4.1 =Free ends= 1 .4.4.4.2 =Fixed ends= 1 .4.4.4.3 =1 free end= 1 .4.5 =Elastic wave velocities= 2 .4.5.1 =In an infinite isotropic solid= 2 .4.5.1.1 =Speed of transverse wave= 1 .4.5.1.2 =Speed of longitudinal wave= 1 .4.5.2 =In a fluid= 1 .4.5.3 =On a thin plate= 2 .4.5.3.1 =Speed of longitudinal (x-axis)= 1 .4.5.3.2 =Speed of transverse wave(y-axis)= 1 .4.5.3.3 =Speed of transverse wave(z-axis)= 1 .4.5.4 =In at thin circular rod= 2 .4.5.4.1 =Speed of longitudinal wave= 1 .4.5.4.2 =Torsional wave velocity= 1 .4.5.4.3 =Speed of transverse wave= 1 .4.6 =Waves in strings and springs= 2 .4.6.1 =In a spring= 1 .4.6.2 =On a stretched string= 1 .4.6.3 =On a stretched sheet= 1 .4.7 =Propagation of elastic waves= 2 .4.7.1 =Acoustic impedance= 1 .4.7.2 =Wave velocity impedance relation= 1 .4.7.3 =Mean energy density= 2 .4.7.3.1 =Calculation based on elastic modulus and wavenumber= 1 .4.7.3.2 =Calculation based on density and angular frequency= 1 .4.7.4 =Energy flux= 1 .4.7.5 =Normal coefficients= 2 .4.7.5.1 =Reflection coefficient= 1 .4.7.5.2 =Transmission coefficient= 1 .4.7.6 =Snell's law= 0 .5 =Rigid body dynamics= 2 .5.1 =Moments of inertia= 2 .5.1.1 =Thin rod= 1 .5.1.10 =Triangular plate= 1 .5.1.2 =Solid sphere= 1 .5.1.3 =Spherical shell= 1 .5.1.4 =Solid cylinder= 2 .5.1.4.1 =I1-axis and I2-axis= 1 .5.1.4.2 =I3-axis= 1 .5.1.5 =Solid cuboid= 2 .5.1.5.1 =I1-axis= 1 .5.1.5.2 =I2-axis= 1 .5.1.5.3 =I3-axis= 1 .5.1.6 =Solid circular cone= 2 .5.1.6.1 =I1-axis and I2-axis= 1 .5.1.6.2 =I3-axis= 1 .5.1.7 =Solid ellipsoid= 2 .5.1.7.1 =I1-axis= 1 .5.1.7.2 =I2-axis= 1 .5.1.7.3 =I3-axis= 1 .5.1.8 =Ellipptical lamina= 2 .5.1.8.1 =I1-axis= 1 .5.1.8.2 =I2-axis= 1 .5.1.8.3 =I3-axis= 1 .5.1.9 =Disk= 2 .5.1.9.1 =I1-axis and I2-axis= 1 .5.1.9.2 =I3-axis= 1 .5.2 =Centres of mass= 2 .5.2.1 =Solid hemisphere= 1 .5.2.2 =Hemispherical shell= 1 .5.2.3 =Sector of disk= 1 .5.2.4 =Arc of circle= 1 .5.2.5 =Arbitrary triangular lamina= 1 .5.2.6 =Solid cone or pyramid= 1 .5.2.7 =Solid spherical cap= 1 .5.2.8 =Shell spherical cap= 1 .5.2.9 =Semi-elliptical lamina= 1 PDrThe position in frame Oam0r^'The position in frame O'bm0tTime in frame Ocs0vVelocity of O' in frame Odm/s0b+c*da-c*d(a-b)/d(a-b)/ctTime in frame Oas0t^'Time in frame O'bs0uVelocity in frame Oam/s0u^'Velocity in frame O'bm/s0vVelocity of O' in frame Ocm/s0b+ca-ca-bpParticle momentum in frame Oakg.m/s0p^'Particle momentum in frame O'bkg.m/s0vVelocity of O' in frame Ocm/s0m Particle massdkg0b+c*da-c*d(a-b)/d(a-b)/cJAngular momentum in frame OaN.m.s0J'Particle momentum in frame O'bN.m.s0m Particle massckg0r'The position in frame O'dm0vVelocity of O' in frame Oem/s0p'Particle momentum in frame O'fkg.m/s0tTime in frame Ogs0 b+c*d*e+e*f*g a-c*d*e-e*f*gTKinetic energy in frame OaN.m.s0T'Kinetic energy in frame O'bN.m.s0m Particle massckg0u'Velocity in frame O'dm/s0vVelocity of O' in frame Oem/s0b+c*d*e+c*e^2/2a-c*d*e-c*e^2/2\gammaLorentz Factora0vVelocity of O' in frame Obm/s0cSpeed of lightcm/s3*10^81/sqrt(1-b^2/c^2)xx-position in frame Oam0\gammaLorentz Factorb0x'x-position in frame O'cm0vVelocity of O' in frame Odm/s0t'Time in frame O'es0 b*(c+d*e) a/(c+d*e)a/b-d*ex'x-position in frame O'am0\gammaLorentz Factorb0xx-position in frame Ocm0vVelocity of O' in frame Odm/s0tTime in frame Oes0 b*(c-d*e) a/(c-d*e)a/b+d*etTime in frame Oas0t'Time in frame O'bs0\gammaLorentz Factorc0vVelocity of O' in frame Odm/s0cSpeed of lightem/s3*10^8x'x-position in frame O'fm0c*(b+d*f/(e^2)) a/c-d*f/(e^2)t'Time in frame O'as0tTime in frame Obs0\gammaLorentz Factorc0vVelocity of O' in frame Odm/s0cSpeed of lightem/s3*10^8xx-position in frame Ofm0c*(b-d*f/(e^2)) a/c+d*f/(e^2)u_x'Particle velocity components in frame Oam/s0{u'}_x(Particle velocity components in frame O'bm/s0vVelocity of O' in frame Ocm/s0cSpeed of lightdm/s3*10^8(b+c)/(1+b*c/d^2){u'}_x(Particle velocity components in frame O'am/s0u_x'Particle velocity components in frame Obm/s0vVelocity of O' in frame Ocm/s0cSpeed of lightdm/s3*10^8(b-c)/(1-b*c/d^2)u_y'Particle velocity components in frame Oam/s0{u'}_y(Particle velocity components in frame O'bm/s0vVelocity of O' in frame Ocm/s0cSpeed of lightdm/s3*10^8{u'}_x(Particle velocity components in frame O'em/s0\gammaLorentz Factorf0b/(1+e*c/d^2)/f{u'}_y(Particle velocity components in frame O'am/s0u_y'Particle velocity components in frame Obm/s0vVelocity of O' in frame Ocm/s0cSpeed of lightdm/s3*10^8u_x'Particle velocity components in frame Oem/s0\gammaLorentz Factorf0b/(1-e*c/d^2)/fu_z'Particle velocity components in frame Oam/s0{u'}_z(Particle velocity components in frame O'bm/s0vVelocity of O' in frame Ocm/s0cSpeed of lightdm/s3*10^8{u'}_x(Particle velocity components in frame O'em/s0\gammaLorentz Factorf0b/(1+e*c/d^2)/f{u'}_z(Particle velocity components in frame O'am/s0u_z'Particle velocity components in frame Obm/s0vVelocity of O' in frame Ocm/s0cSpeed of lightdm/s3*10^8u_x'Particle velocity components in frame Oem/s0\gammaLorentz Factorf0b/(1-e*c/d^2)/fp_xx components of momentum in Oakgm/s0{p'}_xx components of momentum in O'bkgm/s0\gammaLorentz Factorc0vVelocity of O' in frame Odm/s0E'Energy in frame O'eJ0cSpeed of lightfm/s3*10^8 c*(b+d*e/f^2) a/c-d*e/f^2{p'}_xx components of momentum in O'akg.m/s0p_xx components of momentum in Obkg.m/s0\gammaLorentz Factorc0vVelocity of O' in frame Odm/s0EEnergy in frame OeJ0cSpeed of lightfm/s3*10^8 c*(b-d*e/f^2) a/c+d*e/f^2EEnergy in frame OaJ0E'Energy in frame O'bJ0vVelocity of O' in frame Ocm/s0{p'}_xx components of momentum in O'ekg.m/s0\gammaLorentz Factord0 d*(b+c*e)a/d-c*eE'Energy in frame O'aJ0EEnergy in frame ObJ0vVelocity of O' in frame Ocm/s0p_xx components of momentum in Oekgm/s0\gammaLorentz Factord0 d*(b-c*e)a/d+c*eE'Energy in frame O'aJ0EEnergy in frame ObJ0cSpeed of lightcm/s3*10^8pTotal momentum in Oekg.m/s0p'Total momentum in O'fkg.m/s0m_0Massgkg0\nuFrequency received in OaHz0\nu'Frequency received in O'bHz0vVelocity of O' in frame Ocm/s0cSpeed of lightdm/s3*10^8\alphaArrival angle in Serad0\gammaLorentz factorf0b*f*(1+c*cos(e)/d)\theta"Emission angle of light in frame Oarad0\theta'#Emission