WPC )Tq#$" *kV^}2# 4,"%0 R%rq@Ph`' ei4`=3#[:^pќdhQ䤲Srh6yEDbd\!qג} t4tFb+Qs^tSJՏXeB C bs>KoEf~w<7 CR?`+ҥo], a NIWr k\B4hy>wL4޼`k!!نJ:W_U S.9/Tѡ[ n¥%$ۼ^;غ:zDG'1%14Y煥Vpg| i@Fl1]nK+@Ѥ5[6b013! )P /"'&_4+bO:6ᷨo| yʻ'{xSM?mk-:n9DEvU@t %H 0^ B` D3}  0^v 0I AMoj@ B9 D3V 0EO A(Ydw!dd d5*_d5sd5dEJd5bd5d5)du= d5d5Pd6dd6dGAHY hd1=dn8(d`6) o_ @G<dc<L=d5=>d5>K>d5_>>d>E>@d@MAd5A Bd4BNOCdOChEdTFIdWILd5MMd5M Nd5NSNdvgN!NoN@ld_l_RndnHodp[qo&'r@dȘdTtk Cߞfa Bh*f9a;On_ 0C OOdע C  hŤw@ˤ4 * `(CG TimesScalableX > )DKUS.,,,..9% b' v;O X9   ______________________________________________________________________________ Rullendekugle %"side  1  /5(62$ n\!  TRW`6&3'6&A43'TDKUS.,,,..9% b' v;O X9   K Kd'dxd Level 1 Level 2 Level 3 Level 4 Level 5(62$ n\!  TRW`6&3'6&A43'TDKUS.,,,..9% b' v;O X9   (!$ Figur     A<< cWPC9513,, 0d{}+G      ?    ???  ?   ? ?  ?3c ?     ?   G????1 ??Ϗ?ϏǏϏǏǏ?Ǐ +`##+ KUGLE1.PCX d 'dxd Level 1 Level 2 Level 3 Level 4 Level 5($ (    ) M << deUUXXaXXXXXVhXXGlXXEXXg a~=~hoverl`gXXsinXX(XXIvXX)XX;XXX-VhXXCGl sin(v)~=~hoverlX#71772 1oversm2X#71772 1oversm2X#72775 2oversm5X)XX?vFcXXcGr vsubcoverrX#71772 1oversm2X#71772 1oversm2X#72775 2oversm5XX(XkXXvFcXXGrXX.)p52 (vsubcoverr)sup2X#71772 1oversm2X#71775 1oversm5vX#X77710 7oversm10vX#X77710 7oversm10vX#710X77XXgh 10oversm7gh > )DKUS.,,,..9% b' v;O X9   ______________________________________________________________________________ Rullendekugle %"side  1  /5gXXacXXXXX v2gRcXXG2XX}Gl 'asubc~=~{vsubcsup2}over{2l}XXacXXXvX&10GF7XXghXXzG2XXGlXX< .asubc~=~{10oversm7gh}over{2l}~<~XX!acXXXF5u7XX^ XXXVhXXGlXX, XX^g ,asubc~=~5oversm7 hoverl gWPCll9513l,, e}:? @?  ?    ? ??????????  ?       ?         ? ?          ?           ?           ?  ?          ?? ?   ?????????????       ? ?  ?     ? ? ? ? ?  ?? ?    ????8???????  ?       ?   ?              ? ? ?   ?          ?  ? ?  ?   ?   ?  ?? ?? ?         ?  ?  ? ? ~~? ? |  |  ?x ? ǁ   ?   = @ @9?   ?   ?  ? 9  y s s 9   ?  ? ??   ` > ?  ?  ? |  O    ?  ??  ??  ??  ?  ?  ? ? ?   ?  ?  ??????#p|?>?<|?? ?? ?    ? ?????/ ????  ? ?          ? ?          ???????????>?>?~????~?|8????????