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Interactive PhysicsT M

www.interactivephysics.com

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Interactive Physics™

Version IP2000

για Windows®95 & NT και Mac™OS

Εγχειρίδιο Χρήστη

KnowledgeRevolution

iii

Εισαγωγή....................................................................................xvii!" #$%&" '( Interactive Physics; ............................................................................xvii)*#'"+, µ# &-'. '( #/*#"0$1"( .............................................................................xxv

Κεφάλαιο 1 Ξενάγηση στο Interactive Physics.............................................1-11.1. 2#+"%3%'&4 '( Interactive Physics .............................................................1-11.2. 56µ"(-0/$& µ"&4 %7&4 80(9(µ($:964 µ# &8;, <=µ&'& ..........................1-41.3. >+'7;#96 #%.4 -8(1#$/µ&'(4 80(9(µ($:964 ...........................................1-51.4. 56µ"(-0/$& µ"&4 &8;=4 80(9(µ($:964 ....................................................1-6

?%("/µ& #%.4 %7(- #//0,@(- .............................................................1-656µ"(-0/$& +A+;(- .............................................................................1-7B;;&/= µ#/7C(-4 +A+;(-....................................................................1-8D#'&+$%696 '(- +A+;(- 9# 96µ#$( #++$%6964..................................1-9E&C(0"9µ.4 '64 &0*"+=4 '&*A'6'&4................................................1-10>+'7;#96 '64 80(9(µ($:964............................................................1-12

1.5. F1".'6'#4 µ#'0=9#:% &8. µ"& 80(9(µ($:96 ..........................................1-1256µ"(-0/$& µ#'06'= '&*A'6'&4 .......................................................1-13B;;&/= µ(0@=4 #µ@,%"964 '(- µ#'06'= .........................................1-15>µ@,%"96 1"&%-9µ,':% ....................................................................1-16

1.6. >µ@,%"96 $*%(-4 .......................................................................................1-171.7. B8(C=+#-96 µ"&4 80(9(µ($:964 ............................................................1-181.8. GH-8%(4 >8#H#0/&9'=4 ............................................................................1-18

!0(8(8($696 '64 /#:µ#'0$&4 '64 9-%1#9µ(;(/$&4 .......................1-22)A%1#96 +&" I:0"9µ.4 .....................................................................1-24

1.9. D"& &8;= 80(9(µ($:96 µ# #0/&;#$& #;7/*(- +&" +(-µ8", +&'&;./(- #8";(/3% .................................................................................1-27

J34 %& 16µ"(-0/=9#'# '( 1"+. 9&4 µ(%'7;( ..................................1-2756µ"(-0/$& #0/&;#$:% #;7/*(-........................................................1-2956µ"(-0/$& +(-µ8"3% +&'&;./(- #8";(/3%..................................1-30>+'#;79"µ#4 10&9'60".'6'#4 ............................................................1-35

1.10. J#0$;6K6 ....................................................................................................1-35

Κεφάλαιο 2 Οδηγός για τα Εργαλεία & τους Καταλόγους Επιλογών .........2-12.1. L" /0&µµ74 #0/&;#$:% '(- Interactive Physics.........................................2-1

Περιεχόµενα

M0&µµ74 #0/&;#$:% ':% Windows .....................................................2-1M0&µµ74 #0/&;#$:% MacOS ................................................................2-6I0=96 ':% #0/&;#$:% .........................................................................2-6>0/&;#$& Interactive Physics ..............................................................2-8E&%(%"+= M0&µµ= #0/&;#$:% (µ.%( /"& Windows) ..........................2-8>0/&;#$& >8#H#0/&9$&4......................................................................2-9>0/&;#$& #;7/*(- #+'7;#964 ............................................................2-10>0/&;#$& 93µ&'(4 ............................................................................2-10G;#/*(4 )A%1#96/I:0"9µ.4 ............................................................2-11>0/&;#$& 96µ#$:% +&" #/+(83%.......................................................2-11>0/&;#$& ,0C0:964 ..........................................................................2-13>0/&;#$& 9-%179µ(- .........................................................................2-14

2.2. E&',;(/(" #8";(/3% '(- Interactive Physics .........................................2-17O +&',;(/(4 #8";(/3% Apple (µ.%( 9# MacOS) ...........................2-18L +&',;(/(4 #8";(/3% B0*#$( ........................................................2-18L +&',;(/(4 #8";(/3% >8#H#0/&9$&..............................................2-19L +&',;(/(4 #8";(/3% D"+0.+(9µ(4 ............................................2-20L +&',;(/(4 #8";(/3% N7&96 .........................................................2-22L +&',;(/(4 #8";(/3% B%'"+#$µ#%( ...............................................2-25L +&',;(/(4 #8";(/3% L0"9µ.4 .....................................................2-27L +&',;(/(4 #8";(/3% D7'0696.....................................................2-28L +&',;(/(4 #8";(/3% J0(/0&µµ&'"9µ.4.....................................2-29L +&',;(/(4 #8";(/3% J&0,C-0( ..................................................2-29L +&',;(/(4 #8";(/3% O(=C#"& (µ.%( /"& Windows) ..................2-32L +&',;(/(4 #8";(/3% Apple QuideTM (µ.%( /"& MacOS) ..........2-32

Κεφάλαιο 3 Σώµατα ........................................................................................3-13.1. 56µ"(-0/$& 9:µ,':%..................................................................................3-1

56µ"(-0/$& L0C(/:%$:% +&" !#'0&/3%:% ......................................3-256µ"(-0/$& 8(;-/3%:% +&" +&µ8A;:% 9:µ,':% ...........................3-6

3.2. F1".'6'#4 )3µ&'(4 ....................................................................................3-10B0*"+= C796 +&" 80(9&%&'(;"9µ.4 .................................................3-12B0*"+= '&*A'6'&................................................................................3-13>;&9'"+.'6'& +&" '0"<=....................................................................3-14>8"@. J-+%.'6'&, P(8=, Q;"+. +&" R(0'$( ...................................3-16!&-'.*0(%6 &;;&/= ':% "1"('=':% 8(;;3% &%'"+#"µ7%:% .........3-19

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v

5"&1(*"+= &;;&/= "1"('=':% ':% &%'"+#"µ7%:%............................3-21I0=96 ':% 'A8:% /"& %& &%&@#0C#$'# 9'"4 F1".'6'#4 93µ&'(4 ..3-23I0=96 ':% 'A8:% /"& '( #0/&;#$( #;7/*(- '64 +$%6964 #%.493µ&'(4..............................................................................................3-23

3.3. >µ@,%"96 93µ&'(4 ...................................................................................3-233.4. M#:µ#'0$& 93µ&'(4...................................................................................3-27

I0=96 ':% 'A8:% /"& '6 /#:µ#'0$& 93µ&'(4 ...............................3-30I0=96 'A8:% /"& '(% (0"9µ. '64 /#:µ#'0$&4 93µ&'(4 ...............3-31)-%'#'&/µ7%#4 /"& 8(;A/:%& +&" +&µ8A;& 93µ&'&.....................3-31B;;&/= 9*=µ&'(4 8(;-/3%:% +&" +&µ8A;:% 9:µ,':% µ#&0"Cµ6'"+. '0.8( ..............................................................................3-32B%'"/0&@= '64 /#:µ#'0$&4 8(;-/3%(- = +&µ8A;(- 93µ&'(4 80.4 +&" &8. ,;;#4 #@&0µ(/74........................................................3-36

3.5. )'&C#0(8($696 9:µ,':% µ# '6% B/+-0& ...............................................3-383.6. >;#/*(4 9-/+0(A9#:% µ#'&HA 9:µ,':% ................................................3-38

Κεφάλαιο 4 Σύνδεσµοι....................................................................................4-14.1. !" #$%&" 9A%1#9µ(4; .....................................................................................4-14.2. >$16 9-%179µ:%...........................................................................................4-2

M0&µµ"+($ 9A%1#9µ(" ..........................................................................4-2J#0"9'0(@"+($ 9A%1#9µ("...................................................................4-25-%,µ#"4 +&" 0(874 .............................................................................4-3B0C039#"4 ............................................................................................4-3

4.3. M#%"+74 "1".'6'#4 ':% 9-%179µ:% .............................................................4-3J&0,C-0( F1".'6'#4 ............................................................................4-4>µ@,%"96 +&" &8.+0-K6 9-%179µ:% ................................................4-9L%(µ&9$& 9-%179µ:% .........................................................................4-9)-%'#'&/µ7%#4 /"& 96µ#"&+, 9'("*#$& 9-%179µ:% ........................4-10B+0"<=4 '(8(C7'696 9-%179µ:% .....................................................4-14>;#/*(4 '64 #8"9A%&K64 9-%179µ(- ...............................................4-20I:0"9µ.4 +&" &@&$0#96 9-%179µ:% ...............................................4-21>%#0/(8($696 +&" &8#%#0/(8($696 9-%179µ:% ............................4-22L0"9µ($ 8(;"+.'6'&4 ..........................................................................4-3