angle of light in frame O'brad0vVelocity of O' in frame Ocm/s0cSpeed of lightdm/s3*10^8!acos((cos(b)+c/d)/(1+c/d*cos(b)))\theta'#Emission angle of light in frame O'arad0\theta"Emission angle of light in frame Obrad0vVelocity of O' in frame Ocm/s0cSpeed of lightdm/s3*10^8!acos((cos(b)-c/d)/(1-c/d*cos(b))) P(\theta)$Angular distribution of photons in Oa0vVelocity of O' in frame Obm/s0cSpeed of lightcm/s3*10^8\theta'#Emission angle of light in frame O'd0\gammaLorentz Factore0sin(d)/(2*e^2*(1-b/c*cos(d))^2)A any vectora0tTime in frame Obs0\omegaAngular velocity of O' in Ocrad/s0\dot{v}Acceleration in Oam/s^20\dot{v'}Acceleration in O'bm/s^20\omegaAngular velocity of O' in Ocrad/s0v'Velocity in frame O'dm/s0r'Position in frame O'em/s0F_{cor}Coriolis forceaN0m Particle massbkg0v'Velocity in frame O'cm/s0\omegaAngular velocity of O' in Odrad/s0F_cenCentrifugal forceaN0m Particle massbkg0r'Position in frame O'cm/s0\omegaAngular velocity of O' in Odrad/s0\ddot{x}x - Easterly axisam0\dot{y}y - Northerly aixsbm0\dot{z}z - Local vertical aixscm0F_xNongravitation forcedN0m Particle massekg0\omega_eEarth's spin ratef0\lambdaLatitudegrad0(d+2*e*f*(b*sin(g)-c*cos(g)))/e\ddot{y}y - Northerly aixsam0\dot{x} Easterly axisbm0F_yNongravitation forcedN0m Particle massekg0\omega_eEarth's spin ratef0\lambdaLatitudegrad0(d-2*e*f*b*sin(g))/e\ddot{z}z - Local vertical aixsam0gGravitational accelerationbm/s^29.86\dot{x}x - Easterly axiscm0F_zNongravitational forcedN0m Particle massekg0\omega_eEarth's spin ratef0\lambdaLatitudegrad0(d-e*b+2*e*f*c*cos(g))/eF_1Force on object 1aN0GConstant of gravitationb Nm^2/kg^2 6.67*10^(-11)m_1Mass of object 1ckg0m_2Mass of object 2dkg0r_{12} Vector from object 1 to object 2em0 \hat r_{12}/Unit vector of vector from object 1 to object 2fm0 b*c*d*f/e^2gGravitational field strengtham/s^20GConstant of gravitationb Nm^2/kg^2 6.67*10^(-11)MMass of sphereckg0rVector from sphere centredm0-b*c/d^2gGravitational field strengtham/s^20GConstant of gravitationb Nm^2/kg^2 6.67*10^(-11)MMass of sphereckg0rVector from sphere centredm0aradius of spherefm0 -b*c*d/f^3\phi(r)Gravitational potentialaJ0GConstant of gravitationb Nm^2/kg^2 6.67*10^(-11)MMass of sphereckg0rVector from sphere centredm0-b*c/d-a*d/b-b*c/a\phi(r)Gravitational potentialaJ0GConstant of gravitationb Nm^2/kg^2 6.67*10^(-11)MMass of sphereckg0rVector from sphere centredm0aRadius of sphereem0b*c/(2*e^3)*(d^2-3*e^2)a*2*e^3/(d^2-3*e^2)/bF&The resultant force acting on particleaN0mMass of particlebkg0aAcceleration of particlecm/s^20b*ca/ca/bpMomentum of particleakgm/s0mMass of particlebkg0vVelocity of particlecm/s0b*ca/ca/bTKinetic energy of particleaJ0mMass of particlebkg0vVelocity of particlecm/s0b*c^2/22*a/c^2 sqrt(2*a/b)JAngular momentum of particleaNms0rParticle position vectorbm0pMomentum of particleckgm/s0b*ca/ca/bGCoupleaN.m0rParticle position vectorbm0pMomentum of particleckg.m/s0b*ca/ca/bR_0!Position vector of centre of massam0m_iMass of i^{th} particlebkg0r_i"Position vector of i^{th} particlecm0\gammaLorentz Factora0vParticle velocitybm/s0cSpeed of lightcm/s3*10^81/sqrt(1-b^2/c^2)p!Relativistic momentum of particleakgm/s0m_0 Particle massbkg0vVelocity of particlecm/s0\gammaLorentz Factord0b*c*da/c/da/b/da/b/cFForce on particleaN0dp"The change of momentum of particlebkgm/s0dtThe change of timecs0b/ca*cb/aE_rParticle rest energyaJ0m_0 Particle massbkg0cSpeed of lightcm/s3*10^8b*c^2a/c^2TRelativistic kinetic energyaJ0m_0 Particle massbkg0cSpeed of lightcm/s3*10^8\gammaLorentz Factord0 b*c^2*(d-1) a/c^2/(d-1) a/b/c^2+1E Total energyaJ0m_0 Particle massbkg0cSpeed of lightcm/s3*10^8\gammaLorentz Factord0b*c^2*da/c^2/da/b/c^2vInitial velocityam/s0uFinal velocitybm/s0a Accelerationcm/s^20tTimeds0b+c*da-c*d(a-b)/d(a-b)/cvInitial velocityam/s0uFinal velocitybm/s0a Accelerationcm/s^20sDistance travelleddm0sqrt(b^2+2*c*d)sqrt(a^2-2*c*d) (a^2-b^2)/2/d (a^2-b^2)/2/csDistance travelledam0uFinal velocitybm/s0a Accelerationcm/s^20tTimeds0 b*d+c*d^2/2 (a-c*d^2/2)/d (a-b*d)*2/d^2sDistance travelledam0uFinal velocitybm/s0vInitial velocitycm/s0tTimeds0 (b+c)*d/22*a/d-c2*a/d-b 2*a/(b+c)\mu Reduced massakg0m_1Mass of first object bkg0m_2Mass of second objectckg0 b*c/(b+c)r_1#Position vector from centre of massam0m_1Mass of first objectbkg0m_2Mass of second objectckg0rPosition vector between massesdm0 c/(c+b)*dr_2#Position vector from centre of massam0m_1Mass of first objectbkg0m_2Mass of second objectckg0rPosition vector between massesdm0 -b/(c+b)*dIMoment of inertiaakg.m^20\mu Reduced massbkg0|r|Distance between massescm0b*c^2a/c^2 sqrt(a/b)JAngular momentumaNms0\mu Reduced massbkg0rPosition vector between massescm0L LagrangianaJ0\mu Reduced massbkg0rPosition vector between massescm0UPotential energy of interactiondJ0v Velocity at tam/s0v_0Initial velocitybm/s0\alphaElevation anglec^o0tTimeds0gGravitational accelerationem/s^29.86#sqrt((b*cos(c))^2+(b*sin(c)-d*e)^2)v Velocity at tam/s0v_0Initial velocitybm/s0y y coordinatecm0gGravitational accelerationdm/s^29.86sqrt(b^2-2*c*d)y y coordinateam0x x coordinatebm0\alphaElevation anglec^o0v_0Initial velocitydm/s0gGravitational accelerationem/s^29.86!b*tan(c)-e*b^2/(2*d^2*(cos(c))^2)h_{max}Maximum heightam0v_0Initial velocitybm/s0\alphaElevation anglec^o0gGravitational accelerationdm/s^29.86b^2*(sin(c))^2/(2*d)lRangeam0v_0Initial velocitybm/s0\alphaElevation anglec^o0gGravitational accelerationdm/s^29.86b^2*sin(2*c)/dsqrt(a*d/sin(2*c))v_{esc}Escape velocityam/s0GConstant of gravitationb Nm^2/kg^2 6.67*10^(-11)MMass of central bodyckg0rCentral body radiusdm0 sqrt(2*b*c/d) a^2*d/2/b 2*b*c/a^2I_{sp}Specific impluseaNs0uEffective exhaust velocitybm/s0gAcceleration due to gravitycm/s^29.86b/ca*cuEffective exhaust velocity am/s0\gammaratio of heat capacitiesb0RMolar gas constantcJ.K^{-1}.kmol^{-1} 8314.51070T_cCombustion temperaturedK0\mu'Effective molecular mass of exhaust gasekg/kmol0sqrt(2*b*c*d/(b-1)/e)\Delta vRocket velocity incrementam/s0uEffective exhaust velocity bm/s0M_iPre-burn rocket massckg0M_fPost-burn rocket massdkg0 b*ln(c/d)\Delta vRocket velocity incrementam/s0u_iexhaust