colin f. gauld- newton's use of the pendulum to investigate fluid resistance: a case study and some...
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8/3/2019 Colin F. Gauld- Newton's Use of the Pendulum to Investigate Fluid Resistance: A Case Study and Some Implications
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LATPBLY QYA BE PFA ZAL@QCQM PB NLUAYPNDGPA ECQN@ _AYNYPGLIA8 G
IGYA YPQ@^ GL@ YBMA NMZCNIGPNBLY EB_ PAGIFNLD GOBQP PFA LGPQ_A BE
YINALIA
IBCNL E, DGQC@
Yifbbc be A`qigpnbl# Qlnuarynp~ be Lat Ybqpf Tgcay# > Mnifgac Irayialp# Kngmg @btly# LYT#
?199# GqyprgcngAmgnc8 idgqc`Jymgrpifgp,lap,gq
Goyprgip, Obbky N gl` NNN be LatpblyZrnlinzng `auacbz Latpbly `~lgmnigc pfabr~ gl` yfbt fbt np avzcgnly g
lqmoar be iacaypngc zfalbmalg, Obbk NN fgy raianua` cnppca gppalpnbl erbm fnypbrngly br a`qigpbry oaigqya np `bay lbp
zcg~ g mghbr rbca nl Latpbly grdqmalp, Fbtauar# np ny nl Obbk NN pfgp ta yaa mbyp icagrc~ Latpbl obpf gy g
pfabrapningl gl` gl avzarnmalpar, Nl pfny zgrp be pfa Zrnlinzng Latpbl agcp tnpf parrayprngc rgpfar pfgl tnpf iacaypngc
zfalbmalg gl` `ayirnoa` g lqmoar be avzarnmalpy fa igrrna` bqp pb aypgocnyf pfa yqiiayy be fny pfabr~ nl avzcgnlnld pfa
zrbzarpnay be ecqn` raynypglia, Np amblyprgpay mbyp icagrc~ pfa gipnunpnay be g yinalpnyp tbrknld gp pfa ebraerblp be
klbtca`da gl` tbrknld tnpf zfalbmalg tfnif fa `n` lbp eqcc~ ql`arypgl`,
Nl pfny zgzar pfa enryp be Latpbly yap be avzarnmalpy nlpb ecqn` raynypglia gra `ayirnoa` gl` pfa pfabr~ tfnif
ql`arcnay fny avzcglgpnbl ny bqpcnla`, G lqmoar be nyyqay grnynld erbm pfny zbrpnbl be pfaZrnlinzng pbdapfar tnpf
nmzcnigpnbly ebr pagifnld gobqp pfa lgpqra be yinalia gra `nyiqyya`,
Ka~tbr`y, Latpbl# ecqn` raynypglia# zal`qcqm# lgpqra be yinalia# fnypbr~ be yinalia
NLP_B@QIPNBL
PfaZrnlinzng razrayalpa` g lat tg~ be pfnlknld gobqp zf~ynigc zfalbmalg gl` nl np Latpblzrayalpa` g lqmoar be glgc~yay be g ugrnap~ be zfalbmalg parrayprngc mbpnbl# ecqn` mbpnbl gl`
raynypglia# ybql`# cndfp# gl` iacaypngc mbpnbl, Pfaya tara mgryfgcca` pb zrbun`a bla cgrda grdqmalpebr fny lat gzzrbgif, Pfa qcpnmgpa payp be pfny gzzrbgif &zrayalpa` nl Obbk N. tgy npy goncnp~ pb
avzcgnl pfa mbpnbl be g ugrnap~ be iacaypngc bohaipy yqif gy zcglapy# ygpaccnpay gl` ibmapy &zrayalpa`
nl Obbk NNN., Nl zrayalpnld gl bqpcnla be Latpbly grdqmalp nl pfaZrnlinzng mgl~ gqpfbry zrbiaa`erbm pfa pfabr~ nl Obbk N pb pfa yqiiayyeqc avzcglgpnbly be iacaypngc zfalbmalg nl Obbk NNN cgrdac~ndlbrnld pfa iblpalpy be Obbk NN &yaa# ebr avgmzca# Ifgl`rgyakfgr 7>>1# 909-900; @alymbra 7>>?.,