4.4. )*("%", .......................................................................................................4-2556µ"(-0/$& 9*("%"(A .........................................................................4-25F1".'6'#4 9*("%"(A .............................................................................4-26

4.5. >;&'=0"& ....................................................................................................4-3156µ"(-0/$& #;&'60$(- ......................................................................4-31F1".'6'#4 #;&'60$(-...........................................................................4-32

4.6. B8(9<#9'=0#4 ............................................................................................4-3556µ"(-0/$& #%.4 &8(9<#9'=0& ........................................................4-36F1".'6'#4 &8(9<#9'=0& .....................................................................4-37

4.7. >;&'=0"& µ# &8.9<#96..............................................................................4-3856µ"(-0/$& #%.4 #;&'60$(- µ# &8.9<#96 .......................................4-39F1".'6'#4 #;&'60$(- µ# &8.9<#96 ....................................................4-40

4.8. J#0"9'0(@"+, #;&'=0"& ............................................................................4-3956µ"(-0/$& 8#0"9'0(@"+(A #;&'60$(-............................................4-40F1".'6'#4 8#0"9'0(@"+(A #;&'60$(-................................................4-41

4.9. J#0"9'0(@"+($ &8(9<#9'=0#4 ..................................................................4-4356µ"(-0/$& 8#0"9'0(@"+(A &8(9<#9'=0& ......................................4-43F1".'6'#4 8#0"9'0(@"+(A &8(9<#9'=0& ..........................................4-44

4.10. !0(*&;$#4 ...................................................................................................4-4656µ"(-0/$& 9-9'=µ&'(4 '0(*&;$&4.................................................4-46F1".'6'#4 9-9'=µ&'(4 '0(*&;$&4.....................................................4-48

4.11. M0&%,S"&.....................................................................................................4-4956µ"(-0/$& /0&%&S"3% .....................................................................4-50F1".'6'#4 /0&%&S"(A ..........................................................................4-56

4.12. P,<1(" .........................................................................................................4-5856µ"(-0/$& µ"&4 0,<1(- ..................................................................4-58F1".'6'#4 0,<1(- ...............................................................................4-59

4.13. 5"&*:0"9'74 ...............................................................................................4-6056µ"(-0/$& 1"&*:0"9'= ....................................................................4-60F1".'6'#4 1"&*:0"9'= ........................................................................4-62

4.14. 5A%&µ6 ........................................................................................................4-6356µ"(-0/$& 1A%&µ64 .........................................................................4-63F1".'6'#4 1A%&µ64 .............................................................................4-65

4.15. P(8= ............................................................................................................4-6756µ"(-0/$& 0(8=4 .............................................................................4-67F1".'6'#4 0(8=4 .................................................................................4-67

4.16. >%#0/(8("6'74............................................................................................4-6856µ"(-0/$& #%.4 #%#0/(8("6'= ........................................................4-70F1".'6'#4 #%#0/(8("6'= .....................................................................4-71

4.17. E"%6'=0#4 ...................................................................................................4-71

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56µ"(-0/$& +"%6'=0& ........................................................................4-72F1".'6'#4 +"%6'=0& ............................................................................4-72

4.18. B0C039#"4 ..................................................................................................4.7556µ"(-0/$& ,0C0:964.......................................................................4-75F1".'6'#4 ,0C0:964...........................................................................4-79D7'0696 1-%,µ#:% &%'$10&964 9'"4 &0C039#"4 ...........................4-81

4.19. B0C039#"4 #/+(8=4 ...................................................................................4-8356µ"(-0/$& µ"&4 ,0C0:964 +&'&+.0-@64 #/+(8=4 ......................4-8356µ"(-0/$& ,0C0:964 +&µ8A;64 #/+(8=4 .....................................4-8456µ"(-0/$& &0C039#:% #/+(8=4 &8. 9'("*#$& .............................4-88>8"9A%&K6 +&" B8.98&96 #%.4 9'("*#$(- #/+(8=4 ......................4-92F1".'6'#4 ,0C0:964 #/+(8=4............................................................4-93D7'0696 1-%,µ#:% &%'$10&964 9# &0C039#"4 #/+(8=4 ...............4-94L0"9µ.4 '64 /#:µ#'0$&4 µ"&4 +&µ8A;64 #/+(8=4 .........................4-96B;;&/= 9*=µ&'(4 µ"&4 +&µ8A;64 #/+(8=4 µ# &0"Cµ6'"+. '0.8( ..................................................................................................4-98B%'"/0&@= µ"&4 +&µ8A;64 #/+(8=4 80(4 +&" &8. ,;;#4 #@&0µ(/74.........................................................................................4-101

Κεφάλαιο 5 Έξυπνος Επεξεργαστής.............................................................5-15.1. )A%1#96 9'("*#$:% +&" *:0"9µ.4 9-%179µ:%..........................................5-1

G;#/*(4 '64 +$%6964 &%'"+#"µ#%(- +&', '6 1"&1"+&9$& 7%:964 ...5-3G;#/*(4 '64 +$%6964 &%'"+#"µ7%:% ...................................................5-5

5.2. )A09"µ( +&" )'0(@= 9-%1#1#µ7%:% 9:µ,':% .........................................5-6E,%(%'&4 +;"+ +&" 9A0(%'&4 .............................................................5-9E;#$1:µ& )6µ#$:% +&" E;#$1:µ& #0/&;#$:% #;7/*(- ...................5-9

5.3. E&'&%.696 '(- GH-8%(- >8#H#0/&9'=..................................................5-11J&0,1#"/µ& 8(1"(A 0(µ8.' .............................................................5-12J&0,1#"/µ& 9-%1#9µ(;(/$&4 ...........................................................5-13)'0(@= 9:µ,':%...............................................................................5-15J.'# *069"µ(8("#$'&" ( GH-8%(4 >8#H#0/&9'=4; .........................5-18I0=96 '(- GH-8%(- >8#H#0/&9'= µ# '& 9*("%",..........................5-20!" /$%#'&" 9'6% 8#0$8':96 8(- &8('A*#" ( GH-8%(4 >8#H#0/&9'=4;....................................................................................5-20G;#/*(4 '64 &+0$<#"&4 '64 #8#H#0/&9$&4 80,H#:%.......................5-22

Κεφάλαιο 6 Χώρος εργασίας .........................................................................6-16.1. R-9"+, &%'"+#$µ#%& +&" &%'"+#$µ#%& '64 1"#8&@=4 ................................6-16.2. >8";(/74 N7&964 .........................................................................................6-2

D#'&+$%696 '64 N7&4 9'( D"+0.+(9µ( ...........................................6-3D#/7%C-%96-)µ$+0-%96 ......................................................................6-4>µ@,%"96 #0/&;#$:% +&" #0/&;#$:% #;7/*(- '(- *30(- #0/&9$&4................................................................................................6-7>µ@,%"96 ':% &H.%:% * +&" K ........................................................6-10>µ@,%"96 ':% *&0,+:% ...................................................................6-10>µ@,%"96 /0&µµ3% 8;7/µ&'(4 ........................................................6-11>-C-/0,µµ"96 &%'"+#"µ7%:% 9'( 8;7/µ& .......................................6-12>-C-/0,µµ"96 &%'"+#"µ7%:% 9# 93µ&'& ........................................6-13>µ@,%"96 '64 M0&µµ=4 9-%'#'&/µ7%:% ..........................................6-13T /0&µµ= +&',9'&964 ......................................................................6-15

6.3. B0"Cµ($ +&" µ(%,1#4 .................................................................................6-17!A8(" +&" µ(%,1#4.............................................................................6-19

6.4. L0"9µ.4 ':% 8&0&µ7'0:% '(- µ"+0.+(9µ(-.........................................6-20O&0A'6'& ............................................................................................6-20B%'$9'&96 '(- &70& ..........................................................................6-22T;#+'0(9'&'"+=.................................................................................6-235A%&µ6 8#1$(-% .................................................................................6-24

6.5. !0(8(8($696 &%'"+#"µ7%:% ......................................................................6-27>8";(/= 8(;;3% &%'"+#"µ7%:% ........................................................6-27>µ@,%"96 .;:% ':% +0-µµ7%:% &%'"+#"µ7%:%..............................6-28B8(+(8=, B%'"/0&@=, >8"+.;;696 +&" 5"&/0&@=/B8&;("@= ......6-29B%&$0#96 '64 '#;#-'&$&4 #%70/#"&4 8(- #+'#;79&'#......................6-30D#'&+$%696 #%.4 &%'"+#"µ7%(- ........................................................6-31)'0(@= &%'"+#"µ7%:%........................................................................6-32D#'&@(0, #%.4 &%'"+#"µ7%(- 8$9: = µ80(9', .............................6-35