velocity of i^{th} burnbm/s0\mathfrak{M_i}Mass ratio for i^{th} burnc0\Delta vRocket velocity incrementam/s0uEffective exhaust velocity bm/s0 \mathfrak{M} Mass ratioc0gAcceleration due to gravitydm/s^29.86t Burn timees0\thetarocket zenith anglefrad0b*ln(c)-d*e*cos(f)UPotential energyaJ0GConstant of gravitationb Nm^2/kg^2 6.67*10^(-11)MMass of central bodyckg0m Orbiting massdkg0rDistance of m from Mem0b*c*d/ea*e/b/da*e/b/cb*c*d/aE Total energyaJ0\alphaPositive constantbNm^20rDistance of m from Mcm0JTotal angular momentumdNms0m Orbiting massekg0-b/c+d^2/(2*e*c^2)E Total energyaJ0\alphaPositive constantbNm^20aSemi-major axiscm0b/2/ca*c*22*a/brDistance of m from Mam0aSemi-major axisbm0e Eccentricityc0\phi Orbital phasedrad0b*(1-c^2)/(1+c*cos(d))aSemi-major axisam0r_0Semi-latus-rectumbm0e Eccentricityc0 b/(1-c^2) a*(1-c^2)aSemi-major axisam0\alphaPositive constantbNm^20|E| Total energycJ0b/(2*c)2*a*cbSemi-minor axisam0r_0Semi-latus-rectumbm0e Eccentricityc0 b/sqrt(1-c^2)bSemi-minor axisam0JTotal angular momentumbNms0m Orbiting massckg0E Total energydJ0 b/sqrt(2*c*d) a*sqrt(2*c*d) (b/a)^2/(2*d) (b/a)^2/(2*c)e Eccentricitya0E Total energybJ0JTotal angular momentumcNms0m Orbiting massdkg0\alphaPositive constanteNm^20sqrt(1+2*b*c^2/d/e^2)(a^2-1)*d*e^2/(2*c^2)sqrt((a^2-1)*d*e^2/(2*b))2*b*c^2/((a-1)*e^2)sqrt(2*b*c^2/((a-1)*d))e Eccentricitya0bSemi-minor axisbm0aSemi-major axiscm0sqrt(1-b^2/c^2) sqrt(1-a^2)*c b/sqrt(1-a^2)r_0Semi-latus-rectumam0JTotal angular momentumbNms0m Orbiting massckg0\alphaPositive constantdNm^20 b^2/(c*d) sqrt(a*c*d) b^2/(a*d) b^2/(a*c)r_0Semi-latus-rectumam0bSemi-minor axisbm0aSemi-major axiscm0b^2/c sqrt(a*c)b^2/ar_0Semi-latus-rectumam0aSemi-major axisbm0e Eccentricityc0 b*(1-c^2) a/(1-c^2) sqrt(1-a/b)r_{min}Pericentre distanceam0r_0Semi-latus-rectumbm0e Eccentricityc0b/(1+c)a*(1+c)b/a-1r_{min}Pericentre distanceam0aSemi-major axisbm0e Eccentricityc0b*(1-c)a/(1-c)1-b/a r_{max}Apocentre distanceam0r_0Semi-latus-rectumbm0e Eccentricityc0b*(1-c)a/(1-c)1-b/a r_{max}Apocentre distanceam0aSemi-major axisbm0e Eccentricityc0b*(1+c)a/(1+c)a/b-1 \phi Orbital phasearad0JTotal angular momentumbN.ms0rDistance of m from Mcm0m Orbiting massdkg0\alphaPositive constanteN.m^20E Total energyfJ0)acos((b/c-d*e/b)/sqrt(2*d*f+d^2*e^2/b^2)) POrbital periodas0\alphaPositive constantbN.m^20m Orbiting massckg0|E| Total energydJ0pi*b*sqrt(c/(2*abs(d^3))) POrbital periodas0aSemi-major axisbm0m Orbiting massckg0\alphaPositive constantdN.m^202*pi*b^(3/2)*sqrt(c/d) \alphaPositive constantaN.m^20GConstant of gravitationb N.m^2/kg^2 6.67*10^(-11)MMass of central bodyckg0m Orbiting massdkg0b*c*dUScattering potential energyaJ0\alphaConstantb0rParticle separationc0-b/c-a*c-b/a\chiScattering anglearad0|\alpha|Constantb0ETotal energy(>0)cJ0bImpact parameterdm02*atan(abs(b)/(2*c*d))abs(b)/(2*d*tan(a/2))abs(b)/(2*c*tan(a/2))r_{min}Closest approacham0\alphaConstantb0ETotal energy(>0)cJ0\chiScattering angledrad0"abs(b)/(2*c)*(1/sin(d/2)-b/abs(b))aHyperbola semi-exisam0|\alpha|Constantb0ETotal energy(>0)cJ0 abs(b)/(2*c)2*a*c abs(b)/(2*a)e Eccentricitya0ETotal energy(>0)bJ0bImpact parametercm0\alphaConstantd0sqrt(4*b^2*c^2/d^2+1)sqrt(a^2-1)*d/(2*c)sqrt(a^2-1)*d/(2*b)2*b*c/sqrt(a^2-1)y+Position y with respect to hyperbola centream0x+Position x with respect to hyperbola centrebm0ETotal energy(>0)cJ0\alphaConstantd0bImpact parameterem0e*sqrt(4*c^2*b^2/d^2-1)sqrt(1+a^2/e^2)*d/(2*c)x+Position x with respect to hyperbola centream0\alphaConstantb0ETotal energy(>0)cJ0bImpact parameterdm0sqrt(b^2/(4*c^2)+1)sqrt((a^2-d^2))/(2*c)b/(2*sqrt(a^2-d^2))sqrt(a^2-b^2/(4*c^2)) v'_2#Post-collision velocity of object 2am/s0v'_1#Post-collision velocity of object 1bm/s0v_2"Pre-collision velocity of object 2cm/s0v_1"Pre-collision velocity of object 1dm/s0\epsilonCoefficient of restitutione0 b+e*(d-c) a-e*(d-c) d-(a-b)/e c+(a-b)/e T5Total kinetic in zero momentum frame before collisionaJ0T'4Total kinetic in zero momentum frame after collisionbJ0\epsilonCoefficient of restitutionc0b/c^2a*c^2 sqrt(b/a) v'_1#Post-collision velocity of object 1am/s0v_1"Pre-collision velocity of object 1bm/s0v_2"Pre-collision velocity of object 2cm/s0m_1Particle masses of object 1dkg0m_2Particle masses of object 2ekg0\epsilonCoefficient of restitutionf0!(d-f*e)*b/(e+d) + (1+f)*e*c/(d+e) v'_2#Post-collision velocity of object 2am/s0v_1"Pre-collision velocity of object 1bm/s0v_2"Pre-collision velocity of object 2cm/s0m_1Particle masses of object 1dkg0m_2Particle masses of object 2ekg0\epsilonCoefficient of restitutionf0(e-f*d)*c/(d+e)+(1+f)*d*b/(d+e)  \theta'_1Final trajectorya^o0m_1Sphere mass of first objectbkg0m_2Sphere mass of second objectckg0\theta/Angle between centre line and incident velocityd^o0atan(c*sin(2*d)/(b-c*cos(2*d))) v'_1Final velocity of first objectam/s0m_1Sphere mass of first objectbkg0m_2Sphere mass of second objectckg0\theta/Angle between centre line and incident velocityd^o0v!Incident velocity of first objectem/s0$sqrt(c^2+b^2-2*b*c*cos(2*d))/(b+c)*e v'_2Final velocity of second objectam/s0m_1Sphere mass of first objectbkg0m_2Sphere mass of second objectckg0\theta/Angle between centre line and incident velocityd^o0v!Incident velocity of first objectem/s02*b*e/(b+c)*cos(d)\taustressaN/m^20F Applied forcebN0ACross-sectional areacm^20b/ca*cb/aeStraina0\delta lChange in lengthbm0lLengthcm0b/ca*cb/aE Young modulusaN/m^20\taustressbN/m^20eStrainc0b/ca*cb/a\sigma Poisson ratioa0\delta wChange in widthbm0wWidthcm0\delta lChange in lengthdm0lLengthem0-b/c/d*e\muLame coefficienta0E Young modulusbN/m^20\sigma Poisson ratioc0 b/(2*(1+c)) a*2*(1+c)b/2/a-1\lambdaLame coefficienta0E Young modulusbN/m^20\sigma Poisson ratioc0b*c/(1-2*c)/(1+c)M_1Longitudinal elastic modulusaN/m^20E Young modulusbN/m^20\sigma Poisson ratioc0b*(1-c)/(1+c)/(1-2*c)t Stress tensoraN/m^20\muLame coefficientsb0eStrain tensorc0\lambdaLame coefficientd0K Bulk modulusaN/m^20E Young modulusbN/m^20\sigma Poisson ratioc0 b/3/(1-2*c) 3*(1-2*c)*a (1-b/3/a)/2pPressureaJ0K Bulk modulusbN/m^20e_v Volume straincm^30-b*c-a/c-a/b\mu Shear modulusaN/m^20E