Fbtauar# bla be Latpbly bohaipnuay tgy pb `amblyprgpa pfgp pfa gcparlgpnua y~ypam zrbzbya`
o~ @ayigrpay ebr avzcgnlnld pfny mbpnbl fny ubrpav mb`ac tgy nlg`awqgpa &Gnpbl# 7>4?; Kb~r
7>51# zz,4>-796; Latpbl 74?>"7>5:# Obbk NN# Yaipnbl NV; Obbk NNN# enlgc Dalargc Yifbcnqm;Taypegcc 7>6:# z,010.7, Gy zgrp be pfny zrbdrgmma np tgy laiayygr~ pb yfbt pfa aeeaip be ecqn`y
yqif gy gnr bl pfa mbpnbl be bohaipy ynlia np tgy lbp zbyynoca nl Latpbly `g~ pb igrr~ bqp
yqoypglpngc avzarnmalpy bl mbpnbl nl g ugiqqm, Obbk NN be pfaZrnlinzng ny zrnmgrnc~ ibliarla`
tnpf gl nluaypndgpnbl nlpb pfa lgpqra gl` zrbzarpnay be pfa raynypglia pb mbpnbl zrb`qia` o~ ecqn`y
gcpfbqdf bpfar pfnldy gra `agcp tnpf nl pfgp obbk gy tacc, Tfnca nl Obbk N Latpbl zrayalpa` pfa
pfabrapnigc ogyny ebr fny gzzrbgif gl` nl Obbk NNN fa gzzcna` fny rayqcpy pb pfa ql`arypgl`nld beiacaypngc zfalbmalg np ny nl Obbk NN pfgp ta yaa icagrc~ Latpbl gy gl avzarnmalpar, Nl ptb zcgiay
&nl pfa Dalargc Yifbcnqm ebccbtnld Zrbzbynpnbl 97 gl` nl pfa `nyiqyynbl be Zrbzbynpnbl 0:. fa
razbrpa` pfa rayqcpy be mgl~ avzarnmalpy nl tfnif fa qya` fny pfabr~ pb avzcgnl mbpnbl pfrbqdf
ecqn`y,Prqay`acc &7>56o. fgy zbnlpa` bqp pfgp nl Obbk NN Latpbl tgy `agcnld tnpf zraunbqyc~
qlavzcbra` parrnpbr~ gl` pfgp pfa mgparngc nl pfny zgrp be pfaZrnlinzng ny gcmbyp alpnrac~ brndnlgc&z,701. ybmapfnld tfnif tgy lbp dalargcc~ pfa igya tnpf Obbky N gl` NNN, Oaigqya be pfny np
iblpgnly g oatnc`arnld gcparlgpnbl be mgpfamgpnigc zrbbe# ornccnglp f~zbpfayny# zqra dqayynld# ocqee#
gl` zcgnl arrbr &z,70>., Fbtauar# Prqay`acc gcyb grdqa` pfgp pfa ornccnglpc~ nldalnbqy oqp# nl pfa
al`# cgrdac~ qlygpnyegipbr~ Obbk NN cgn` bqp pfa gragy gl` `aenla` pfa zrbocamy ebr mgl~ be pfa
maifglnigc rayagrifay be pfa lavp ialpqr~ &7>56g# z,>7.,Nl pfny zgzar Latpbly gzzrbgif pb ypq`~nld ecqn` raynypglia qynld zal`qcqm mbpnbl ny
bqpcnla` gl` fny rayqcpy gra `nyiqyya`, Pfa ynmzcninp~ be fny avzarnmalpy mgka pfam n`agc
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8/3/2019 Colin F. Gauld- Newton's Use of the Pendulum to Investigate Fluid Resistance: A Case Study and Some Implications
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igl`n`gpay ebr razcnigpnbl o~ zrayalp `g~ ypq`alpy gl` fny pfabr~ zrbun`ay g icagr avgmzca be pfa
tg~ Latpbl pfbqdfp gobqp zf~ynigc zfalbmalg, Pfa racgpnblyfnz oaptaal pfabr~ gl` avzarnmalp
nl Latpbly pfnlknld ny gcyb icagrc~ nccqyprgpa`,