6.6. I0=96 8&0&CA0:% /"& &;;&/= ':% "1"('=':% #%.4 &%'"+#"µ7%(- ......6-36

Κεφάλαιο 7 ∆ιεπαφές προσοµοίωσης ...........................................................7-17.1. D#'06'74 ......................................................................................................7-1

!" µ8(0#$ %& µ#'06C#$ µ# '(-4 µ#'06'74; ..........................................7-1D7'0696 &;;6;#8"10,9#:% µ#'&HA 1A( &%'"+#"µ7%:% ...................7-356µ"(-0/$& µ#'06'3%..........................................................................7-4

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>µ@,%"96 9# µ(0@= K6@"&+=, /0&@"+= +&" :4 0&<1./0&µµ& .......7-4!0(8(8($696 µ#'06'= /"& #µ@,%"96 '0(8(8("6µ7%:% "1"('=':% .7-6B;;&/= C7964 +&" µ#/7C(-4 '(- µ#'06'= ..........................................7-6>8";#+'"+= #µ@,%"96 +&" &8.+0-K6 "1"('=':%...............................7-8B;;&/= '64 +;$µ&+&4 '64 /0&@"+=4 8&0,9'&964 (min +&" max) ..7-8B;;&/= '(- *03µ&'(4 ':% /0&µµ3% 9# µ"& /0&@"+= 8&0,9'&96..........................................................................................7-11)A/+0"96 &8('#;#9µ,':% 8(;;&8;3% 80(9(µ("39#:% ..............7-11

7.2. >0/&;#$& #;7/*(- ......................................................................................7-1456µ"(-0/$& #0/&;#$:% #;7/*(-........................................................7-17!0(8(8($696 C7964 +&" µ#/7C(-4'(- #0/&;#$(- #;7/*(- ......................................................................7-17>$16 #0/&;#$:% #;7/*(- +&" "1".'6'#4 ............................................7-18

7.3. E(-µ8", +&'&;./(- #8";(/3% ................................................................7-22)A%1#96 8(;;3% #//0&@3% µ# +(-µ8", +&'&;./(- #8";(/3% ....7-24

7.4. 5"&%A9µ&'& ................................................................................................7-26>µ@,%"96 1"&%-9µ,':% ....................................................................7-27J0(9&0µ(/= '(- µ=+(-4 ':% 1"&%-9µ,':%...................................7-28PACµ"96 #8";(/3% #µ@,%"964 1"&%-9µ,':% ..................................7-29>µ@,%"96 1"&%-9µ,':% '64 1A%&µ64 &%'$10&964 ,0C0:964 ......7-31

7.5. E#$µ#%( .......................................................................................................7-32I0=96 '(- #0/&;#$(- E#$µ#%( .........................................................7-32>8";(/= &%'"+#"µ7%(- +#"µ7%(-.......................................................7-325"&/0&@= #%.4 &%'"+#"µ7%(- +#"µ7%(-............................................7-33>"9&/:/= %7(- +#"µ7%(- ...................................................................7-33B;;&/= /0&µµ&'(9#"0,4, µ#/7C(-4 +&" 9'-; +#"µ7%(-.................7-34L%(µ&9$& &%'"+#"µ7%:%....................................................................7-35

7.6. >"+.%#4 .......................................................................................................7-3556µ"(-0/$& &%'"+#"µ7%:% #"+.%&4...................................................7-35>8"9A%&K6 &%'"+#"µ7%:% #"+.%&4 9# 93µ&'&................................7-36

Κεφάλαιο 8 Εκτέλεση προσοµοιώσεων .........................................................8-18.1. >+'7;#96 µ"&4 80(9(µ($:964....................................................................8-18.2. 5"&+(8= µ"&4 80(9(µ($:964 .....................................................................8-28.3. I0=96 ':% #0/&;#$:% #;7/*(- '(- +&9#'(@3%(-...................................8-3

O6µ&'"+= #+'7;#96 80(9(µ($:964 ....................................................8-4

Q8#08=1696 8;&"9$:% ........................................................................8-4B%&8&0&/:/= '64 80(9(µ($:964 80(4 '& 8$9:.............................8-5D#',<&96 9# 7%& 9-/+#+0"µ7%( 8;&$9"( ..........................................8-6>8"',*-%96 '64 &%&8&0&/:/=4.........................................................8-7J0(9:0"%= 8&A96 ...............................................................................8-8

8.4. J0('"µ=9#"4..................................................................................................8-9>8#H#0/&9$& &%'"+#"µ7%:% :4 8#0"/0,µµ&'& = :4 &%'"+#$µ#%& .8-10)A09"µ( '(- 1"&%A9µ&'(4 '64 '&*A'6'&4.......................................8-10B-'.µ&'(4 -8(;(/"9µ.4 &0*"+3% 9-%C6+3%.................................8-10B8(@-/= #8#H#0/&9$&4 870&% ':% &0*"+3% 9-%C6+3%................8-11B;;&/= 10(µ7& 9# 9=µ& 9'(8 +&', '6 1",0+#"& #+'7;#964 '6480(9(µ($:964....................................................................................8-11B%&+A+;:96 (<0./*(4) .'&% '( +&9#'.@:%( #$%&" /#µ,'( .........8-12B-'.µ&'#4 #H$9:9#"4 96µ#$(- 9'6% J0(9,0'696 &%'"+#"µ7%(-...8-12B8(C=+#-96 80('"µ=9#:% ................................................................8-13

8.5. >//0&@= µ"&4 80(9(µ($:964 ..................................................................8-14B8&"'=9#"4 µ%=µ64 µ"&4 #//0&@=4 ..................................................8-14>+'7;#96 µ"&4 80(9(µ($:964 870& &8. .,'" µ8(0#$ %& #//0&@#$ ........................................................................................8-15I0=96 ':% 0-Cµ$9#:% µ"&4 -8,0*(-9&4 80(9(µ($:964 &%'$ µ"&4,;;64 %7&4 ..........................................................................................8-17

8.6. >+'7;#96 80(/0&µµ&'"9µ3% ....................................................................8-178.7. J#0"<,;;(%'&/!0.8(" 80(9(µ($:964.....................................................8-18

J#0"<,;;(% #8#H#0/&9$&4 ................................................................8-18J#0"<,;;(% #+'7;#964.......................................................................8-19

8.8. )A9'6µ& &%&@(0,4 ...................................................................................8-20I0=96 ':% 9-9'6µ,':% &%&@(0,4 .................................................8-215"&/0&@= 9-9'=µ&'(4 &%&@(0,4 ....................................................8-23N7&96 µ"&4 80(9(µ($:964 &8. '( 9A9'6µ& &%&@(0,4 '64 µ,S&4 ............................................................................................8-24

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9.7. B%'&;;&/= 1#1(µ7%:% 9# 80&/µ&'"+. *0.%( µ# #H:'#0"+74 #@&0µ(/74...................................................................................................9-14

B%'&;;&/= 1#1(µ7%:% µ79: Apple Events = DDE ........................9-16

Κεφάλαιο 10 Xρήση τύπων.............................................................................10-110.1. D(%,1#4 9# 'A8(-4 ...................................................................................10-1

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Παράρτηµα Α Τεχνικές Πληροφορίες...............................................................Α-1B.1. E&;A'#06 &H"(8($696 '64 1"&C79"µ64 µ%=µ64.........................................B-1

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Παράρτηµα Γ Χρήσιµες συµβουλές και συντοµεύσεις ....................................Γ-1M.1. I0=96 8;=+'0:% 1"&@(0(8($6964 ...........................................................M-1

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constraintforce(n).x

constraintforce(n).y

|constraintforce(n)|

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point[4].force.x

point[4].force.y

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Point[3]

Point[4]

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10.3. X-.'/ ()*0" B47 (/" (-!*!*!5/'/7"(4@%4µ8"0"

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if (time < 100) mass = 20000 - 100*timeelse mass = 10000

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if (time<100, 20000-100*time, 10000)

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if (time<100, 5000, 0)

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x: body[3].vertex[2].xy: body[3].vertex[2].y

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Leslie Fox: Numerical Solution of Ordinary and Partial DifferentialEquations, Addison-Wesley, 1962.

Robert W. Hornbeck: Numerical Methods, Prentice-Hall, 1961.

Gene H. Golub and James M. Ortega: Scientific Computing and Differential Equations: An Introduction to Numerical Methods, Academic Press, Inc., 1992.

C. William Gear: Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, 1991.

Germund Dahlquist and Ake Bjorck: Numerical Methods, Prentice-Hal, 1974.