Young modulusbN/m^20\sigma Poisson ratioc0 b/2/(1+c) 2*a*(1+c) b/(2*a)-1\tau_TTransverse stressarad0\mu Shear modulusbN/m^20\theta_{\small sh} Shear straincrad0b*ca/ca/bE Young modulusaN/m^20\mu Shear modulusbN/m^20K Bulk moduluscN/m^20 9*b*c/(b+3*c) \sigma Poisson ratioa0\mu Shear modulusbN/m^20K Bulk moduluscN/m^20(3*c-2*b)/(2*(3*c+b))GTwisting coupleaN.m.rad0\phiTwist angle in lengthbrad0l Rod lengthcm0CTorsional rigiditydN.m^20d*b/ca*c/db*d/aa*c/bCTorsional rigidityaN.m^20aRadiusbm0\mu Shear modoluscN/m^20tWall thicknessdm0 2*pi*b^3*c*d(a/(2*pi*c*d))^(1/3)a/(2*pi*b^3*d)a/(2*pi*b^3*c)CTorsional rigidityaN.m^20\mu Shear modolusbN/m^20a_1 Inner Radiuscm0a_2 Outer Radiusdm0b*pi*(d^4-c^4)/22*a/(pi*(d^4-c^4))CTorsional rigidityaN.m^20ACross-sectional areabm^20\mu Shear modoluscN/m^20tWall thicknessdm0P Perimeterem0 4*b^2*c*d/esqrt(a*e/(4*c*d)) a*e/(4*b^2*d) a*e/(4*b^2*c) 4*b^2*c*d/aCTorsional rigidityaN.m^20\mu Shear modolusbN/m^s0wCross-sectional widthcm0tWall thicknessdm0 b*c*d^3/3 3*a/c/d^3 3*a/b/d^3(3*a/(b*c))^(1/3)G_bBending momentaN.m0E Young modulusbN/m^20IMoment of areacm^40R_cRadius of curvaturedmb*c/da*d/ca*d/bb*c/ayDisplacement from horizontalam0W End weightbkg0E Young moduluscN/m^20IMoment of areadm^40l Beam length em0xDistance along beamfm0b/(2*c*d)*(e-f/3)*f^2a*(2*c*d)/((e-f/3)*f^2)b/(2*a*d)*(e-f/3)*f^2b/(2*c*a)*(e-f/3)*f^2wBeam weight per unit lengthaN.m^20E Young modulusbN/m^20IMoment of areacm^40xDistance along beamdm0yDisplacement from horizontalem0F_cCritical compression forceaN0E Young modulusbN/m^20IMoment of areacm^40l Beam length dm0 pi^2*b*c/d^2a*d^2/(pi^2*c)a*d^2/(pi^2*b)sqrt(pi^2*b*c/a)F_cCritical compression forceaN0E Young modulusbN/m^20IMoment of areacm^40l Beam length dm04*pi^2*b*c/d^2a*d^2/(pi^2*c*4)a*d^2/(4*pi^2*b)sqrt(4*pi^2*b*c/a)F_cCritical compression forceaN0E Young modulusbN/m^20IMoment of areacm^40l Beam length dm0pi^2*b*c/d^2/4a*4*d^2/(pi^2*c)4*a*d^2/(pi^2*b)sqrt(pi^2*b*c/a/4)v_tSpeed of transverse waveam/s0\mu Shear modulusbN/m^20\rhoDensityckg/m^30 sqrt(b/c)a^2*cb/a^2v_lSpeed of longitudinal waveam/s0M_lLongitudinal modulusbN/m^20\rhoDensityckg/m^30 sqrt(b/c)a^2*cb/a^2v_lSpeed of longitudinal waveam/s0K Bulk modulusbN/m^20\rhoDensityckg/m^30 sqrt(b/c)a^2*cb/a^2 v_l^{(x)}Speed of longitudinalam/s0E Young modulusbN/m^20\rhoDensityckg/m^30\sigma Poisson ratiod0sqrt(b/c/(1-d^2)) a^2*c*(1-d^2) b/a^2/(1-d^2)Fm%% v_t^{(y)}Speed of transverse waveam/s0\mu Shear modulusbN/m^20\rhoDensityckg/m^30 sqrt(b/c)a^2*cb/a^2 v_t^{(z)}Speed of transverse waveam/s0E Young modulusbN/m^20tPlate thicknesscm0\rhoDensitydkg/m^30\sigma Poisson ratioe0k Wave numberfm^{-1}0f*sqrt(b*c^2/(12*d(1-e^2)))v_lSpeed of longitudinal waveam/s0E Young modulusbN/m^20\rhoDensityckg/m^30 sqrt(b/c)a^2*cb/a^2v_\phiTorsional wave velocityam/s0\mu Shear modulusbN/m^20\rhoDensityckg/m^30 sqrt(b/c)a^2*cb/a^2v_tSpeed of transverse waveam/s0k Wavenumberbm^{-1}0a Rod radiuscm0E Young modulusdN/m^20\rhoDensityekg/m^30b*c/2*sqrt(d/e)v_lSpeed of longitudinal waveam/s0kSpring constantbN/m0l Spring lengthcm0\rho_lMass per unit lengthdkg/m0 sqrt(b*c/d)a^2*d/c (d*a^2)/bb*c/a^2v_tSpeed of transverse waveam/s0TTensionbN0\rho_lMass per unit lengthckg/m0 sqrt(b/c)a^2*cb/a^2v_tSpeed of transverse waveam/s0\tauTension per unit widthbN/m0\rho_AMass per unit areackg/m^20 sqrt(b/c)a^2*cb/a^2Z ImpedanceaN.m/s^30E'Elastic modulusbN/m^20\rhoDensityckg/m^30 sqrt(b*c)a^2/ca^2/bZ ImpedanceaN.m/s^30\rhoDensitybkg/m^30vWave phase velocitycm/s0b*ca/ca/b \mathfrak{U}Energy densityaN/m^20E'Elastic modulusbN/m^20k Wavenumbercm^{-1}0u_0Maximum displacementdm0 b*c^2*d^2/2 2*a/(c^2*d^2)sqrt(2*a/(b*d^2))sqrt(2*a/(b*c^2)) \mathfrak{U}Energy densityaN/m^20\rhoDensitybkg/m^30\omegaAngurlar frequencycHz0u_0Maximum displacementdm0 b*c^2*d^2/2 2*a/(c^2*d^2)sqrt(2*a/(b*d^2))sqrt(2*a/(b*c^2))PMean energy fluxaN/(m.s)0 \mathfrak{U}Energy densitybN/m^20vWave phase velocitycm/s0b*ca/ca/brReflection coefficienta0Z&The impedance of an acoustic componentbN.m/s^30Z_0%The characteristic acoustic impedancecN.m/s^30 (b-c)/(b+c)tTransmission coeffictiona0Z&The impedance of an acoustic componentbN.m/s^30Z_0%The characteristic acoustic impedancecN.m/s^30 2*b/(b+c)\theta_iAngle of incidencearad0\theta_rAngle of relectionbrad0\theta_tAngle of refractioncrad0v_i Wave phase velocity of incidencearad0v_r Wave phase velocity of relectionbrad0v_t!Wave phase velocity of refractioncrad0 I_1 = I_2'I1-axis and I2-axis moments of inertia akg.m^20mMass of thin rodckg0lLength of thin roddm0c*d^2/12a*12/d^2 sqrt(a*12/c) I_1=I_2=I_3Ii-axis moments of inertia akg.m^20mMass of solid speredkg0rRadius of solid sphereem0 d*e^2*2/5 5*a/2/e^2 sqrt(5*a/2/d)I_1 = I_2 = I_3Ii-axis moments of inertia akg.m^20mMass of spherical shelldkg0rRadius of spherical shellem0 d*e^2*2/3 3*a/2/e^2 sqrt(3*a/2/d) I_1 = I_2Ii-axis moments of inertia akg.m^20mMass of solid cylinderckg0rRadius of solid cylinderdm0lLength of solid cylinderem0c/4*(d^2+e^2/3)I_3I3-axis moments of inertia akg.m^20mMass of solid cylinderbkg0rRadius of solid cylindercm0b*c^2/22*a/c^2 sqrt(2*a/b)I_1I1-axis moments of inertia akg.m^20mMass of the solid cuboidbkg0bside b of the solid cuboidcm0cside c of the solid cuboiddm0b*(c^2+d^2)/12I_2I2-axis moments of inertia akg.m^20mMass of the solid cuboidbkg0cside c of the solid cuboidcm0aside a of the solid cuboiddm0b*(c^2+d^2)/12I_3I3-axis moments of inertia akg.m^20mMass of the solid cuboidbkg0aside a of the solid cuboidcm0bside b of the solid cuboiddm0b*(c^2+d^2)/12 I_1 = I_2Ii-axis moments of inertia akg.m^20mMass of solid circular coneckg0r#Based radius of solid circular conedm0hHeight of solid circular coneem03/20*c*(d^2+e^2/4)I_3I3-axis moments of inertia akg.m^20mMass of solid circular conebkg0r"Base radius of solid circular conecm0 3/10*b*c^2I_1I1-axis moments of inertia akg.m^20mMass of the solid ellipsoidbkg0b"semi axis b of the solid ellipsoidcm0c"semi axis c of the solid ellipsoiddm0 b*(c^2+d^2)/5I_2I2-axis moments of inertia akg.m^20mMass of the solid ellipsoidbkg0c"semi axis c of the solid ellipsoidcm0a"semi axis a of the solid ellipsoiddm0 