Pfa enryp ptalp~-lnla zrbzbynpnbly nl Obbk NN zrbun`a gl glgc~yny be bla gl` ptb `nmalynblgc
mbpnbl ql`ar pfa nlecqalia be raynypgliay tnpf `neearalp `azal`aliay bl pfa uacbinp~ be pfa mbunld
bohaip, Nl Zrbzbynpnbl 9: Latpbl zrayalpa` pfa pfabr~ tfnif tbqc` algoca fnm pb qya pfa `aig~ nl
pfa ytnld be g zal`qcqm pb enl` bqp pfa dalargc zrbzarpnay be ecqn` raynypglia,
LATPBLY PFAB_^ BE FNY ECQN@ _AYNYPGLIA AVZA_NMALPY
Ypgda 7 Latpbly avzaipgpnbl
Nl Yaipnbl N be Obbk NN &Zrbzbynpnbly 7 pb 0. be fnyZrnlinzng Latpbl `agcp tnpf pfa zrbzarpnay beraynypglia tfnif ugrnay `nraipc~ tnpf pfa uacbinp~, Nl pfa Yifbcnqm gp pfa al` be pfgp yaipnbl Latpbl
trbpa
Fbtauar# pfgp pfa raynypglia be ob`nay ny nl pfa rgpnb be pfa uacbinp~# ny mbra g mgpfamgpnigc f~zbpfaynypfgl g zf~ynigc bla, Nl ma`nqmy ubn` be gcc palginp~# pfa raynypgliay mg`a pb ob`nay gra gy pfa ywqgra be
pfa uacbinpnay, &Latpbl 74?>"7>5:# z,?00.
Nl bpfar tbr`y Latpbl `n` lbp oacnaua pfgp raynypglia tgy ragcc~ zrbzbrpnblgc pb uacbinp~ oqp# nl pfa
n`agc igya# pb pfa ywqgra be pfa uacbinp~, Fa grdqa` pfgp pfa mbunld ob`~ ibccn`a` tnpf pfa
ypgpnblgr~ ma`nqm &br zgrpnicay be pfa ma`nqm.# avarpa` g ebria bl np gl` ypgrpa` np mbunld, Ne bla
iblyn`ary pfa ma`nqm pb oa g ibccaipnbl be nl`nun`qgc zgrpnicay pfa uacbinp~# u# nmzgrpa` pb bla bepfaya zgrpnicay ny zrbzbrpnblgc pb pfa uacbinp~# U# be pfa mbunld ob`~, Ne pfa ibccnynbl ny zareaipc~
acgypni pfa zgrpnica giwqnray gcmbyp ptnia pfa uacbinp~ be pfa mbunld ob`~; ne pfa ibccnynbl ny
zareaipc~ nlacgypni pfa uacbinp~ be pfa zgrpnica ny gcmbyp awqgc pb pfgp be pfa mbunld ob`~ &yaa
Latpbly `nyiqyynbl be Zrbzbynpnbly 99 gl` 91., Pfqy g mbra rgzn`c~ mbunld ob`~ zrb`qiay gzrbzbrpnblgcc~ cgrdar uacbinp~ nl pfa zgrpnicay be pfa ma`nqm pfgl g mbra ycbtc~ mbunld ob`~, Pfa
pnma pgkal pb prglyear g dnual pbpgc gmbqlp be mbpnbl &mbmalpqm. ny nluaryac~ zrbzbrpnblgc pb pfa
uacbinp~, Pfqy pfa raynypglia# tfnif ny zrbzbrpnblgc pb pfa gmbqlp be mbpnbl cbyp o~ pfa mbunld
ob`~ zar qlnp pnma ny zrbzbrpnblgc pb ifglda nl mbpnbl zar qlnp pnma# pfgp ny pb mu"&7"U. brU? tfara
m ny pfa mgyy be pfa ma`nqm yap nl mbpnbl,
-- ENDQ_A 7 --
_aearalia pb Endqra 7 yfbty pfgp nl bla yaibl` pfa ob`~ mbuay g `nypglia Uyb pfgp pfa
mbmalpqm prglyear zar yaibl` ny awqgc pb mu tfara m ny pfa gmbqlp be ma`nqm iblpgnla` nl pfa
ubcqma prguarya` o~ pfa mbunld ob`~ nl bla yaibl`, Pfqy# erbm Latpbly pfnr` cgt# pfa raynypglia
pb pfa mbpnbl be pfa mbunld ob`~ ny
_ = mu &`GU.U`GU?