>?? :8KUGLE2.PCX,X/XXdvcXXEreffXXz XdXXcvcXXzrXX XXsinXX(GX.BG72XX) Bvsubcoverrsubeff~=~vsubcover{r sin(thetaoversm2)}X#71772 1oversm2X#71772 1oversm2X#72775 2oversm5XXhXXjXXpjXX+jXXjXXiXX(oXX(qXX(pqXX(+qXX(qXX(p XXXvcXXrXX XX(sinXX(GX\pu72XX)~2 ;left(vsubcover{r sin(thetaoversm2)}right)sup2X)=1=5XXzmXX#zv2EcXXz j XXX1XX1sin2XX](GX72XX) C1oversm5mvsubcsup2 1over{sinsup2(thetaoversm2)}X#71772 1oversm2X)=1=5XXzmXX#zv2EcXXz j XXX1XX1sin2XX](GX72XX) D1oversm5mvsubcsup2 1over{sinsup2(thetaoversm2)}XX?hXXjXXjXXpjXX+jXXjXXiXX ?oXX qXX qXX pqXX +qXX qXX pXXXK2XXdKXX4K5XXK XXKsin2XXK(GX@yTY2XXK)XX<10XX XX6sin 2XXb(GX72XX ) ^left({2~+~5 sinsup2(thetaoversm2)}over{10 sinsup2(thetaoversm2)}right)M&X)XX?NhXX? jXX?jXX?jXX?:jXX?jXX?&iXX4NoXX4 qXX4qXX4qXX4:qXX4qXX4&pXXXZ2XXZXX\Z5XXZ XXZsin2XXZ(GXh|2XXZ)XXd10XX, XX^sinH2XX(GXF2XXH)1XX; XXmghXXlG2XXGl z{left({2~+~5 sinsup2(thetaoversm2)}over{10 sinsup2(thetaoversm2)}right)sup{1} gh}over{2l}XX#zaXXzXX?hXXjXXjXXpjXX+jXXjXXiXX?oXXqXXqXXpqXX+qXXqXXpX*XXK5XXjK XXKsin2XXK(GXy&+2XXK)XX@2XXXX5XX0 XXbsinL2XX(GX72XXL)XXz XWXXmhXXlXXz XXzgXXzXX zXX z XS XXi hXX lXX z XX zg a~=~left({5 sinsup2(thetaoversm2)}over{2~+~5 sinsup2(thetaoversm2)}right) hoverl g~=~beta hoverl gX#75777 5oversm7X#75779 5oversm9X#71772 1oversm2XXaXXX5XXv2XX/vlXXGt$2 a~=~{2l}overtsup2WPC$$9513$,, V }4=  ??  ?   ?      ???     ?  ?     ?   ?     ?     ? ?  ?          ? ?? ?              ? ?  ?            ?????????????????>?~?? ? ?         ? ? ? ? ?  ?  ?       ?  ?               ?        ?   ?       ?     ? ?                 ?  ?         ?  ?               ??  ?      1 ?         c ??     ??   ?      ?  ? w ~? / C     ~^  >^ ? ^? ??  ?  ?| ?| >? >? x >?x < <  8 9 ? ?  ? ? ? ??< ??| ?  ?  ? ? ? ? ?? ? ?8 ?8 ?x >x >? >? <? < 88 1x ! # 7 ?? ?? ? ? ???? ?? ??? ?? ?? ??? ?? ? ???? >???? >???8 ???4XKUGLE3.PCXXX!zcosXX+z(GXu2XXz)XXzX_Xy12XXK XX$Kd iX}172XX XXdhkXXz< Ucos(thetaoversm2)~=~{1oversm2 dsubi}over{1oversm2 dsubk}~~<XXsin 2XXK(GX72XX )XXXXs1XXXXcos2XX(GX=QV72XX) >sinsup2(thetaoversm2)~=~1~cossup2(thetaoversm2)XXsinf2XXG(GXFw2XX)XXXXo1XXXXhXXpjXX+jXXjXXiXXoXXpqXX+qXXqXXpXFXX?dwiXX4xd7kb92 Qsinsup2(thetaoversm2)~=~1~left({dsubi}over{dsubk}right)sup2WPC( ( 9513( ,, &J};??   ???         χ?  &x  p       $D  O      ; ,  ?    ?    ?    ??         ?          ?    ?       ? ? ??          ??  ?? ? ? ?          ? ?     s s s 3 3  7               #   1   9   y   x   x            y    y        ?   ?      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" ,/x@ ?