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Thomas R. Kane and Dvid A. Levinson: Dynamics: Thoery and Applications, McGraw-Hill Publishing Company, 1985.

Ferdinand P. Beer and E. Russel Johnston, Jr.: Vector Mechanics forEngineers, McGraw-Hill Book Company, 1977.

Nicholas P. Chronis: Mechanisms and Mechanical Devices Sourcebook. McGraw-Hill Inc., 1991.

Edward J. Haug: Computer-Aided Kinematics and Dynamics of Mechanical Systems, Allyn and Bacon, 1989.

A. Bedford and W. Fowler: Engineering Dynamics-Mechanics, Addison-Wesley Publishing Company, 1995.

R. E. Roberson and R. Schwertassek: Dynamics of Multybody Systems, Springer-Verlag, 1988.

R. C. Hibbeler: Engineering Mechanics-Dynamics, Macmillan Publishing Co., Inc., 1983.

J. L. Meriam and L. G. Kraige: Engineering Mechanics, Volume 2, Dynamics, John Wiley and Sons, 1987.

David J. McGill and Wilton W. King: Engineering Mechanics:an introduction to Dynamics, Brooks/Cole Engineering Division, 1984.

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T! ".54! !ffset ."/*&$,3./ &! 5/%0(*µ# "!( ".$/,-./ &'0 &$,-!(*#5/#µ)$39*' (.x, .y, .r) &!( *'µ.4!( *&# "2#4*/# &!( BA (*;*&'µ##0#3!$%+) &!( *8µ#&!+ *&! !"!4! .40#/ "$!*#$&'µ,0! &! *'µ.4!(&!"/1,+ *(0&.&#6µ,0.+).

K0 &! *'µ.4! .40#/ "$!*#$&'µ,0! *&! 3)0&!, &! ".54! !ffset/*!5(0#µ.4 µ. &! ".54! .p &!( *'µ.4!(. @'2#57:

point[n] .p.x = point [n] .offset.x

S#/ "#$!µ!49+ 6/# &# ".54# .y 1#/ .r.

P! ".54! body ."/*&$,3./ &! *8µ# *&! !"!4! .40#/ "$!*#$&'µ,0!&! *'µ.4!. ?.$/**)&.$.+ "2'$!3!$4.+ 6/# &# *-.&/1% ".54# :#>$.4&. *&# "?.54# *9µ%&90" *&' *.245# B-7.

P! ".54! force ."/*&$,3./ ,0# 5/%0(*µ# "!( #0&/"$!*9".;./ &'5;0#µ' "!( #*1.4&#/ *&! *'µ.4! - 1#/ 6/# &'0 #1$4>./#, &' 5;0#µ'"!( #*1.4&#/ *&! *8µ# *&' :,*' "!( >$4*1.&#/ &! *'µ.4!. P#*&!/-.4# 540!0&#/ *&# "2#4*/# &90 "#61)*µ/90 *(0&.&#6µ,090,#0.<%$&'&# #") &! "!; .40#/ "$!*#$&'µ,0! &! *'µ.4!.

p,v,a

M%$!$:+)-' (*ffset)

A<µ! (body)

=#2!µ' (force)

7%/,! -92/1-µC2

K(&7 .40#/ ' &$,-!(*# #")*&#*' #0%µ.*# *&# 5;! %1$# &!(*(05,*µ!(. C/# 0# >$.4&. &! &$,-!0 µ71!+ .0)+ .2#&'$4!(, :#"$,"./ 0# 58*.&.:

constraint [3] .length µ27%' 6-1><6µ%- #3

A(&,+ .40#/ !/ &$,-!(*.+ &/µ,+ "!( #3!$!;0 &' 5/#3!$% :,*'+,&#-;&'&#+ 1#/ ."/&%-(0*'+ #0%µ.*# *&# 5;! %1$# &!(*(05,*µ!(. S%:. ,0# #") #(&% &# ".54# ."/*&$,3./ &'0 &/µ7 .0)+.45!(+ 5/#0;*µ#&!+.

H/ &/µ,+ #(&,+ µ.&$80&#/ *&! *;*&'µ# #0#3!$%+ &!( *(05,*µ!(.L &/µ7 x µ.&$%&#/ 1#&% µ71!+ &'+ 6$#µµ7+ "!( *(05,./ &# 5;!%1$# .0)+ *(05,*µ!( *'µ.4! "$!+ *'µ.4!.

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constraint [3] .dv.x

K%:. ,0# #") #(&% &# ".54# ."/*&$,3./ &! *'µ.4! "!( 2./&!($6.49+ %1$! .0)+ *(05,*µ!(. P! ".54! p1 ."/*&$,3./ &! *'µ.4! "!(5'µ/!($67:'1. "$8&!. @.4&. &! :,µ# "?.54# *'µ.490" *&' *.245#B-9 6/# &# *-.&/1% ".54#.

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B-10 7!&;&$'µ! B-G!0'µ!$)63 M"<--!

M36*. (length)

dp, dv, da

p1, p2

=#2!µ' (force)

B.5 ?%"%-$1. B-11

7%/,! %(:/*9

K(&7 .40#/ ' &/µ7 "!( .µ3#04=.&#/ *&!0 %<!0# x 7 &'0 &.&µ'µ,0'µ/#+ 6$#3/17+ "#$%*&#*'+ .<)5!(.

!utput [6].x ."µ2 4%- )µ?,1+@).," 6.%1 #G%1, x .3' )G8>%- 6

A(&,+ .40#/ !/ &/µ,+ "!( .µ3#04=!0&#/ *&!0 %<!0# y µ/#+6$#3/17+ "#$%*&#*'+ .<)5!(.

!utput [6].y1 ."µ2 4%- )µ?,1+@).," 6.%1 #G%1, y1 .3' )G8>%- 6!utput [6].y2 ."µ2 4%- )µ?,1+@).," 6.%1 #G%1, y2 .3' )G8>%- 6!utput [6].y3 ."µ2 4%- )µ?,1+@).," 6.%1 #G%1, y3 .3' )G8>%- 6!utput [6].y4 ."µ2 4%- )µ?,1+@).," 6.%1 #G%1, y4 .3' )G8>%- 6

B.5. T%"%-$1.

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x

y1, y2, y3, y4

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^ (5;0#µ') a ^ b ab

> a > b 1 7 0< a < b 1 7 0>= a >= b 1 7 0<= a <= b 1 7 0= (/*!;&#/) a = b 1 7 0<> (5.0 /*!;&#/) a <> b 1 7 0

H/ &.2.*&,+ #(&!4 5,-!0&#/ #$/:µ!;+ *&/+ ./*)5!(+ &!(+. S%&/&,&!/! *'µ#40./ "9+ 5.0 µ"!$.4&. 0# ./*%6.&. *&!/-.4# &;"90 "!(5.0 .40#/ #$/:µ!4.

F%:!+:

body [3] + point [3] ,>/1,.3 3 4*86=)63 6Dµ,.%' µ) 63µ)+%

body [3] .p - 34.5 ,>/1,.3 3 ,?,+*)63 ,*"=µ%/ ,48 >"#1-6µ,

point [7] .v + body [3] ,>/1,.3 3 4*86=)63 >",1/6µ,.%' µ) 6Dµ,

body [3] .p > 44.0 ,>/1,.3 3 6/&7*"63 >",1/6µ,.%' µ) ,*"=µ8

B-12 7!&;&$'µ! B-G!0'µ!$)63 M"<--!

B.5 ?%"%-$1. B-13

B9*&):

body [3].p.x + point [3].p.x

body [3].p.x - 34.5

point [7] .v.y - body [3] .v.y

body [3] .p.y > 44.0

body[3] .p.y = 44.0

body [3] .p.y !=44.0

?#4$0./ ,0#0 #$/:µ) 1#/ ."/*&$,3./ &!0 #$0'&/1) #(&!; &!(#$/:µ!;.

?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ &! %:$!/*µ#.

?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ &' 5/#3!$%.

?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ &! 6/0)µ.0!.

?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ &! "'241!.

?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ &! (")2!/"! &'+ 5/#4$.*'+&'+ "$8&'+ &/µ7+ µ. &' 5.;&.$'.

?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ &'0 "$8&' &/µ7 (A9µ,0'*&' 5;0#µ' &'+ 5.;&.$'+ &/µ7+.

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?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ &'0 &/µ7 1 #0 ' "$8&' &/µ7.40#/ µ/1$)&.$' #") &' 5.;&.$' &/µ7. @/#3!$.&/1%, ."/*&$,3./&/µ7 0.

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?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ &'0 &/µ7 1 #0 ' "$8&' &/µ7.40#/ µ/1$)&.$' #") 7 4*' µ. &' 5.;&.$' &/µ7. @/#3!$.&/1%,."/*&$,3./ &/µ7 0.