tfara `ny pfa `alynp~ be pfa ma`nqm gl`G ny pfa irbyy-yaipnblgc grag zarzal`niqcgr pb pfa `nraipnblbe mbpnbl,
Latpbl yfbta` nl Obbk NN pfgp fa avzaipa` pfa mghbr iblprnoqpnbl pb raynypglia pb oa
zrbzbrpnblgc pb pfa ywqgra be pfa uacbinp~# pfa grag be pfa mbunld ob`~ zarzal`niqcgr pb pfa `nraipnbl
be mbpnbl gl` pfa `alynp~ be pfa ma`nqm &Yifbcnqm ebccbtnld Zrbzbynpnbl 70; Zrbzbynpnbl 99;
Zrbzbynpnbl 91# Igyay 7-9# Ibrbccgrnay 7-1.,Pfny be ibqrya ny blc~ nl ma nqmy ubn` be gcc palginp~, Nl pfa Yifbcnqm gp pfa al` be YaipnblNNN be Obbk NN Latpbl trbpa
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8/3/2019 Colin F. Gauld- Newton's Use of the Pendulum to Investigate Fluid Resistance: A Case Study and Some Implications
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Pfa raynypglia be yzfarnigc ob`nay nl ecqn`y grnyay zgrpc~ erbm pfa palginp~# zgrpc~ erbm pfa gpprnpnbl# gl`
zgrpc~ erbm pfa `alynp~ be pfa ma`nqm, Gl` pfgp zgrp be pfa raynypglia tfnif grnyay erbm pfa `alynp~ be pfa
ecqn` ny# gy N ygn`# gy pfa ywqgra be pfa uacbinp~; pfa bpfar zgrp# tfnif grnyay erbm pfa palginp~ be pfa ecqn`# ny
qlnebrm, &Latpbl 74?>"7>5:# z,?6:.
Latpbl pragpa` qlnebrm raynypglia nl g ynmncgr egyfnbl pb pfa aeeaip be drgunp~ bl gl gyial`nld
ob`~ gl` fa yqddaypa` pfgp bla mndfp gcyb db bl pb pfa mbpnbl be ob`nay tfnif gra raynypa` nl zgrp
qlnebrmc~# nl zgrp nl pfa rgpnb be pfa uacbinp~# gl` nl zgrp nl pfa `qzcnigpa rgpnb be pfa ygma uacbinp~,
Fa zrbun`a` avgmzcay be fbt pfny mndfp oa `bla nl fny `nyiqyynbl be Zrbzbynpnbly 77 pb 70,
Ypgda ? Latpbly iblyprqipnbl
Nl `nyiqyynld Zrbzbynpnbl 9: Latpbl oadgl fny glgc~yny be pfa zrbocam be raynypglia pb pfa mbpnbl
be g i~icbn`gc zal`qcqm o~ qynld pfa ygma ergmatbrk fa qya` tfal `agcnld tnpf pfa uacbinp~ be
pfa zal`qcqm obo nl pfa igya tfara pfara ny lb raynypglia &Obbk N# Zrbzbynpnbl 1?., Nl Endqra ? pfa
`ngmapar#OG# be g yamn-inrica razrayalpy pfa zbynpnbl be pfa obo gy np mbuay erbm npy oadnllnld
zbnlp#O# pb npy al` zbnlp#G &yaa gcyb Endqra 5&g.., Pfa ialpra be pfny zgpf ny Igl`@ gl` `grazbnlpy ybmatfara bl pfa zgpf be pfa zal`qcqm obo yazgrgpa` o~ g ymgcc pnma nlparugc# p, Latpbl
yfbta`# nl Zrbzbynpnbly 96 gl` 1? be Obbk N pfgp pfa uarpnigc cnla@Any zrbzbrpnblgc pb pfa uacbinp~be pfa obo gp pfa zbnlp@ gl` `a ny zrbzbrpnblgc pb pfa uacbinp~ be pfa obo gp zbnlp `&yaa Dgqc`
?::0., Nl Zrbzbynpnbl 17 be Obbk N Latpbl fg` gcrag`~ yfbtl pfgp
obopfabeTandfp
zgpfnpygcbldobozal`qcqmblEbria
C
I@=
-- ENDQ_A ? --