>??;`KUGLE4.PCXXX'zcosXX1z(GX{2XXz)XXzu XeXX{ XXBCXX XX XXABXX XXzXeXy12XXK XXKd yXXKXy 1 2XX\ K XX Kd sX172XX XXdhkXXX 1 72XX\  XX d hsXX z< cos(thetaoversm2)~=~{lineBCline}over{lineABline}~=~{1oversm2 dsuby~~1oversm2 dsubs}over{1oversm2 dsubk~+~1oversm2 dsubs}~~<XXsinf2XXG(GXFw2XX)XXXXo1XXXXhXXpjXX+jXXjXXiXXoXXpqXX+qXXqXXp XFXX4dwyXX/XXdhwsXX4xd7kXX/xXXxdh7s'92 jsinsup2(thetaoversm2)~=~1~left({dsuby~~dsubs}over{dsubk~+~dsubs}right)sup2 dTable_A&0 d dTable_B3|L* `(CG TimesScalableXXw P7XP(\$  TRW`6&3'6&A43'TDKUS.,,,..9% b' v;O X9   ~left.stack{l~=~1oversm2 asubc tsup2#vsubc~=~asubc t}rightrbrace~8~  XX!XX$XX$XX$XX"XX$XX$XX~$XX#XXlXXX1D2XX XXaucXX XXtL2XXjxv7cXXYxXX)xa7cXXx XXxtXX88dj  D  E   P Er"LSR" n\!  TRW`6&3'6&A43'TDKUS.,,,..9% b' v;O X9   0 Rullendekugle   Forml: 5  0 b Atundersgeenrullendekuglesacceleration,specieltnrunderlagetdannerenvinkel. b%"b%" Apparatur:  H 0 b 2lange(LIGE)stativstnger,modellervoks,lilleniveaubord,stlkugler(stor&lille),skydelre,meterstok,stopur,gardinskinnebane(monteretptrunderlag),farvettape. b%"b%" Symboler:   0 b dk=kuglensdiameter.m b%"b%" 0 b ds=stativstangensdiameterogdy=ydrebreddeafstativstangsbanen(sefigur4).Yb%"b%" 0 b di=indrebreddeafgardinskinnebanen(sefigur3).Eb%"b%" Teori:  l 0 b 1) Renglidningafklodspskrplan :Xb%"b%" x B2.z `@@@Ea|KK-TDx %"%"0 b VFBz X L/ 0 @Xdddddddd@Ebttm߀daXHDz X I 0 @dddddddd@E dd: }H4bb%"  %"%"0 b 0' b%"b%"0 ' %"' %"0 %" %"0v%"%"0;v%"v%"0;%";%"figur1%"%" 0 b 2) Renrulningafkuglepfladtskrplan :v b%"b%" 0 b Kuglenssamledekinetiskeenergibestrafdenalmindeligetranslatoriskeenergiogrotationsenergien.Dvs: b%"b%" 0 b 0' b%"b%"Ekin=Etrans+Erot=XHDz X ] 0 @dddddddd@Ei*$hZdd#0"mvc2+XHDz X ] 0 @dddddddd@E*$hZdd#0"I32*$' %"' %" 0 b hvorvcbetegnermassemidtpunktets(centrums)hastighed,ogIkaldesinertimomentet &!  og3ervinkelhastighedenirotationen.o'"!b%"b%" 0 b ForenkugleerI=XHDz X ] 0 @dddddddd@EH (hZdd f(0"mr2og3=XHDz X < 0 @dddddddd@Ej[(dd?'.Derforfs:[(#"b%"b%"  0 b 0' b%"b%"Ekin=XHDz X ] 0 @dddddddd@E +hZdd l+0"mvc2+! XHDz X ] 0 @dddddddd@Eg+hZddl+0" #"XHDz X ] 0 @dddddddd@E+hZddl+0" mr2 %$XHDz X < 0 @dddddddd@Ea+dd?*S ߀='&XHDz X ] 0 @dddddddd@E\+hZddl+0" mvc2+)(XHDz X ] 0 @dddddddd@E+hZddrl+0" mvc2<a+&#' %"' %" 0 b 0' b%"b%"Ekin=+*XHDz X  0 @dddddddd@E Zdd 9r" mvc2' %"' %"  0 b Energiomstningenfrastarttilsluterda: b%"b%" 0 b 0' b%"b%"Ekin=Epot<E' %"' %" 0 b 0' b%"b%"-,XHDz X  0 @dddddddd@E' Zdd jr"mvc2=mgh<vc2=/.XHDz X a 0 @dddddddd@EQrZddj:"󀀀[*]1' %"' %" 0 b Forenkonstantaccelereretbevgelseglder: b%"b%" 0 b 0' b%"b%"pqYIEz X  p @dddddddd@E'  vddA >߀32 XHDz X Y 0 @dddddddd@Eb bttz *߀[**] v' %"' %" 0 b Af[*]og[**]fs:54XHDz X I 0 @dddddddd@EP idd /1߀76 XHDz X I 0 @dddddddd@EAtt b%"b%" 0 b Accelerationenermindreenditilfldet"denreneglidning".Detskyldes,atrotationen'sluger'energi. b%"b%" 0 b 3) Renrulningafkuglepvinkelskrplan : b%"b%"  9D40z `l @@Ea LK, Ta %"%"0 b Deneffektiverulleradiusrefferher bb%" 0 b mindreendr! bb%" 0 b 0' bb%"3=;:XHDz X  0 @dddddddd@E sdd# bs' ' %" 0 b Rotationsenergienbliverda: bb%" 0 b 0' bb%"Erot==<XHDz X ] 0 @dddddddd@E hZdd 0"I32=[' ' %" 0 b ?