b*(c^2+d^2)/5I_3I3-axis moments of inertia akg.m^20mMass of the solid ellipsoidbkg0a"semi axis a of the solid ellipsoidcm0b"semi axis b of the solid ellipsoiddm0 b*(c^2+d^2)/5I_1I1-axis moments of inertia akg.m^20mMass of the ellipptical laminabkg0b%semi axis b of the ellipptical laminacm0b*c^2/44*a/c^2 sqrt(4*a/b)I_2I2-axis moments of inertia akg.m^20mMass of the ellipptical laminabkg0a%semi axis a of the ellipptical laminacm0b*c^2/44*a/c^2 sqrt(4*a/b)I_3I3-axis moments of inertia akg.m^20mMass of the ellipptical laminabkg0a%semi axis a of the ellipptical laminacm0b%semi axis b of the ellipptical laminadm0 b*(c^2+d^2)/4  I_1 = I_2Ii-axis moments of inertia akg.m^20mMass of the diskckg0rBase radius of the diskdm0c*d^2/44*a/d^2 sqrt(4*a/c) I_3I3-axis moments of inertia akg.m^20mMass of the diskckg0rBase radius of the diskdm0c*d^2/22*a/d^2 sqrt(2*a/c) I_3I3-axis moments of inertia akg.m^20mMass of the diskbkg0aSide a of triangular platecm0bSide b of triangular platedm0cSide c of triangular plateem0b/36*(c^2+d^2+e^2)d-Distance from centre of mass to sphere centream0rRadius of solid hemispherebm03*b/88*a/3d-Distance from centre of mass to sphere centream0rRadius of hemispherical shellbm0b/22*ad+Distance from centre of mass to disk centream0rRadius of the diskbm0\thetaAnglecrad02/3*b*sin(c)/cd-Distance from centre of mass to circle centream0rRadius of circlebm0\thetaAnglecrad0 b*sin(c)/cd,Distance from the centre of mass to the baseam0h%Height of arbitrary triangular laminabm0b/3a*3d,Distance from the centre of mass to the baseam0h#Height of the solid cone or pyramidm0b/4a*4d1Distance from centre of mass to the sphere centream0r Sphere radiusbm0h!Height of the solid spherical capcm03*(2*b-c)^2/4/(3*b-c)d1Distance from centre of mass to the sphere centream0r Sphere radiusbm0h!Height of the shell sphere centrecm0b-c/2a+c/22*(b-a) d,Distance from the centre of mass to the baseam0h$Height of the semi-elliptical laminabm0 4*b/(3*pi)3*pi*a/4[Position\text r = r^' + vt Dynamics and mechanics_1_1_1.PNGTime \text t = t^' Dynamics and mechanics_1_1_1.PNGVelocity\text u = u^' + v Dynamics and mechanics_1_1_1.PNGMomentum\text p = p^' + mv Dynamics and mechanics_1_1_1.PNGAngular momentum\text J = J' + mr'v + vp't Dynamics and mechanics_1_1_1.PNGKinetic energy.$\text T = T' + mu'v + \frac{mv^2}{2} Dynamics and mechanics_1_1_1.PNGLorentz Factor1\text \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} Dynamics and mechanics_1_2_1.PNG Position x\text x = \gamma(x'+vt') Dynamics and mechanics_1_2_1.PNG Position x'\text x' = \gamma(x-vt) Dynamics and mechanics_1_2_1.PNGTime t$\text t = \gamma(t'+\frac{v}{c^2}x') Dynamics and mechanics_1_2_1.PNGTime t'#\text t' = \gamma(t-\frac{v}{c^2}x) Dynamics and mechanics_1_2_1.PNGVelocity x-axis in frame O2\text u_x = \frac{{u'}_x+v}{1+\frac{v}{c^2}{u'}_x} Dynamics and mechanics_1_3_1.PNGVelocity x'-axis in frame O'/\text {u'}_x = \frac{u_x-v}{1-\frac{v}{c^2}u_x} Dynamics and mechanics_1_3_1.PNGVelocity y-axis in frame O8\text u_y = \frac{{u'}_y}{\gamma(1+\frac{v}{c^2}{u'}_x)}Velocity y'-axis in frame O'5\text {u'}_y = \frac{u_y}{\gamma(1+\frac{v}{c^2}u_x)}Velocity z-axis in frame O8\text u_z = \frac{{u'}_z}{\gamma(1+\frac{v}{c^2}{u'}_x)}Velocity z'-axis in frame O'5\text {u'}_z = \frac{u_z}{\gamma(1+\frac{v}{c^2}u_x)} Momentum x*\text p_x = \gamma({p'}_x+v\frac{E'}{c^2}) Dynamics and mechanics_1_4_1.PNG Momentum x')\text {p'}_x = \gamma(p_x-v\frac{E}{c^2}) Dynamics and mechanics_1_4_1.PNGEnergy x\text E = \gamma(E'+v{p'}_x) Dynamics and mechanics_1_4_1.PNG Energy x'\text E' = \gamma(E-vp_x) Dynamics and mechanics_1_4_1.PNGEnergy equality.0\text E^2 - p^2c^2 = E'^2 - {p'}^2c^2 = m^2_0c^4 Dynamics and mechanics_1_4_1.PNGDopplerB\text \frac{\nu}{\nu'} = \gamma\left(1+\frac{v}{c}cos\alpha\right) Dynamics and mechanics_1_5_1.PNG Aberration xH\text cos\theta = \frac{cos\theta'+\frac{v}{c}}{1+\frac{v}{c}cos\theta'} Dynamics and mechanics_1_5_2.PNG Aberration x'G\text cos\theta' = \frac{cos\theta-\frac{v}{c}}{1-\frac{v}{c}cos\theta} Dynamics and mechanics_1_5_2.PNGRelativistic beamingC\text P(\theta) = \frac{sin\theta}{2\gamma^2{[1-(v/c)cos\theta]}^2} Dynamics and mechanics_1_5_2.PNGVector transformationsP\text \left[\frac{dA}{dt}\right]_s = \left[\frac{dA}{dt}\right]_{s'} + \omega A Accelerationm\text \dot{v} = \dot{v'}+2\vec{\omega}\;\time\;\vec{v'}+\vec{\omega}\;\time\; (\vec{\omega}\;\time\;\vec{r'}) Dynamics and mechanics_1_6_1.PNGCoriolis force8\text \vec {F'}_{cor} = -2m\vec{\omega}\;\time\;\vec{v'} Dynamics and mechanics_1_6_1.PNGCentrifugal forceG\text F'_{cen} = -m\vec{\omega}\;\time\;(\vec{\omega}\;\time\;\vec{r'}) Dynamics and mechanics_1_6_1.PNGx-axisI\text m\ddot{x} = F_x + 2m\omega_e(\dot{y}sin\lambda - \dot{z}cos\lambda)"Dynamics and mechanics_1_6_5_1.PNGy-axis3\text m\ddot{y} = F_y - 2m\omega_e\dot{x}sin\lambda"Dynamics and mechanics_1_6_5_1.PNGz-axis8\text m\ddot{z} = F_z - mg + 2m\omega_e\dot{x}cos\lambda"Dynamics and mechanics_1_6_5_1.PNGNewton's law of gravitation1\text F_1 = \frac{G m_1 m_2}{r^2_{12}}\hat r_{12}If r > a"\text g(r) = -\frac{GM}{r^2}\hat r Dynamics and mechanics_2_2_1.PNGIf r < a#\text g(r) = -\frac{GMr}{a^3}\hat r Dynamics and mechanics_2_2_1.PNGIf r > a\text \phi(r) = -\frac{GM}{r} Dynamics and mechanics_2_2_1.PNGIf r < a)\text \phi(r) = \frac{GM}{2a^3}(r^2-3a^2) Dynamics and mechanics_2_2_1.PNGNewtonian forceThe rate of change of momentum of a body is proportional to the resultant force acting on the body and is in the same direction. \text F = maMomentum \text p = mvKinetic energy\text T = \frac{mv^2}{2}Angular momentum \text J = rpCouple \text G = rFCentre of mass9\text R_0 = \frac{\sum_{i=1}^N m_i r_i}{\sum_{i=1}^N m_i}Lorentz Factor1\text \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}Momentum\text p= \gamma m_0 vForce\text F = \frac{dp}{dt} Rest energy\text E_r = m_0 c^2Kinetic energy\text T = m_0 c^2(\gamma - 1) Total energy\text E =\gamma m_0 c^2Initial velocity\text v = u + atSquare initial velocity\text v^2 = u^2 + 2asHDistance travelled calculates based on initial velocity and acceleration\text s = ut + \frac{at^2}{2}JDistance travelled calculates based on initial velocity and final velocity\text s = \frac{u+v}{2}t Reduced mass#\text \mu = \frac{m_1 m_2}{m_1+m_2} Dynamics and mechanics_3_4_1.PNG8Distances from centre of mass to centre of first object!