Oaigqya be pfa raynypglia pfa al` zbnlp be pfa enryp ytnld be pfa obo ny pfa zbnlp g tfnif egccyyfbrp beG o~ pfa `nypgliaGg, Gp pfa zbnlp@ pfa uacbinp~# nlypag` be oanld razrayalpa` o~@Any
razrayalpa` o~ pfa ymgccar caldpf@Egl`# ne erbm pfny zbnlp bl# pfa obo iblpnlqa` qlraynypa` nptbqc` ragif pfa zbnlp M gp pfa caep yn`a be npy zgpf, Ne raynypglia avnypy oaptaal@ gl` `pfa uacbinp~
gp `ny razrayalpa` o~ pfa cnla `h yb pfgp pfa cnlaeh razrayalpy pfa `airagya nl uacbinp~ oaptaal pfazbnlpy@ gl` ``qa pb pfa raynypglia erbm pfa ma`nqm pfrbqdf tfnif pfa obo ny prguaccnld, Pfa
zbnlpLrazrayalpy pfa zbynpnbl pfa obo tbqc` lbt ragif ne raynypglia tara xarb gepar pfa zbnlp `gl`B ny pfa ialpra be pfa mbpnbl be pfa obo tfal raynypa` ebr pfa tfbca be npy mbuamalp erbmO pb g,
Pfa cnla@Kny iblyprqipa` yb pfgp
obopfabeTandfp
@gpraynypgliabeEbria
C
@K=
Ypgda 9 Latpbly glgc~yny
Endqra 9 yfbty mbra `apgnc be pfa radnbl be Endqra 7 oaptaal Igl`@, IEgl` Idgra rg`ngc cnlay
erbm pfa ialpra# I# gl`dny pfa zbnlp tfara pfa griLh avpal`a` nlparyaipy pfa cnla@E, Pfa rg`nqyIEiqpy pfa grihdgp f, Em ny zgrgccac pb I@ gl` yb ny awqgc pb@`, Ibmzgrnld Endqray ? gl` 9# IL
= Id# ML = Ef# IM = IEgl`em = `e - @E,
-- ENDQ_A 9 --
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8/3/2019 Colin F. Gauld- Newton's Use of the Pendulum to Investigate Fluid Resistance: A Case Study and Some Implications
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Np igl oa yaal pfgp# ne pfa `nypglia@`ny ymgcc# pfa pfraa prngldcay#Eme#Efdgl`E@Igra
ynmncgr, OaigqyaEmegl`E@Igra ynmncgr
em
Em
em
@`
I@
@E= =
Edny zrbzbrpnblgc pb pfa `airagya nl uacbinp~ `qa pb pfa raynypglia gl`em ny zrbzbrpnblgc pb pfa
nliragya nl uacbinp~ `qa pb drgunp~ yb pfgp# erbm Latpbly enryp cgt
drgunp~pb`qaoboblEbria
raynypgliapb`qaoboblEbria
drgunp~beebriapb`qauacbinp~nlNliragya
raynypgliapb`qauacbinp~nl@airagya=
Pfgp nyI@
@K
em
Ed=
Pfqy@E
I@
I@
@K
@`
em
em
Ed,, =
yb pfgpEd
@`
@K
@E=
Oaigqya prngldcayEfdgl`E@Igra ynmncgr
IE
@E
ed
Ef=
yb pfgp@E
@K
IE
@E
@`
ed
ed
Ef,, =
PfqyEf
@`
@K
IE=
brML
@`
@K
IM=
Tfal pfa tfbca mbpnbl ny raynypa` ML,IM= @`,@K, Pfa grag be pfa yfg`a` radnbl@`kK
ny awqgc pb@`,@Kyb pfgp @`,@Kny pfa grag be pfa tfbca radnblOIgUK, Pfa grag be pfa yfg`a`
radnbl ML_Y# tfara IM= M_# ny awqgc pb ML,IMyb pfgp ML,IMny awqgc pb pfa grag tfbca
prgzaxbn`gc radnblGgQT, Pfa grag beGgQTny awqgc pbGg,&IO + Ig. =Gg, gO =Gg,BO =
grag beOIgUK,
Ypgda 0 Latpbly gzzcnigpnbl
Igya 78_ = iblypglp
Nl pfa ynpqgpnbl tfara_ ny g iblypglp pfa radnblOIgUKny g raipgldca tnpf g caldpf awqgc pbOg gl`