>XHDz X ] 0 @dddddddd@E hZdd\ 0" A@XHDz X ] 0 @dddddddd@EZ hZdd 0"mr2 CBXHDz X  0 @dddddddd@E@ Q dd Q bb%" 0 b 0' bb%"=EDXHDz X , 0 @dddddddd@E $Hidd{ #1 $' ' %"  %"%"0 b Densamledekinetiskeenergi: v  ;     figur2'"b%"b%"  0 b 0' b%"b%"Ekin=Etrans+Erot=GFXHDz X ] 0 @dddddddd@Ei(hZddN(0"mvc2+IHXHDz X , 0 @dddddddd@Et(Hidd(1߀=t(#' %"' %" 0 b 0' b%"b%"KJXHDz X  0 @dddddddd@E' `dd< 9( mvc2' %"' %"  0 b Derforfsidettetilflde: b%"b%" 0 b ac=MLXHDz X <e 0 @dddddddd@E $$dd\\l( ߀<ON XHDz X  0 @dddddddd@EV/ l $tt < 4  b%"b%" 0 b 0' b%"b%"0 ' %"' %"0 %" %"0v%"%"0;v%"v%"0;%";%"0%"%"0%"%"[formel1] %"%" 0 b NB:0' b%"b%"Faktorenforrestiformlenkaldesher.y ' %"' %" 0 b 0' b%"b%"Hvis=180$(fladtunderlag)giverformlen=QPXHDz X ] 0 @dddddddd@EqhZdd 0" .e' %"' %" 0 b 0' b%"b%"Hvis=90$(vinkelskinne)giverformlen=SRXHDz X ] 0 @dddddddd@E[hZddO0"!.[' %"' %" 0 b 4) Bestemmelseafa : b%"b%" 0 b 0' b%"b%"l=UTXHDz X ] 0 @dddddddd@E hZdd z0""a t2<WV XHDz X `u 0 @dddddddd@Evtt.,>#߀[formel2] ' %"' %"  %"%"0 b 5) Bestemmelseaf : bb%"  YD40z `$ @@EaKK-8T^$0 b GARDINSKINNEBANE bb%" 0 b 0' bb%"[ZXHDz X  0 @dddddddd@E' Y dd)4 %Y ' ' %" 0 b da]\XHDz X  0 @dddddddd@ET$Zdd#"&߀fsT$bb%" 0 b 0' bb%"_^ XHDz X E 0 @dddddddd@E' &tt I&gs'&"' ' %" 0 b 0' bb%"[formel3] ' ' %"  .j)   0 b 0' bb%"     0' ' %"figur3 ).j)%" )  0 b 0' b%"b%"0 ' %"' %" 0v%"0@vb%"b%" 0;v%"v%"0;%";%"0%"%"figur3 ).j)vv%"v ) aD40z `(  @@E& XKKh ^ ( ^^%"%"0 b STATIVSTANGSBANE b^b%" 0 b 0' b^b%"cbXHDz X 7 0 @dddddddd@E' 7 d 4 % )' ^' %" 0 b 0' b^b%"ed XHDz X  0 @dddddddd@E' d tt| , s* ' ^' %" 0 b 0' b^b%"0 ' ^' %"[formel4]  ^ %" Forsgetsudfrelse&resultatbehandling:    %"%"^^0 b 0' b%"b%"0 ' %"' %"0 %" %"0v%"%"0;v%"v%"0;%";%"0%"%"figur40%"%" %"%" 0 b I:GARDINSKINNEBANE z b%"b%" 0 b Banenskalhlde(stenklodsunder).Afmrkstartogslutmedfarvettape. b%"b%" 0 b Mldenindrebreddediafskinnen('sprkken'),kuglensdiameterdk,hjdeforskellen R hogvejlngdenl.>b%"b%" 0 b Denstorestlkugleslippesudenhastighed,oglbetidentmles.*b%"b%" 0 b Forsgetgentagesmeddenlillestlkugle,ogderkanevt.lavesennyhjdeforskel. b%"b%" *qgh ddd Xdd Xdd X%"%"q,d ,d ,td ,td ,t +   .iir Q . 