\text r_1 = \frac{m_2}{m_1+m_2}r Dynamics and mechanics_3_4_1.PNG9Distances from centre of mass to centre of second object!\text r_2 = -\frac{m_1}{m_1+m_2}r Dynamics and mechanics_3_4_1.PNGMoment of inertia\text I = \mu |r|^2 Dynamics and mechanics_3_4_1.PNGTotal angular momentum.\text J = \mu r\ddot{r} Dynamics and mechanics_3_4_1.PNG Lagrangian,\text L = \frac{\mu |\dot{r}|^2}{2} - U(|r|) Dynamics and mechanics_3_4_1.PNG#Velocity (Based on elevation angle)7\text v = \sqrt{(v_0 cos\alpha)^2+(v_0 sin\alpha-gt)^2} Dynamics and mechanics_3_5_1.PNG Velocity (Based on y coordinate)\text v^2 = v^2_0 - 2gy Dynamics and mechanics_3_5_1.PNG Trajectory7\text y = x\;tan\alpha - \frac{gx^2}{2v^2_0cos^2\alpha} Dynamics and mechanics_3_5_1.PNGMaximum height+\text h_{max} = \frac{v^2_0}{2g}sin^2\alpha Dynamics and mechanics_3_5_1.PNGHorizontal range#\text l = \frac{v^2_0}{g}sin2\alpha Dynamics and mechanics_3_5_1.PNGEscape velocity#\text v_{esc}= \sqrt{\frac{2GM}{r}}Specific impluse\text I_{sp} = \frac{u}{g}*Effective exhaust velocity (into a vacuum)5\text u = \sqrt{\frac{2\gamma RT_c}{(\gamma -1)\mu}}Rocket equation2\text \Delta v = u\;ln\left(\frac{M_i}{M_f}\right)Multistage rocket4\text \Delta v = \sum_{i=1}^N u_i\;ln \mathfrak{M_i}!In a constant gravitational field2\text \Delta v = u\;ln\mathfrak{M} - gt\;cos\thetaPotential energy of interaction\text U(r) = -\frac{GMm}{r} Dynamics and mechanics_3_7_1.PNG)Calculation based on distance of m from M-\text E = -\frac{\alpha}{r}+\frac{J^2}{2mr^2} Dynamics and mechanics_3_7_1.PNG$Calculation based on semi-major axis\text E = -\frac{\alpha}{2a} Dynamics and mechanics_3_7_1.PNGOrbital equation&\text r = \frac{a(1-e^2)}{1+e cos\phi} Dynamics and mechanics_3_7_1.PNG&Calculation based on semi-latus rectum\text a = \frac{r_0}{1-e^2} Dynamics and mechanics_3_7_1.PNG!Calculation based on total energy\text a = \frac{\alpha}{2|E|} Dynamics and mechanics_3_7_1.PNG&Calculation based on semi-latus rectum"\text b = \frac{r_0}{\sqrt{1-e^2}} Dynamics and mechanics_3_7_1.PNG<Calculation based on total energy and total angular momentum \text b = \frac{J}{\sqrt{2m|E|}} Dynamics and mechanics_3_7_1.PNG<Calculation based on total energy and total angular momentum-\text e = \sqrt{1 + \frac{2EJ^2}{m \alpha^2}} Dynamics and mechanics_3_7_1.PNG8Calculation based on semi-major axis and semi-minor axis$\text e = \sqrt{1 - \frac{b^2}{a^2}} Dynamics and mechanics_3_7_1.PNG=Calculation based on total angular momentum and orbiting mass\text r_0 = \frac{J^2}{m\alpha} Dynamics and mechanics_3_7_1.PNG8Calculation based on semi-major axis and semi-minor axis\text r_0 = \frac{b^2}{a} Dynamics and mechanics_3_7_1.PNG5Calculation based on semi-major axis and eccentricity\text r_0 = a(1-e^2) Dynamics and mechanics_3_7_1.PNG&Calculation based on semi-latus-rectum\text r_{min} = \frac{r_0}{1+e} Dynamics and mechanics_3_7_1.PNG$Calculation based on semi-major axis\text r_{min} = a(1-e) Dynamics and mechanics_3_7_1.PNG &Calculation based on semi-latus-rectum\text r_{max} = \frac{r_0}{1-e} Dynamics and mechanics_3_7_1.PNG $Calculation based on semi-major axis\text r_{max} = a(1+e) Dynamics and mechanics_3_7_1.PNG PhaseG\text cos\phi = \frac{(J/r)-(m\alpha/J)}{\sqrt{2mE+(m^2 \alpha^2/J^2)}} Dynamics and mechanics_3_7_1.PNG Calculation m and E*\text P = \pi\alpha\sqrt{\frac{m}{2|E|^3}} Dynamics and mechanics_3_7_1.PNG Calculation m and a5\text P = 2\pi a^{\frac{3}{2}}\sqrt{\frac{m}{\alpha}} Dynamics and mechanics_3_7_1.PNG Positive constant\text \alpha = GMm Dynamics and mechanics_3_7_1.PNGScattering potential energy\text U(r) = -\frac{\alpha}{r} Dynamics and mechanics_3_8_1.PNGScattering angle1\text tan(\frac{\chi}{2}) = \frac{|\alpha|}{2Eb} Dynamics and mechanics_3_8_1.PNGClosest approachY\text r_{min} = \frac{|\alpha|}{2E}\left(csc\frac{\chi}{2}-\frac{\alpha}{|\alpha|}\right) Dynamics and mechanics_3_8_1.PNG Semi axis\text a = \frac{|\alpha|}{2E} Dynamics and mechanics_3_8_1.PNG Eccentricity+\text e = \sqrt{\frac{4E^2b^2}{\alpha^2}+1} Dynamics and mechanics_3_8_1.PNGMotion trajectory2\text \frac{4E^2x^2}{\alpha^2}-\frac{y^2}{b^2} = 1 Dynamics and mechanics_3_8_1.PNGScattering centre-\text x = \pm\sqrt{\frac{\alpha^2}{4E^2}+b^2} Dynamics and mechanics_3_8_1.PNG Coefficien of restitution%\text v'_2 - v'_1 = \epsilon(v_1-v_2) Dynamics and mechanics_3_9_1.PNG Loss of kinetic energy$\text \frac{T-T'}{T} = 1- \epsilon^2 Dynamics and mechanics_3_9_1.PNG Final velocity of object 1S\text v'_1 = \frac{m_1-\epsilon m_2}{m_1+m_2}v_1+\frac{(1+\epsilon)m_2}{m_1+m_2}v_2 Dynamics and mechanics_3_9_1.PNG Final velocity of object 2S\text v'_2 = \frac{m_2-\epsilon m_1}{m_1+m_2}v_2+\frac{(1+\epsilon)m_1}{m_1+m_2}v_1 Dynamics and mechanics_3_9_1.PNG Direction of motion?\text tan\theta'_1 = \frac{m_2 sin 2\theta}{m_1-m_2cos 2\theta}"Dynamics and mechanics_3_10_1a.PNG"Dynamics and mechanics_3_10_1b.PNG Final velocity of first objectH\text v'_1 = \frac{\sqrt{m^2_1 + m^2_2 - 2m_1m_2 cos 2\theta}}{m_1+m_2}v"Dynamics and mechanics_3_10_1a.PNG"Dynamics and mechanics_3_10_1b.PNG Final velocity of second object+\text v'_2 = \frac{2m_1v}{m_1+m_2}cos\theta"Dynamics and mechanics_3_10_1a.PNG"Dynamics and mechanics_3_10_1b.PNGStress\text \tau = \frac{F}{A} Dynamics and mechanics_4_1_1.PNGStrain\text e = \frac{\delta l}{l} Dynamics and mechanics_4_1_1.PNGYoung modulus(Hooke's law)\text E = \frac{\tau}{e} Dynamics and mechanics_4_1_1.PNG Poisson ratio-\text \sigma = -\frac{\delta w/w}{\delta l/l} Dynamics and mechanics_4_1_1.PNGLame coefficients 1"\text \mu = \frac{E}{2(1+\sigma )} Dynamics and mechanics_4_2_1.PNGLame coefficients 25\text \lambda = \frac{E\sigma}{(1+\sigma)(1-2\sigma)}Longitudinal modulus5\text M_1 = \frac{E(1-\sigma)}{(1+\sigma)(1-2\sigma)} Stress tensor \text t = 2\mu e + \lambda1tr(e) Bulk modulus \text K = \frac{E}{3(1-2\sigma)}Pressure\text -p = Ke_v Shear modulus!