g fandfp awqgc pb@Kyb npy grag nyOg,@K, Pfqy
Gg, gO =@K,gO
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8/3/2019 Colin F. Gauld- Newton's Use of the Pendulum to Investigate Fluid Resistance: A Case Study and Some Implications
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Yb pfgp BU=@K= Gg,
tfara BUrazrayalpy pfa raynypglia gp pfa ialpra be pfa ytnld,
PfqyobopfabeTandfp
Bgp_aynypglia
C
.Gg& ?7= [7]
Igya ?8_ uacbinp~
Latpbl grdqa` pfgp ne pfa raynypglia tara ymgcc pfa uacbinp~ oaptaalO gl` g ibqc` oa razrayalpa`o~ pfa cnla@Enl Endqra 0 tfaraOEg ny g yamninrica tnpf `ngmaparOg, Fa fg` yfbtl nl pfa
Ibrbccgr~ pb Zrbzbynpnbl ?1 pfgp# ebr g iblypglp raynypglia# pfa egypayp uacbinp~ biiqrra` gp B# pfaialpra be pfa cnlaOg, Ne pfa raynypglia tara zrbzbrpnblgc pb pfa uacbinp~ pfal pfa fandfp be pfa iqrua
OKkUg tbqc` oa g iblypglp ergipnbl be pfa fandfp be pfa yamninricaOEg, Nl bpfar tbr`yGKkUg ny gyamn-accnzya tnpf gl grag awqgc pb BU,BO = BU, Og,
-- ENDQ_A 0 --
Pfqy BU, Og =Gg, gO
Yb pfgp GgGg
BU 774
?'=
/
Nl bpfar tbr`yobopfabeTandfp
Bgp_aynypglia
C
Gg=
.& 774 [?]
Igya 98_ uacbinp~?
Ne pfa raynypglia tara zrbzbrpnblgc pb pfa ywqgra be pfa uacbinp~ gl` pfa iqruaGEg ny g yamninricapfal pfa iqruaGKkUg ny g zgrgobcg tnpf gl grag 9? Og,BU, Pfqy nl pfny igya BU= Gg gl`
obopfabeTandfp
Bgp_aynypglia
C
Gg=
.& 09 [9]
Latpbl giklbtca`da` pfgp pfa iqruaOEg mndfp lbp oa g yamninrica gl` yb nl igyay ? gl` 9 pfa
iqruay mndfp lbp oa avgipc~ gl accnzya gl` g zgrgobcg, Fbtauar# fa iblyn`ara` fny iblicqynbly [7]#[?] gl` [9] pb oa icbya albqdf ebr fny zrgipnigc zqrzbyay qynld pfa tbr` lagrc~ ebr gzzrbvnmgpac~
prqa,
LATPBLY AVZA_NMALPY NLPB ECQN@ _AYNYPGLIA
Nl br`ar pb qya fny pfabr~ pb `aparmnla pfa zrbzarpnay be ecqn` raynypglia gy pfa~ gra geeaipa` o~ pfa
uacbinp~ gl` ynxa be pfa ob`~ tfnif ny mbunld gl` pfa `alynp~ be pfa ecqn` Latpbl yap qz g cbldzal`qcqm gl` ugrna` pfa zrbzarpnay be pfa ob`nay tfnif mg`a qz pfa obo, Fa `ayirnoa` pfa igra
tnpf tfnif fa `ayndla` pfa yqzzbrp pbtgr`y pfa al` be pfa Yifbcnqm gl` avzcgnla` pfa laa` ebr g
yqzzbrp yqif gy g klnea a`da tfnif `n` lbp nlfnonp pfa zal`qcqm ytnld, Fa tgy goca pb ugr~ pfauacbinp~ gp pfa cbtayp zbnlp o~ gcparnld pfa `nypglia pfa obo tgy zqcca` ogik erbm npy cbtayp zbnlp
&Dgqc` ?::0., Fny zrbia`qra tgy pb zqcc ogik pfa obo g klbtl `nypglia gl` ibqlp pfa lqmoar be
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8/3/2019 Colin F. Gauld- Newton's Use of the Pendulum to Investigate Fluid Resistance: A Case Study and Some Implications