1ii" = 1Forsgnr. -iiriir - 0ii!iir 0@_ _ - 1 -i riir - 0i !iir 0@77Ai2 -i ri r - 0i !i r 0@UQ3 -i iri r - 4i i!i r 4@h!4 Hii9!  i ir HЀt(s) 0ii!*ii 0 0i !*ii 0 0i !*i  0 4i i!*i  4 H i9!*  i i sHЀl(m) 0 i! i 0 0 ! i 0 0 !   0 4 i!!  4 H i9!" s  i sHЀh(m) 0 i!!u# i 0 0 !!u$ i 0 0 !!u%  0 4 i!!u&  4 H i9!!u' s  i sHЀdk(mm) 0 i!"( i 0 0 !") i 0 0 !"*  0 4 i!"+  4 H i9!", s  i sHЀdi(mm) 0 i!#]- i 0 0 !#]. i 0 0 !#]/  0 4 i!#]0  4 Hii9!#]1 s  i |HЀsin2(/2) % 2 Ѐ[formel3] 0ii!z&!3ii 0 0i !% 4ii 0 0i !% 5i  0 4i i!% 6i  4 H i9!% 7 | i i \HЀfaktoren 'Q#8 Ѐ[formel1] 0 i!(=$9 i 0 0 !'Q#: i 0 0 !'Q#;  0 4 i!'Q#<  4 H i9!'Q#= \  i \HЀamlt(m/s2) N*%> Ѐ[formel2] 0 i!:+&? i 0 0 !N*%@ i 0 0 !N*%A  0 4 i!N*%B  4 F ii5!N*%C \  i FЀateori(m/s2) ,(D Ѐ[formel1] 2 ii!-(E ii 2 2 i!,(F ii 2 2 i!,(G i 2 4 ii!,(H i 4 1'%,(I 0   ii 10 b II:STATIVSTANGSBANE b%"b%"  0 b Underenderneaf2stativstngerplaceresenlilleklatmodellervoksskanstngerneliggestilleudenatrulle!Stngernelggesparalleltmeddeneneendepniveaubordet,hvorvedderdannesenhjdeforskelh.Afmrkstartogslutmedfarvettape.ab%"b%" 0 b Medskydelrensrgesforkonstantafstandmellemstngerne(dyerdenydrebredde).Mb%"b%" 0 b UndersgsomiforsgIaccelerationenafdenstorestlkuglevedatmlevejlngdeogtid,nrkuglenrullernedadbanen.Huskatmlestrrelserne:t,l,h,dk,ds,dy. tb%"b%" 0 b Forsgetvarieresvedatprvemedforskelligeafstandmellemstngerne! b%"b%" 0 b NB:modellervoksetkangenbruges,hvisdetpakkesindigen! b%"b%" *jk d dd d td td t gh%"%",d ,d ,td ,td ,t +   .iir   . 1ii"   1Forsgnr. -iirA iir - 0ii!A iir 0@_ _ - 1 -i rA iir - 0i !A iir 0@77Ai2 -i rA i r - 0i !A i r 0@UQ3 -i irA i r - 4i i!A i r 4@h!4 Hii9!A   i ir HЀt(s) 0ii!8ii 0 0i !8ii 0 0i !8i  0 4i i!8i  4 H i9!8  i i sHЀl(m) 0 i!I i 0 0 !I i 0 0 !I  0 4 i!I  4 H i9!I s  i sHЀh(m) 0 i!  i 0 0 !  i 0 0 !   0 4 i!    4 H i9! ! s  i sHЀdk(mm) 0 i!1" i 0 0 !1# i 0 0 !1$  0 4 i!1%  4 H i9!1& s  i sHЀds(mm) 0 i!' i 0 0 !( i 0 0 !)  0 4 i!*  4 H i9!+ s  i sHЀdy(mm) 0 i!|, i 0 0 !|- i 0 0 !|.  0 4 i!|/  4 Hii9!|0 s  i |HЀsin2(/2) 1 Ѐ[formel4] 0ii!2ii 0 0i !3ii 0 0i !4i  0 4i i!5i  4 H i9!6 | i i \HЀfaktoren  p7 Ѐ[formel1] 0 i!\8 i 0 0 ! p9 i 0 0 ! p:  0 4 i! p;  4 H i9! p< \  i \HЀamlt(m/s2) m = Ѐ[formel2] 0 i!Y!> i 0 0 !m ? i 0 0 !m @  0 4 i!m A  4 F ii5!m B \  i FЀateori(m/s2) "0C Ѐ[formel1] 2 ii!#D ii 2 2 i!"0E ii 2 2 i!"0F i 2 4 ii!"0G i 41'%"0H 0   ii 1Bestemaccelerationenmeddeangivneformler,ogvurderforsgetsnjagtighedogfejlkilder.f%"3FY,SJ,1997