?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ &'0 &/µ7 1 #0 !/ 5;! &/µ,+.40#/ 4*.+. @/#3!$.&/1%, ."/*&$,3./ &/µ7 0. H &.2.*&7+ #(&)+ 5.0#"!545./ 1#µ4# &/µ7 *&! #$/*&.$) µ,$!+ &'+ .<4*9*'+. O &;"!+:

body [3] .p.y = 3

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?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ &'0 &/µ7 1 #0 !/ 5;! &/µ,+5.0 .40#/ 4*.+. @/#3!$.&/1%, ."/*&$,3./ &/µ7 0.

7&*$%&!):$'$! $%"%-$<2

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B.5 ?%"%-$1. B-15

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and (time>1, body[2].v.y>10)

."/*&$,3./ &'0 &/µ7 1 .%0 &! time .40#/ µ.6#2;&.$! #") 1 1#/ '&#-;&'&# y &!( *8µ#&!+ #2 .40#/ µ.6#2;&.$' #") 10.

?#4$0./ ,0# 5/%0(*µ# 1#/ ."/*&$,3./ &' 6904# "!( 5'µ/!($6.4 &!5/%0(*µ# *&! ."4".5! &'+ *(0&.&#6µ,0'+. C/# "#$%5./6µ#, #0 ,0#*8µ# ,-./ &#-;&'&# 0 *&'0 1#&.;:(0*' x 1#/ 10 *&'0 1#&.;:(0*'y, &! *8µ# ,-./ &#-;&'&# "!( >$4*1.&#/ *&'0 1#&.;:(0*' 90 7 "/2*&! ."4".5! &'+ *(0&.&#6µ,0'+. H &;"!+

angle(body[3].v)

:# .",*&$.3. &'0 &/µ7 "/2.

?#4$0./ ,0#0 #$/:µ) 1#/ ."/*&$,3./ &! #0&4*&$!3! *(0'µ4&!0!&!( #$/:µ!;. H/ &/µ,+ "!( ."/*&$,3!0&#/ ".$/,-!0&#/ *&'0124µ#1# [0,"].

?#4$0./ ,0#0 #$/:µ) 1#/ ."/*&$,3./ &! #0&4*&$!3! 'µ4&!0! &!(#$/:µ!;. H/ &/µ,+ "!( ."/*&$,3!0&#/ ".$/,-!0&#/ *&'0 124µ#1# [-"/2, "/2].

?#4$0./ ,0#0 #$/:µ) 1#/ ."/*&$,3./ &'0 #0&4*&$!3' .3#"&!µ,0'&!( #$/:µ!;. H/ &/µ,+ "!( ."/*&$,3!0&#/ ".$/,-!0&#/ *&'0124µ#1# [-"/2, "/2].

B-20 7!&;&$'µ! B-G!0'µ!$)63 M"<--!

abs(x)

and(x,y)

angle(v)

acos(x)

asin(x)

atan(x)

B.6 A92!&$3-%). B-21

?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ &'0 #0&4*&$!3' .3#"&!µ,0'&!( y/x. L *(0%$&'*' #(&7 .40#/ -$7*/µ' 5/)&/, #0&4:.&# µ. &'*(0%$&'*' atan, µ"!$.4 0# 58*./ µ/# 6904# *&! *9*&)&.&#$&'µ)$/!. H/ &/µ,+ "!( ."/*&$,3!0&#/ ".$/,-!0&#/ *&'0124µ#1# [-", "].

?#4$0./ ,0#0 #$/:µ) 1#/ ."/*&$,3./ &! µ/1$)&.$! #1,$#/! ()-/µ/1$)&.$! #") &!0 #$/:µ)).

?#4$0./ ,0#0 #$/:µ) 1#/ ."/*&$,3./ &! *(0'µ4&!0! &!( #$/:µ!;.

?#4$0./ ,0#0 #$/:µ) 1#/ ."/*&$,3./ &! .1:.&/1) #"!&,2.*µ# &!(#$/:µ!; (e (A9µ,0! *&'0 &/µ7 &!( #$/:µ!;).

?#4$0./ ,0#0 #$/:µ) 1#/ ."/*&$,3./ &!0 µ.6#2;&.$! #1,$#/! ()-/µ.6#2;&.$! #") &!0 #$/:µ)).

@,-.&#/ &$./+ #$/:µ!;+. K0 ' &/µ7 &!( "$8&!( #$/:µ!; (x) 5.0.40#/ 4*' µ. 0, &)&. ."/*&$,3./ &'0 &/µ7 &!( 5.;&.$!( #$/:µ!; (y).@/#3!$.&/1%, ."/*&$,3./ &'0 &/µ7 &!( &$4&!( #$/:µ!; (z). C/#"#$%5./6µ#:

if (time>1, 20, 0

."/*&$,3./ &'0 &/µ7 20 #0 &! time .40#/ µ.6#2;&.$! #") 1,5/#3!$.&/1% ."/*&$,3./ &/µ7 0.

P("/1%, &! "$8&! )$/*µ# µ/#+ *(0%$&'*'+ if .40#/ µ/# *-,*'()"9+ x>y), 7 µ/# 2!6/17 "$%<' ()"9+ and(a, b). M"!$.4&. 0#.0*9µ#&8*.&. 5'28*./+ if, 1%0!0&#+ #0#5$!µ/17 -$7*' %2290*(0#$&7*.90 if, 9+ )$/*µ#.

atan2(x,y)

ceil(x)

cos(x)

exp(x)

floor(x)

if(x,y,z)

C/# "#$%5./6µ#, >2,".&. "#$#1%&9 ,0# *-.&/1% #"2) &µ7µ#185/1# C &! !"!4! ."/*&$,3./ &! µ,6/*&! &$/80 #$/:µ80 a, b 1#/c:

{if (a>b) {

if (a>c)return a ;

elsereturn c ;

}else {

if (b>c)return b ;

elsereturn c ;

}}

B&' µ#:'µ#&/17 628**# &!( Interactive Physics, &! "#$#"%091!µµ%&/ µ"!$.4 0# µ.&#&$#".4 *. µ/# µ)0! 6$#µµ7 9+#1!2!;:9+:

if(a>b, if (a>c,a,c, if (b>c,b,c))

?#4$0./ ,0#0 #$/:µ) 1#/ ."/*&$,3./ &! 3(*/1) 2!6%$/:µ! &!(#$/:µ!;.

?#4$0./ ,0#0 #$/:µ) 1#/ ."/*&$,3./ &! 2!6%$/:µ! >%*'+ 10 &!(#$/:µ!;.

?#4$0./ ,0# 5/%0(*µ# 1#/ ."/*&$,3./ &! µ,6.:!+ &!(5/#0;*µ#&!+. T! #"!&,2.*µ# .40#/ &! 45/! µ. #(&) &!( |v|.

?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ &! µ.6#2;&.$! #") &!(+

B-22 7!&;&$'µ! B-G!0'µ!$)63 M"<--!

ln(x)

log(x)

mag(v)

B.6 A92!&$3-%). B-23

#$/:µ!;+ #(&!;+. C/# "#$%5./6µ#:

max(body[1].a.x , body[2].a.x)

."/*&$,3./ &' µ.6#2;&.$' x ."/&%-(0*' .4&. &!( *8µ#&!+ #1 .4&.&!( *8µ#&!+ #2.

K0 :,2.&. 0# >$.4&. &! µ,6/*&! &$/80 #$/:µ80 a, b 1#/ c, µ"!$.4&.0# -$'*/µ!"!/7*.&. #0#5$!µ/1% &/+ *(0#$&7*./+ max( ) 9+#1!2!;:9+:

max (max(a,b),c)

?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ &! µ/1$)&.$! #") &!(+ 5;!.C/# "#$%5./6µ#:

minbody[1].v.x, body[2].v.x)."/*&$,3./ &' µ/1$)&.$' &#-;&'&# x .4&. &!( *8µ#&!+ #1 .4&. &!(*8µ#&!+ #2.

D"9+ *&' *(0%$&'*' max( ), µ"!$.4&. 0# >$.4&. &! .2%-/*&!&$/80 #$/:µ80 a, b 1#/ c 9+ .<7+:

min (min(a,b),c)mod(x,y)

?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ &! (")2!/"! "!( "$!1;"&./#") &' 5/#4$.*' &'+ "$8&'+ &/µ7+ µ. &' 5.;&.$'.

F!6/17 "$%<' NOT. ?#4$0./ ,0#0 #$/:µ) 1#/ ."/*&$,3./ &'0 &/µ70 .%0 ! #$/:µ)+ 5.0 .40#/ 0. @/#3!$.&/1% ."/*&$,3./ &/µ7 1.