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ytnldy &erbm bla yn`a pb pfa bpfar. ebr pfa gmzcnpq`a pb ra`qia o~ bla andfpf br bla wqgrpar, Fa
pfal razagpa` pfa avzarnmalp o~ `bqocnld pfa nlnpngc gmzcnpq`a qlpnc fa fg` g yap be rayqcpy nl tfnif
pfa nlnpngc gmzcnpq`ay ebrma` g dabmaprni yarnay, Latpbl `aparmnla` pfa ynxa beGg ebr pfa enryp
ytnld o~ yqoprgipnld pfa ynxa be pfa enlgc gyialp erbm pfa ynxa be pfa nlnpngc `ayialp gl` `nun`nld o~
pfa lqmoar be ytnldy, Qynld fny avzrayynbl [7]# [?] br [9] algoca` fnm pb qya fny ugcqa beGg pb
enl` pfa raynypglia gp pfa cbtayp zbnlp nl pfgp ytnld,
Latpbl grdqa` pfgp pfa arrbr nl pragpnld pfa inriqcgr zal`qcqm gy g i~icbn`gc zal`qcqm tgyladcndnoca, Ebr g i~icbn`gc zal`qcqm pfa uacbinp~ gp pfa cbtayp zbnlp be pfa ytnld ny zrbzbrpnblgc pbpfa `nypglia gcbld pfa griOItfnca ebr g inriqcgr zal`qcqm np ny zrbzbrpnblgc pb pfa caldpf be pfa
ifbr`OI&Dgqc` 7>>6; 7>>>., Pfa zgpf be pfa obo be g inriqcgr zal`qcqm fgy g iblypglp iqrugpqratfnca pfgp be g i~icbn`gc zal`qcqm be pfa ygma caldpf oadnly gp pfa obppbm tnpf pfa ygma iqrugpqra
gy pfa inriqcgr zal`qcqm oqp pfny iqrugpqra `airagyay gy pfa gldca be pfa yprnld nliragyay, Pfa
byinccgpnld yprnld caldpf `airagyay gy pfa yprnld mbuay gdgnlyp pfa i~icbn`gc ifaaky, Pfqy# ne pfa
caldpf be pfa gri erbm tfnif pfa obo ypgrpy npy `btltgr` hbqrla~ ny pfa ygma ebr pfa ptb zal`qcqmy
pfa ypgrpnld zbnlp ebr pfa i~icbn`gc zal`qcqm ny fndfar gl` pfa uacbinp~ be pfa i~icbn`gc zal`qcqm ny
cgrdar tfal pfa obo ragifay pfa cbtayp zbnlp,Ebr g inriqcgr zal`qcqm pfa ypgrpnld fandfp ny dnual o~
C
ifbr`f
?
.& ?=
Ebr g i~icbn`gc zal`qcqm pfa ypgrpnld fandfp ny dnual o~
C
grif
?
.& ?=
Ynlia pfa ypgrpnld fandfp ny zrbzbrpnblgc pb pfa ywqgra be pfa uacbinp~ gp pfa obppbm pfa rgpnb be pfa
uacbinp~ ebr pfa inriqcgr zal`qcqm pb pfgp be pfa i~icbn`gc zal`qcqm ny awqgc pb &ifbr`"gri.,Latpbl icgnma` pfgp pfa rgpnb be pfa zarnb`y ebr pfaya zal`qcqmy ny &gri"ifbr`., Fbtauar# ne
bla ibmzgray pfaya rgpnby bla enl`y pfa~ gra blc~ gzzrbvnmgpac~ awqgc?, Ebr avgmzca# ebr g ypgrpnld
gldca be 9::
pfa rgpnby `neear o~ gobqp :,1! &Kn`` % Ebdd ?::?., Latpbl `n` lbp hqypne~ fny icgnm
oqp gl grdqmalp fa mndfp fgua qya` ny pfa ebccbtnld, Pfa zal`qcqm oboy obpf ypgrp gp rayp gl` pfa~
prguac pfa ygma `nypglia gcbld gcmbyp n`alpnigc zgpfy, &Pfa fandfp gobua pfa cbtayp zbnlp gp pfa
oadnllnld ebr pfa inriqcgr zal`qcqm ny 74,?7 nlif tfnca pfgp ebr pfa i~icbn`gc zal`qcqm ny 74,5?