\text \mu = \frac{E}{2(1+\sigma)}Transverse stress$\text \tau_T = \mu\theta_{\small sh} Young modulus!\text E = \frac{9\mu K}{\mu + 3K}  Poisson ratio*\text \sigma = \frac{3K - 2\mu}{2(3K+\mu)}Torsional rigidity\text G = C\frac{\phi}{l} Dynamics and mechanics_4_3_1.PNGThin circular cylinder\text C = 2\pi a^3 \mu tThick circular cylinder0C = \frac{1}{2}\mu\pi \left(a^4_2 - a^4_1\right)Arbitrary thin-walled tube\text C = \frac{4A^2\mu t}{P}Long flat ribbon\text C = \frac{1}{3}\mu w t^3Bending moment\text G_b = \frac{EI}{R_c} Dynamics and mechanics_4_4_1.PNG Light beam6\text y = \frac{W}{2EI}\left(l - \frac{x}{3}\right)x^2 Dynamics and mechanics_4_4_2.PNG Heavy beam \text w(x) = EI\frac{d^4y}{dx^4} Free ends\text F_c = \pi^2\frac{EI}{l^2}"Dynamics and mechanics_4_4_4_1.PNG Fixed ends \text F_c = 4\pi^2\frac{EI}{l^2} 1 free end \text F_c = \pi^2\frac{EI}{4l^2}Speed of transverse wave#\text v_t = \sqrt{\frac{\mu}{\rho}}Speed of longitudinal wave#\text v_l = \sqrt{\frac{M_l}{\rho}} In a fluid!\text v_l = \sqrt{\frac{k}{\rho}}Speed of longitudinal (x-axis)3\text v_l^{(x)} = \sqrt{\frac{E}{\rho(1-\sigma^2)}}"Dynamics and mechanics_4_5_3_1.PNG Speed of transverse wave(y-axis))\text v_t^{(y)} = \sqrt{\frac{\mu}{\rho}} Speed of transverse wave(z-axis)9\text v_t^{(z)} = k\sqrt{\frac{Et^2}{12\rho(1-\sigma^2)}}Speed of longitudinal wave!\text v_1 = \sqrt{\frac{E}{\rho}}Torsional wave velocity&\text v_\phi = \sqrt{\frac{\mu}{\rho}}Speed of transverse wave-\text v_t = \frac{ka}{2}\sqrt{\frac{E}{\rho}} In a spring$\text v_1 = \sqrt{\frac{kl}{\rho_l}}On a stretched string#\text v_t = \sqrt{\frac{T}{\rho_l}}On a stretched sheet&\text v_t = \sqrt{\frac{\tau}{\rho_A}}Acoustic impedance\text Z = \sqrt{E'\rho} Wave velocity impedance relation\text Z = \rho v3Calculation based on elastic modulus and wavenumber*\text \mathfrak{U} = \frac{1}{2}E'k^2u^2_02Calculation based on density and angular frequency3\text \mathfrak{U} = \frac{1}{2}\rho \omega^2 u^2_0 Energy flux\text P = \mathfrak{U}vReflection coefficient!\text r = \frac{Z - Z_0}{Z + Z_0}Transmission coefficient\text t = \frac{2Z}{Z+Z_0} Snell's lawZ\text \Large \frac{sin \theta_i}{v_i} = \frac{sin \theta_r}{v_r}= \frac{sin \theta_t}{v_t}Thin rod!\text I_1 = I_2 = \frac{ml^2}{12}!Dynamics and mechanics_3_11_1.PNG Solid sphere"\text I_1=I_2=I_3= \frac{2}{5}mr^2!Dynamics and mechanics_3_11_2.PNGSpherical shell"\text I_1=I_2=I_3= \frac{2}{3}mr^2!Dynamics and mechanics_3_11_2.PNGI1-axis and I2-axis;\text I_1 = I_2 = \frac{m}{4}\left(r^2+\frac{l^2}{3}\right)#Dynamics and mechanics_3_11_4_1.PNGI3-axis\text I_3 = \frac{mr^2}{2}#Dynamics and mechanics_3_11_4_1.PNGI1-axis!\text I_1 = \frac{m(b^2+c^2)}{12}#Dynamics and mechanics_3_11_5_1.PNGI2-axis!\text I_2 = \frac{m(c^2+a^2)}{12}#Dynamics and mechanics_3_11_5_1.PNGI3-axis!\text I_3 =\frac{ m(a^2+b^2)}{12}#Dynamics and mechanics_3_11_5_1.PNGI1-axis and I2-axis=\text I_1 = I_2 = \frac{3}{20}m\left(r^2+\frac{h^2}{4}\right)#Dynamics and mechanics_3_11_6_1.PNGI3-axis\text I_3 = \frac{3}{10}mr^2#Dynamics and mechanics_3_11_6_1.PNGI1-axis \text I_1 = \frac{m(b^2+c^2)}{5}#Dynamics and mechanics_3_11_7_1.PNGI2-axis \text I_2 = \frac{m(c^2+a^2)}{5}#Dynamics and mechanics_3_11_7_1.PNGI3-axis \text I_3 = \frac{m(a^2+b^2)}{5}#Dynamics and mechanics_3_11_7_1.PNGI1-axis\text I_1 =\frac{mb^2}{4}#Dynamics and mechanics_3_11_8_1.PNGI2-axis\text I_2 = \frac{ma^2}{4}#Dynamics and mechanics_3_11_8_1.PNGI3-axis \text I_3 = \frac{m(a^2+b^2)}{4}#Dynamics and mechanics_3_11_8_1.PNG I1-axis and I2-axis \text I_1 = I_2 = \frac{mr^2}{4}#Dynamics and mechanics_3_11_9_1.PNG I3-axis\text I_3 = \frac{mr^2}{2}#Dynamics and mechanics_3_11_9_1.PNG Triangular plate%\text I_3 = \frac{m}{36}(a^2+b^2+c^2)"Dynamics and mechanics_3_11_10.PNGSolid hemisphere\text d = \frac{3r}{8}Hemispherical shell\text d = \frac{r}{2}Sector of disk.\text d = \frac{2}{3}r\frac{sin\theta}{\theta} Arc of circle#\text d = r\frac{sin\theta}{\theta}Arbitrary triangular lamina\text d = \frac{h}{3}Solid cone or pyramid\text d = \frac{h}{4}Solid spherical cap#\text d = \frac{3(2r-h)^2}{4(3r-h)}Shell spherical cap\text d = r - \frac{h}{2} Semi-elliptical lamina\text d = \frac{4h}{3\pi}PNG  IHDR :0PLTE!!!999JJJccccccssse l pHYsgRtIME  RcStEXtAuthorH tEXtDescription !# tEXtCopyright:tEXtCreation time5 tEXtSoftware]p: tEXtDisclaimertEXtWarningtEXtSourcetEXtComment̖tEXtTitle'IDAThoDReIz/pfCdL!"x?YP[k:#=֖Ǔb |B}/b<.>Y~>US{jGqFÈ=FSbqK|V= /|6¼_/_y"2Y#1OTDow\7Pv 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M8I38XJG |r|FlȻ%Y=LIENDB`8:o"l2NA/#;^6M ` Rz  H C * ? 6 :J s   N  B Y+ lh x 3 SP b]!"" 3" `# G$ % 9% v& ' =v( ) Z+ 3,.1 =Thermodynamic properties= 2 .1.1 =Specific volume= 1 .1.2 =Density= 1 .10 =Monatomic gas= 2 .10.1 =Pressure= 1 .10.2 =Equation of state of an ideal gas= 1 .10.3 =Internal energy= 1 .10.4 =Heat capacities= 0 .11 =Thermal expansion= 2 .11.1 =Linear expansion= 1 .11.2 =Superficial expansion= 1 .11.3 =Cubic expansion= 1 .12 =The measurement of temperature= 2 .12.1 =Resistance thermometers= 1 .3 =Liquid - Vapor mixture= 2 .3.1 =The quality x= 1 .3.2 =The moistute content= 1 .3.3 =Specific volume of wet steam at constant pressure= 1 .3.4 =Enthalpy of wet steam at constant pressure= 1 .3.5 =Entropy of wet steam at constant temperature= 1 .5 =Efficiency of Carnot's cycle= 1 .6 =Ideal gas= 2 .6.1 =Ideal gas isobaric transformation= 2 .6.1.1 =Equation= 1 .6.1.2 =Technical work= 0 .6.1.3 =Heat supplied to system= 1 .6.1.4 =Change in internal energy= 1 .6.1.5 =Change in enthalpy= 1 .6.1.6 =Change in entropy= 1 .6.2 =Ideal gas isothermal transformation= 2 .6.2.1 =Equation= 1 .6.2.2 =Technical work= 1 .6.2.3 =Heat supplied to system= 0 .6.2.4 =Change in internal energy= 0 .6.2.5 =Change in enthalpy= 0 .6.2.6 =Change in entropy= 1 .6.3 =Ideal gas isochoric transformation= 2 .6.3.1 =Equation= 1 .6.3.2 =Technical work= 1 .6.3.3 =Heat supplied to system= 1 .6.3.4 =Change in internal energy= 0 .6.3.5 =Change in enthalpy= 1 .6.3.6 =Change in entropy= 1 .6.4 =Ideal gas adiabatic equation= 2 .6.4.1 =Equations= 0 .6.4.2 =Technical work= 1 .6.4.3 =Heat supplied to system= 0 .6.4.4 =Change in internal energy= 1 .6.4.5 =Change in enthalpy= 1 .6.4.6 =Change in entropy= 0 .6.4.7 =Ratio of heat capacities= 1 .6.5 =Equation of state (Ideal gas law)= 1 .6.6 =Specific gas constant= 1 .7 =Van der Waals gas= 2 .7.1 =Equation of state= 1 .7.2 =Critical point= 2 .7.2.1 =Critical temperature= 1 .7.2.2 =Critical pressure= 1 .7.2.3 =Critical molar volume= 1 .8 =Thermodynamic potentials= 2 .8.1 =Internal energy= 0 .8.2 =Enthalpy= 1 .8.3 =Helmholtz free energy= 1 .8.4 =Gibbs free energy= 1 .8.5 =Grand potential= 0 .8.6 =Gibbs-Duhem relation= 0 .9 =Phase transitions= 2 .9.1 =Heat absorbed= 1 .9.2 =Clausius-Clapeyron equation= 0 .9.3 =Gibbs's phase rule= 1 vSpecific volumeam^3/kg0VVolumebm^30mMassckg0b/ca*cb/a\rhoDensityakg/m^30mMassbkg0VVolumecm^30b/ca*cb/ax The qualitya0m_{vap} Mass of vaporbkg0m_{liq}Mass of liquidckg0b/(b+c) a*c/(1-a)b/a-bMMosstute contentM0x The qualityx01-x1-MvSpecific volumeam^30xQualityb0v_gSpecific volume of vaporcm^30v_fSpecific volume of liquiddm^30 b*c+(1-b)*d (a-d)/(c-d) (a-(1-b)*d)/b (a-b*c)/(1-b)hSpecific enthalpyakJ/kg0xQualityb0h_g$Specific enthalpy of saturated vaporckJ/kg0h_f$Specific enthalpy of saturate liquiddkJ/kg0 b*c+(1-b)*d (a-d)/(c-d) (a-(1-b)*d)/b (a-b*c)/(1-b)sSpecific entropyakJ/kgK0xQualityb0s_g#Specific entropy saturated of vaporckJ/kgK0s_f$Specific entropy saturated of liquiddkJ/kgK0 b*c+(1-b)*d (a-d)/(c-d) (a-(1-b)*d)/b (a-b*c)/(1-b)\etaEfficiency of Carnot cyclea%0T_CThe low-temperature reservoirbK0T_HThe high-temperature reservoircK01-b/c(1-a)*cb/(1-a)V_1Volume at initial statusV1m^30V_2Volume at final statusV2m^30T_1Temperature at initial statusT1K0T_2Temperature at final statusT2K0T1/T2*V2V1*T2/T1V1/V2*T2T1*V2/V1qHeatqkJ/kg0C_pHeat capacity, p constantCpkJ/kgK0T_1Temperature at initial statusT1K0T_2Temperature at final statusT2K0 Cp*(T2-T1)T2-q/CpT1+q/Cp\Delta uChange in internal energyukJ/kg0C_vHeat capacity, V constantCvkJ/kgK0T_1Temperature at initial statusT1K0T_2Temperature at final statusT2K0 Cv*(T2-T1)T2-u/CvT1+u/Cv\Delta iChange in enthalpydikJ/kg0C_pHeat capacity, p constantCpkJ/kgK0T_1Temperature at initial statusT1K0T_2Temperature at final statusT2K0 Cp*(T2-T1)T2-di/CpT1+di/Cp\Delta sChange in entropyskJ/kgK0C_pHeat capacity, p constantCpkJ/kgK0T_1Temperature at initial statusT1K0T_2Temperature at final statusT2K0 Cp*ln(T2/T1) T2/e^(s/Cp) T1*e^(s/Cp)p_1Pressure at initial statusp1Pa0V_1Volume at initial statusV1m^30p_2Pressure at final statusp2Pa0V_2Volume at final statusV2m^30p2*V2/V1p2*V2/p1p1*V1/V2p1*V1/p2 l_{kt12}Technical worklkkJ/kg0RSpecific gas constantRkJ/kgK0T TemperatureTK0v_1Volume at initial statusv1m^30v_2Volume at final statusv2m^30 R*T*ln(v2/v1)lk/R/ln(v2/v1) v2/e^(lk/R/T) v1*e^(lk/R/T)l_{kt12}Technical work0\Delta sChange in entropyskJ/kgK0RSpecific gas constantRkJ/kgK0v_1Volume at initial statusv1m^30v_2Volume at final statusv2m^30 R*ln(v2/v1) v2/e^(s/R) v1*e^(s/R)p_1Pressure at initial statusp1Pa0p_2Pressure at final statusp2Pa0T_1Temperature at initial statusT1K0T_2Temperature at final statusT2K0p2*T1/T2p1*T2/T1T2*p1/p2T1*p2/p1 l_{kt12}Technical worklkJ/kg0vVolumevm^30p_1Pressure at initial statusp1Pa0p_2Pressure at final statusp2Pa0 v*(p1-p2) l/(p1-p2)l/v+p2p1-l/vqHeatqkJ/kg0C_vHeat capacity, V constantCvkJ/kgK0T_1Temperature at initial statusT1K0T_2Temperature at final statusT2K0 Cv*(T2-T1) q/(T2-T1)T2-q/CvT1+q/CvqHeat supplied to system \Delta iChange in enthalpydikJ/kg0C_pHeat capacity, p constantCpkJ/kgK0T_1Temperature at initial statusT1K0T_2Temperature at final statusT2K0 Cp*(T2-T1)T2-di/CpT1+di/Cp\Delta sChange in entropyskJ/kgK0C_vHeat capacity, V constantCvkJ/kgK0p_1Pressure at initial statusp1Pa0p_2Pressure at final statusp2Pa0 Cv*ln(p2/p1) p2/e^(s/Cv) p1*e^(s/Cv)pAbsolute pressurepPa0V Volume of gasVm^30T TemperatureTK0\gamma!Ratio of heat capacities(C_p/C_V)0\Delta WWork done on system0 l_{kt12}Technical worklkJ/kg0\gamma!Ratio of heat capacities(C_p/C_v)k0v_1Volume at initial statusv1m^30p_1Pressure at initial statusp1Pa0p_2Pressure at final statusp2Pa0%k/(1-k)*p1*v1*((p2/p1)^((k-1)/k) - 1)\Delta iChange in enthalpydikJ/kg0 l_{kt12}Technical worklkJ/kg0\Delta uChange in internal energyukJ/kg0C_vHeat capacity, V constantCvkJ/kgK0T_1Temperature at initial statusT1K0T_2Temperature at final statusT2K0 Cv*(T2-T1)T2-u/CvT1+u/Cv\Delta iChange in enthalpydikJ/kg0C_pHeat capacity, p constantCpkJ/kgK0T_1Temperature at initial statusT1K0T_2Temperature at final statusT2K0 Cp*(T2-T1)T2-di/CpT1+di/Cp\gammaRatio of heat capacitiesg0C_pHeat capacity, p constantCpkJ/kgK0C_vHeat capacity, V constantCvkJ/kgK0Cp/CvpAbsolute pressurepPa0V Volume of gasVm^30nNumber of molesnmol0 R^{\prime}Universal gas constantRJ/molK8.314472T TemperatureTK0n*R*T/Vn*R*T/p p*V/(R*T) p*V/(n*R)RSpecific gas constantRkJ/kgK0M Molar massMkg0 8.314472/MpPressurepPa0V_m Molar volumeVm^30RSpecific gas constantRkJ/kgK0TAbsolute temperatureTK0aVan der Waals' constants aa0bVan der Waals' constants bb0R*T/(V-b)-a/V^2(p+a/V^2)*(V-b)/T(p+a/V^2)*(V-b)/RT_c Critical temperatureTK0aVan der Waals' constants aa0bVan der Waals' constants bb0RSpecific gas constantRkJ/kgK0 8*a/(27*R*b)p_cCritical pressurepPa0aVan der Waals' constants aa0bVan der Waals' constants bb0 a/(27*b^2)V_{mc}Critical molar volumeVm^30bVan der Waals' constants bb03*bUInternal energy T Temperature SEntropy \muChemical potential VVolume H EnlthalpyHkJ0UInternal energyUkJ0pPressurepN/m^20VVolumeVm^30U+p*VH-p*V(H-U)/V(H-U)/pFHelmholtz free energyFkJ0UInternal energyUkJ0SEntropySkJ/K0T TemperatureTK0U-T*SF+T*S(U-F)/T(U-F)/SGGibbs free energyGkJ0UInternal energyUkJ0T temperatureTK0SEntropySkJ/K0pPressurepN/m^20VVolumeVm^30 U-T*S+p*V G+T*S-p*V (U+p*V-G)/S (U+p*V-G)/T (G+T*S-U)/V (G+T*S-U)/p\PhiGrand potential  L(latent) heat absorbedLkJ0TTemperature of phase changeTK0S_1Entropy at status 1S1kJ0S_2Entropy at status 2S2kJ0 T*(S2-S1) L/(S2-S1) pPressure VVolume SEntropy 1,2 Phase states  PNumber of phase in equilibriumP0FNumber of degrees of freedomF0CNumber of componentsC0C+2-FC+2-PP+F-2