F!6/17 "$%<' OR. ?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ &'0 &/µ71 .%0 &!(2%-/*&!0 ,0#+ #") &!(+ #$/:µ!;+ 5.0 .40#/ 0.E"/*&$,3./ 0 .%0 1#/ µ)0! 1#/ !/ 5;! #$/:µ!4 .40#/ 0. C/#"#$%5./6µ#:

or(time>1, body[2].v.r>10)

."/*&$,3./ &'0 &/µ7 1 #0 &! time .40#/ µ.6#2;&.$! #") 1 7 .%0 '690/#17 &#-;&'&# &!( *8µ#&!+ #2 .40#/ µ.6#2;&.$' #") 10.

max(x,y)

min(x,y)

not(x)

or(x,y)

mi*/(x,y)

?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ &'0 &/µ7 &!( x (A9µ,0'*&'0 &/µ7 &!( y· 5'2#57, ."/*&$,3./ xy.

E"/*&$,3./ &'0 &/µ7 &!( ".

E"/*&$,3./ µ/# &(-#4# &/µ7 #") 0 ,9+ 1.

?#4$0./ ,0#0 #$/:µ) 1#/ ."/*&$,3./ &'0 &/µ7 1 .%0 ! #$/:µ)+ .40#/µ.6#2;&.$!+ #") 7 4*!+ µ. 0. @/#3!$.&/1%, ."/*&$,3./ &'0 &/µ7 -1.

?#4$0./ ,0#0 #$/:µ) 1#/ ."/*&$,3./ &! 'µ4&!0! &!( #$/:µ!;.

?#4$0./ ,0#0 #$/:µ) 7 ,0# 5/%0(*µ#. A0 ' .4*!5!+ .40#/ #$/:µ)+,."/*&$,3./ &! &.&$%690! (x*x) &!( #$/:µ!;. A0 ' .4*!5!+ .40#/5/%0(*µ#, ."/*&$,3./ &! %:$!/*µ# &!( ".54!( .x &.&$#690/*µ,0!1#/ &! ".54!( .y &.&$#690/*µ,0!.

?#4$0./ ,0#0 #$/:µ) 1#/ ."/*&$,3./ &'0 &.&$#690/17 $4=# &!(#$/:µ!;.

?#4$0./ ,0#0 #$/:µ) 1#/ ."/*&$,3./ &'0 &.&$#690/17 $4=# &!(#$/:µ!;.

?#4$0./ 5;! #$/:µ!;+ 1#/ ."/*&$,3./ ,0# 5/%0(*µ# "!(#"!&.2.4&#/ #") &!(+ #$/:µ!;+ #(&!;+. H "$8&!+ #$/:µ)+ (x)640.&#/ &! ".54! .x &!( 5/#0;*µ#&!+. H 5.;&.$!+ #$/:µ)+ (y)640.&#/ &! ".54! .y &!( 5/#0;*µ#&!+.

B-24 7!&;&$'µ! B-G!0'µ!$)63 M"<--!

pow(x,y)

pi ()

rand( )

sign(x)

sin(x)

sqr(x)

sqrt(x)

tan(x)

vector(x,y)

B.6 A92!&$3-%). B-25

A92!&$3-%). +&*-*µ*)<-%C2

O/ *(0#$&7*./+ "$!*!µ!/8*.90 -$'*/µ!"!/!;0&#/ 6/# 0#.<%6!(0 5.5!µ,0# #") "$!*!µ!/8*./+. H/ *(0#$&7*./+ #(&,+-$'*/µ!"!/!;0&#/ *&!(+ 5/%3!$!(+ µ.&$'&,+ 1#/ &# 5/%3!$#5/#0;*µ#&# &!( Interactive Physics.

C0(µ& E/9(1(' A)(1('

constraintforce #$/:µ)+ 5/%0(*µ##$/:µ)+, #$/:µ)+ 5/%0(*µ#

#$/:µ)+, #$/:µ)+, #$/:µ)+ 5/%0(*µ#

frame #$/:µ)+

frictionforce #$/:µ)+, #$/:µ)+ 5/%0(*µ#

groupcofm #$/:µ)+ 5/%0(*µ#

kinetic #$/:µ)+

length #$/:µ)+, #$/:µ)+ #$/:µ)+

normalforce #$/:µ)+, #$/:µ)+ 5/%0(*µ#

section #$/:µ)+, 5/%0(*µ# #$/:µ)+

?#4$0./ &!0 #$/:µ) &#(&)&'&#+ &!( *(05,*µ!( (x), 1#/ ."/*&$,3./,0# 5/%0(*µ# "!( "$!*5/!$4=./ &'0 &$,-!(*# 5;0#µ' "!(.3#$µ)=.&#/ #") &! *;05.*µ!. C/# 0# >$.4&. &'0 "4.*' "!(#*1.4&#/ *. ,0# .2#&7$/!, -$'*/µ!"!/7*&. &!0 &;"!:

constraintforce(3).x

B&!(+ *(05,*µ!(+ *'µ.4! "$!+ *'µ.4!, &! &µ7µ# .x &'+ 5;0#µ'+&!( 5/#0;*µ#&!+ µ.&$%&#/ "%0&# 1#&% µ71!+ &'+ 6$#µµ7+ "!(.080./ &# 5;! %1$#. B. *(05,*µ!(+ "!( #*1!;0 µ/# $!"7, &!&µ7µ# .r &'+ 5;0#µ'+ &!( *(05,*µ!( ".$/,-./ &'0 &/µ7 &'+#*1!;µ.0'+ $!"7+.

C/# #$:$8*./+ µ. ".4$!, &# &µ7µ#&# x 1#/ y 540!0&#/ *&# "2#4*/#&90 "#61)*µ/90 *(0&.&#6µ,090 #<)090.

constraintforce(x)

?#4$0./ &!0 #$/:µ) &#(&)&'&#+ &!( *(05,*µ!( (x) 1#/ &!0 #$/:µ)&#(&)&'&#+ &!( *8µ#&!+ (y). E"/*&$,3./ &! "!*) &'+ 5;0#µ'+"!( .3#$µ)=.&#/ *&! *8µ# #") &! *;05.*µ! 9+ 5/%0(*µ#. L*(0%$&'*' #(&7 -$'*/µ!"!/.4&#/ #") µ.&$'&,+ "!( µ.&$!;0>#$;&'&#, #0&4*&#*' #,$#, ".54# '2.1&$!*&#&/1% 1#/"$!*#$µ!=)µ.0.+ 5(0%µ./+ ".54!(. H/ #$/:µ!4 &#(&)&'&#+ #(&80&90 &.**%$90 *(05,*µ90 .40#/ *&#:.$!4 1#/ ' ".$/6$#37 &!(+540.&#/ *&'0 .")µ.0' .0)&'&#.

H *;05.*µ!+ &'+ >#$;&'&#+ -$'*/µ!"!/.4 "%0&# &'0 &#(&)&'&#*(05,*µ!( #10002. C/# &' µ,&$'*' &'+ 5;0#µ'+ "!( #*1.4&#/ *.1%"!/! *8µ# #") &' 6$#µµ/17 >#$;&'&#, -$'*/µ!"!/7*&. &!0&;"!:

constraintforce(10002, 3).y

B&'0 ".$4"&9*' #(&7, ' 1#&%2'<' .y -$'*/µ!"!/.4&#/ 6/# &' 27A'&'+ &/µ7+ &'+ 5;0#µ'+ *&'0 1#&.;:(0*' y ("%09 1#/ 1%&9).

?#4$0./ &!0 #$/:µ) &#(&)&'&#+ &!( *(05,*µ!( (x), 1#/ &!(+#$/:µ!;+ &#(&)&'&#+ 5;! *9µ%&90 (y, z). E"/*&$,3./ &'0"!*)&'&# &'+ 5;0#µ'+ "!( .3#$µ)=.&#/ #") &! *;05.*µ!#0%µ.*# *&# 5;! *8µ#&#.

L *(0%$&'*' #(&7 ."/*&$,3./ µ)0! &/µ,+ 6/# 5(0%µ./+ !/ !"!4.+#*1!;0&#/ *. 1%:. =.;6!+ *9µ%&90 ("2#0'&/17 >#$;&'&#,'2.1&$!*&#&/17 1#/ ".54# 1#:!$/=)µ.0'+ 5;0#µ'+). H/ #$/:µ!4&#(&)&'&#+ #(&80 &90 *(05,*µ90 .40#/ *&#:.$!4 1#/ '".$/6$#37 &!(+ 540.&#/ *&'0 .")µ.0' .0)&'&#. L >#$;&'&#-$'*/µ!"!/.4 "%0&# &'0 &#(&)&'&# *(05,*µ!( #10002. C/# &'µ,&$'*' &'+ 5;0#µ'+ >#$;&'&#+ µ.&#<; 5;! *(61.1$/µ,090*9µ%&90 *. ,0# "2#0'&/1) *;*&'µ#, -$'*/µ!"!/7*&. &!0 &;"!:

constraintforce(10002,3,5) .x

D"9+ 1#/ µ. &!(+ *(05,*µ!(+ *'µ.4! "$!+ *'µ.4! (point topoint), ' &/µ7 .x &!( 5/#0;*µ#&!+ µ.&$%./ &' 5;0#µ' "!( #*1.4&#/1#&% µ71!+ &'+ 6$#µµ7+ "!( *(05,./ &! 1,0&$! µ%=#+ &90 5;!*9µ%&90.

B-26 7!&;&$'µ! B-G!0'µ!$)63 M"<--!

constraintforce(x,y)

constraintforce(x,y,z)

B.6 A92!&$3-%). B-27

E"/*&$,3./ &!0 &$,-!0&# #$/:µ) "2#/*4!(. H/ #$-/1,+ *(0:71.+"$!*5/!$4=!0&#/ 9+ µ'5.0/1) "2#4*/!.

?#4$0./ &!0 #$/:µ) &#(&)&'&#+ 5;! *9µ%&90 (x,y) 1#/ ."/*&$,3./&' 5;0#µ' &$/>7+ "!( #*1.4&#/ #") &! "$8&! *8µ# *&! 5.;&.$!.L &/µ7 ."/*&$,3.&#/ 9+ 5/%0(*µ#.

?#4$0./ &!0 #$/:µ) &#(&)&'&#+ µ/#+ !µ%5#+ (x) 1#/ ."/*&$,3./ &!1,0&$! µ%=#+ )290 &90 *9µ%&90 &'+ !µ%5#+. S#&% &' 5.5!µ,0'*&/6µ7, ' µ)0' !µ%5# "!( ,-./ "$!*5/!$/*&.4 .40#/ ' !µ%5# #0, '!"!4# ".$/,-./ )2# &# *8µ#&#.

E"/*&$,3./ 9+ #$/:µ) &'0 !2/17 1/0'&/17 .0,$6./# )290 &90*9µ%&90 &'+ "$!*!µ!49*'+.

?#4$0./ &!(+ #$/:µ!;+ &#(&)&'&#+ 5;! *9µ%&90 (x, y) 1#/."/*&$,3./ &! µ71!+ &'+ 6$#µµ7+ "!( *(05,./ &# 1,0&$# µ%=#+&!(+.

?#4$0./ &!(+ #$/:µ!;+ &#(&)&'&#+ 5;! *9µ%&90 (x, y) 1#/."/*&$,3./ &' 5;0#µ' ."#37+ "!( #*1.4&#/ #") &! "$8&!#0&/1.4µ.0! *&! 5.;&.$!. L &/µ7 ."/*&$,3.&#/ 9+ 5/%0(*µ#.P! Interactive Physics .12#µ>%0./ &' 5;0#µ' ."#37+ 9+ &!%:$!/*µ# &'+ 1#0!0/17+ 5;0#µ'+ 1#/ &'+ 5;0#µ'+ 1$!;*.9+.?.$/**)&.$.+ "2'$!3!$4.+ :# >$.4&. *&! :,µ# A.7"?-(9(µ(/+95 !-(:9"+0".

?#4$0./ &!0 #$/:µ) &#(&)&'&#+ .0)+ *8µ#&!+ (x) 1#/ µ/#"!*)&'&# 5/#0;*µ#&!+ (y). E"/*&$,3./ &! "2%&!+ 5/#&!µ7+ &!(*8µ#&!+ *&'0 1#&.;:(0*' &!( 5/#0;*µ#&!+. C/# "#$%5./6µ#, &!section(body[1], vector(1,0) :# ."/*&$,A./ &! 1#&#1)$(3! "2%&!+5/#&!µ7+ &!( body[1]. L *(0%$&'*' #(&7 -$'*/µ!"!/.4&#/ #") &!".54! 5;0#µ'+ &'+ #0&4*&#*'+ &!( #,$# 6/# &'0 "$!*,66/*' &'+,2<'+ &90 *9µ%&90.

frame( )

frictionforce(x,y)

groupcofm(x)

kinetic( )

length(x,y)

normalforce(x,y)

section(x,v)

B.7 7&*6!0*&)-µ12%. $)µ1.

G%$!E"'$1.

R"%$-!(0 5/%3!$.+ "$!1#:!$/*µ,0.+ µ.&#>2'&,+ &/+ !"!4.+µ"!$.4&. 0# -$'*/µ!"!/7*.&. *. &;"!(+.

C0(µ& T:3(+time 7 t #$/:µ)+self µ%=#other µ%=#

ground µ%=#

E"/*&$,3./ &!0 &$,-!0&# -$)0! &'+ "$!*!µ!49*'+. P! time<.1/0%./ "%0&# µ. #0.0 *&! "2#4*/! #0.

E"/*&$,3./ ,0# &;"! *8µ#&!+ )&#0 &!"!:.&.4&#/ µ,*# *. µ/#.<4*9*' ".54!( 5(0%µ.9+. S#&% &!0 ("!2!6/*µ) .</*8*.90".5490 5(0%µ.9+ 6/# 1%:. *8µ# &'+ "$!*!µ!49*'+, &! "self"2#µ>%0./ 9+ 5.5!µ,0' &'0 &/µ7 &!( &$,-!0&!+ *8µ#&!+ *&! !"!4!#*1.4&#/ &! ".54! 5(0%µ.9+. C/# "#$%5./6µ#, ' .<4*9*' 6/# ,0#".54! 6$#µµ/17+ >#$;&'&#+ .40#/:

Fy : - self.mass * 9.81

K*1.4&#/ µ/# 5;0#µ' *. 1%:. *8µ# &'+ "$!*!µ!49*'+. L &/µ7 &!("self" 2#µ>%0./ 9+ 5.5!µ,0' &'0 &/µ7 &!( *(61.1$/µ,0!( *8µ#&!+"%09 *&! !"!4! #*1.4&#/ ' 5;0#µ'. I"!µ,09+, *. #(&70 &'0".$4"&9*', "%09 *. 1%:. *8µ# #*1.4&#/ µ/# 5;0#µ' "!(/*!5(0#µ.4 µ. -9.81 3!$,+ &' µ%=# &!(.

K#&% &!0 ("!2!6/*µ) &90 ".5490 5(0%µ.9+ 1%:. =.;6!(+*9µ%&90 (".54# µ. =.;6'), ' &/µ7 &!( "self" 2#µ>%0./ 9+ 5.5!µ,0!&! *8µ# &!( "$8&!( *8µ#&!+ 1%:. =.;6!(+.

E"/*&$,3./ ,0# &;"! *8µ#&!+ 1#&% &'0 &!"!:,&'*7 &!( µ,*# *.µ/# .<4*9*' ".54!( 5(0%µ.9+. K#&% &!0 ("!2!6/*µ) &90 ".54905(0%µ.9+ 1%:. =.;6!(+ *9µ%&90 &'+ "$!*!µ!49*'+ (".54# µ.=.;6'), ' &/µ7 &!( "other " 2#µ>%0./ 9+ 5.5!µ,0' &'0 &/µ7 &!(

B-28 7!&;&$'µ! B-G!0'µ!$)63 M"<--!

time 3 t

self

other

B.7 7&*6!0*&)-µ12%. ?)µ1. B-29

5.;&.$!( *8µ#&!+ 1%:. =.;6!(+.

C/# "#$%5./6µ#, ' .<4*9*' ".54!( 5(0%µ.9+ 6/# &'0 "2#0'&/17>#$;&'&# .40#/:

-self.mass*6.67e-11 / sqr(self.p -other.p) * other.mass

7 *(0':,*&.$#: Gm1m2/r2

H .<4*9*' #(&7 /*-;./ 6/# 1%:. =.;6!+ *9µ%&90 &'+"$!*!µ!49*'+. S#:8+ ' .<4*9*' /*-;./ 6/# 1%:. =.;6!+*9µ%&90, &! "self" 2#µ>%0./ 9+ 5.5!µ,0' &'0 &/µ7 &!( "$8&!(*8µ#&!+ 1#/ &! “other” &'0 &/µ7 &!( 5.;&.$!( *8µ#&!+ 1%:.=.;6!(+.

E"/*&$,3./ ,0# &;"! *8µ#&!+ 6/# &! 3)0&!. K(&) .40#/!(*/#*&/1% ,0# *8µ#, *&' :,*' 0.0, &! !"!4! 5.0 µ.&#1/0.4&#/"!&,.

A$!0%&1.R"%$-!(0 &,**.$/+ #$/:µ!4 &#(&)&'&#+ 6/# &# "#61)*µ/# ".54#5;0#µ'+ >#$;&'&#+, '2.1&$!*&#&/17+, #0&4*&#*'+ #,$# 1#/ &!>#*/1) ".54! 5;0#µ'+. O# 5.4&. #(&!;+ &!(+ #$/:µ!;+&#(&)&'&#+ *. &;"!(+ "!( -$'*/µ!"!/!;0&#/ #") µ.&$'&,+ "!(µ.&$!;0 5(0%µ./+ !/ !"!4.+ #*1!;0&#/ #") &!(+ *(05,*µ!(+#(&!;+. H/ #$/:µ!4 #(&!4 ,-!(0 9+ #1!2!;:9+:

=:0&µ5 ?"1/(@ ="9µ"@µ20(6 &-';µ76 4&@47454&6W#$;&'&# 10002L2.1&$!*&#&/17 10004K0&4*&#*' &!( #,$# 10006

?.54! 1#:!$/=)µ.0'+ 5(0%µ.9+ 10008

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Interactive PhysicsT M

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Rainbow Computer AE∞Ó¿‰Ô¯Ô˜ ÊÔÚ¤·˜

∏Ï›· ∏ÏÈÔ‡ 75, 117 44, ¡. ∫fiÛÌÔ˜web site: www.rainbow.gre-mail: [email protected]

∆Ô ÏÔÁÈÛÌÈÎfi Interactive Physics ÂÍÂÏÏËÓ›ÛÙËÎÂ Î·È ‰È·Ù¤ıËÎÂ Î·Ù·Ú¯‹Ó ÁÈ· ¯Ú‹ÛË ÛÙ·°˘ÌÓ¿ÛÈ·, §‡ÎÂÈ· Î·È ∆∂∂ Ô˘ Û˘ÌÌÂÙ¤¯Ô˘Ó ÛÙËÓ √‰‡ÛÛÂÈ· – «∂ÏÏËÓÈÎ¿ ™¯ÔÏÂ›· ÛÙËÓ∫ÔÈÓˆÓ›· ÙË˜ ¶ÏËÚÔÊÔÚ›·˜». ¶ÚfiÎÂÈÙ·È ÁÈ· ÙÔ Ì·ÎÚfi¯ÚÔÓÔ (1996-2006) Û¯Â‰È·ÛÌ¤ÓÔÂıÓÈÎfi ÚfiÁÚ·ÌÌ· ·È‰·ÁˆÁÈÎ‹˜ ¤ÓÙ·ÍË˜ ÙˆÓ ∆Â¯ÓÔÏÔÁÈÒÓ ÙË˜ ¶ÏËÚÔÊÔÚ›·˜ Î·È ÙË˜∂ÈÎÔÈÓˆÓ›·˜ ÛÂ fiÏÔ ÙÔ Â‡ÚÔ˜ ÙÔ˘ ÂÎ·È‰Â˘ÙÈÎÔ‡ Û˘ÛÙ‹Ì·ÙÔ˜. ™ÙÔ ™¯ÔÏÈÎfi ∂ÚÁ·ÛÙ‹ÚÈ

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- ‰ÈÂÚÂ˘ÓËÙÈÎfi ‰È·ıÂÌ·ÙÈÎfi ÂÎ·È‰Â˘ÙÈÎfi ÏÔÁÈÛÌÈÎfi (30 ·Î¤Ù· ·Ó·Ù‡¯ÙËÎ·Ó ÂÍ·Ú¯‹˜ Î·È 10 ÚÔÛ·ÚÌfiÛÙËÎ·Ó ·fi‰ÈÂıÓ¤˜, ÁÈ· fiÏÂ˜ ÙÈ˜ Ù¿ÍÂÈ˜ °˘ÌÓ·Û›Ô˘ Î·È §˘ÎÂ›Ô˘ Î·È ÙÈ˜ ‰È¿ÊÔÚÂ˜ ÂÈ‰ÈÎfiÙËÙÂ˜ Î·ıËÁËÙÒÓ) Ô˘ ÏÂÈÙÔ˘ÚÁÂ› ÛÙ·

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√ ÂÍÂÏÏËÓÈÛÌfi˜ Î·È Ë ÚÔÛ·ÚÌÔÁ‹ ÙÔ˘ ÂÎ·È‰Â˘ÙÈÎÔ‡ ÏÔÁÈÛÌÈÎÔ‡ Interactive Physics ÛÙÔ ÂÏÏËÓÈÎfi ÂÎ·È‰Â˘ÙÈÎfi Û‡ÛÙËÌ·˘ÏÔÔÈ‹ıËÎÂ ÛÙÔ Ï·›ÛÈÔ ÙË˜ ∫›ÚÎË˜, ¤ÚÁÔ ÙË˜ ¢Ú¿ÛË˜ ππ: «∂Î·È‰Â˘ÙÈÎfi ÏÔÁÈÛÌÈÎfi» ÙË˜ √‰‡ÛÛÂÈ·˜. ∏ ‰ËÌfiÛÈ· ¯ÚËÌ·ÙÔ‰fiÙËÛËÙË˜ ÚÔÛ·ÚÌÔÁ‹˜ ÂÍ·ÛÊ·Ï›˙ÂÈ fiÙÈ ÙÔ ÎfiÛÙÔ˜ ÙÔ˘ ·ÚfiÓÙÔ˜ ÏÔÁÈÛÌÈÎÔ‡ ‰ÂÓ ˘ÂÚ‚·›ÓÂÈ ÙÔ ·ÓÙ›ÛÙÔÈ¯Ô ‰ÈÂıÓ¤˜.

∏ √‰‡ÛÛÂÈ· ¯ÚËÌ·ÙÔ‰ÔÙÂ›Ù·È ·fi ÙÔ ∂È¯ÂÈÚËÛÈ·Îfi ¶ÚfiÁÚ·ÌÌ· ∂Î·›‰Â˘ÛË˜ Î·È ∞Ú¯ÈÎ‹˜ ∂·ÁÁÂÏÌ·ÙÈÎ‹˜ ∫·Ù¿ÚÙÈÛË˜(∂¶∂∞∂∫) ÙÔ˘ ÀÔ˘ÚÁÂ›Ô˘ ∂ıÓÈÎ‹˜ ¶·È‰Â›·˜ Î·È £ÚËÛÎÂ˘Ì¿ÙˆÓ (∂›‚ÏÂ„Ë: ¢ÈÂ˘ı‡ÓÛÂÈ˜ ™Ô˘‰ÒÓ ¢Â˘ÙÂÚÔ‚¿ıÌÈ·˜∂Î·›‰Â˘ÛË˜ Î·È ∫ÔÈÓÔÙÈÎÔ‡ ¶Ï·ÈÛ›Ô˘ ™Ù‹ÚÈÍË˜, ÈÛÙÔÔ›ËÛË: ¶·È‰·ÁˆÁÈÎfi πÓÛÙÈÙÔ‡ÙÔ). √ Û¯Â‰È·ÛÌfi˜, Ë ÂÈÛÙËÌÔÓÈÎ‹ ÙÂ¯ÓÈÎ‹ÛÙ‹ÚÈÍË, Ô Û˘ÓÙÔÓÈÛÌfi˜ Î·È Ë ‰ÈÔÈÎËÙÈÎ‹ Î·È ÔÈÎÔÓÔÌÈÎ‹ ‰È·¯Â›ÚÈÛË Á›ÓÔÓÙ·È ·fi ÙÔ πÓÛÙÈÙÔ‡ÙÔ ∆Â¯ÓÔÏÔÁ›·˜ ÀÔÏÔÁÈÛÙÒÓ (π∆À).™ÙËÓ ÂÎÙ¤ÏÂÛ‹ ÙË˜ Û˘ÌÌÂÙ¤¯Ô˘Ó ¿Óˆ ·fi 2500 ÏËÚÔÊÔÚÈÎÔ›, ÂÎ·È‰Â˘ÙÈÎÔ›, ÙÂ¯ÓÈÎÔ›, Î·È ‰ÈÔÈÎËÙÈÎÔ› ÂÚÁ·˙fiÌÂÓÔÈ ÛÂ 50·ÓÂÈÛÙËÌÈ·Î¿ ÙÌ‹Ì·Ù·, 60 ÂÙ·ÈÚ›Â˜ Î·È 15 ÌÔ˘ÛÂ›·, È‰Ú‡Ì·Ù·, ÂÚÂ˘ÓËÙÈÎ¿ Î¤ÓÙÚ· Î¿.

∫¤ÓÙÚÔ ¶ÏËÚÔÊfiÚËÛË˜ √‰‡ÛÛÂÈ·˜: [email protected]

http://Odysseia.cti.gr/kirki/

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