nlif g `neearalia be blc~ :,07 nlif,. Latpbl mg~ fgua pfal ynmzc~ iblicq`a` pfgp pfa pnma pgkal ny
nluaryac~ zrbzbrpnblgc pb pfa enlgc uacbinp~,
Latpbly iblicqynbl erbm fny `nyiqyynbl be Zrbzbynpnbl 9: tgy pfgp pfa `airagya nl gmzcnpq`a
gepar bla ytnld be pfa zal`qcqm `qa pb raynypglia tgy zrbzbrpnblgc pb pfa raynypglia v caldpf be pfazal`qcqm &awqgpnbly [7]# [?] gl` [9]., Nl pfa Dalargc Yifbcnqm fa raypgpa` pfny nl g ebrm tfnif
raearra` blc~ pb pfa zrbzarpnay be pfa mbpnbl be pfa obo gl` lbp pb pfa zal`qcqm, Nl pfny ebrm pfa
`airagya nl gmzcnpq`a ny zrbzbrpnblgc pb pfa raynypglia v &zarnb`.?, Gyyqmnld pfgp pfa raynypglia gp
pfa obppbm tgy zrbzbrpnblgc pb U?
pfny magly pfgp pfa `airagya nl gmzcnpq`a tgy zrbzbrpnblgc pb U?
vP
?, Fbtauar# Latpbl fg` gcrag`~ ypgpa` pfgp U&i~icbn`."U&inrica. = &gri"ifbr`. gl` pfgp
P&i~icbn`."P&inrica. = &ifbr`"gri. yb fny iblicqynbl tgy pfgp pfa `airagyay nl pfa gmzcnpq`a ebr ginriqcgr zal`qcqm gl` g i~icbn`gc zal`qcqm tnpf pfa ygma caldpf gl` pfa ygma gmzcnpq`a tara
lagrc~ awqgc, Pfqy fa oacnaua` pfgp fny qya be inriqcgr rgpfar pfgl i~icbn`gc zal`qcqmy `n` lbp
nlprb`qia yqoypglpngc arrbry nlpb fny avzarnmalpy,
-- PGOCA 7 --
Latpbly `gpg ebr pfa 09 avzarnmalpy fa razbrpa` nl pfa Yifbcnqm ebccbtnld fny `nyiqyynbl be
Zrbzbynpnbl 97 gra dnual nl Pgoca 7 gl` g drgzfnigc yqmmgr~ ny zrayalpa` nl Endqra 1, Fny qlnp be
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8/3/2019 Colin F. Gauld- Newton's Use of the Pendulum to Investigate Fluid Resistance: A Case Study and Some Implications
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`nypglia tgy pfa Aldcnyf nlif &pb oa `nypnldqnyfa` erbm pfa Zgrny nlif. 7? be tfnif mgka qz gl
Aldcnyf ebbp tfnif ny awqgc pb :,9:1 mapra, Fny qlnp be tandfp &br mgyy. tgy pfa prb~ bqlia tfnif
ny awqgc pb ?6,91 drgmy, Fa igrrna` bqp pfraa mgnl yarnay be avzarnmalpy pb amblyprgpa
rayzaipnuac~ pfa `azal`alia be pfa raynypglia bl &g. pfa uacbinp~ be pfa obo# &o. pfa `ngmapar be pfa
obo gl` &i. pfa `alynp~ be pfa ecqn`,
-- ENDQ_A 1 --
@AZAL@ALIA BL PFA UACBINP^ BE PFA OBO &Igyay 7-7?.
Latpbl qya` igyay 7-5# nl tfnif fa magyqra` pfa lqmoar be ytnldy rawqnra` ebr pfa zal`qcqm
gmzcnpq`a pb `airagya o~ bla andfpf# pb nluaypndgpa pfa `azal`alia be pfa raynypglia bl uacbinp~,
Pfa zal`qcqm caldpf#C# tgy 7?5 nlifay gl` pfa zal`qcqm obo fg` g `ngmapar be 5,641 nlifay gl`g mgyy be 14,976 bqliay, Latpbl zcgia` g mgrk bl pfa tnra yqzzbrpnld pfa obo gp g zbnlp 7?7
nlifay &C. erbm pfa yqzzbrp, Oafnl` pfny mgrk fa zcgia` g rqcar tfnif tgy qya` pb cbigpa pfa
zbynpnbl be pfa mgrk gy pfa zal`qcqm byinccgpa`, Pfa ugcqay nl pfa yaibl` gl` pfnr` ibcqmly be
Pgoca 7 gra pfa `nypgliay mbua` o~ pfny mgrk tfal `ayial`nld gl` gyial`nld, Np igl oa yaal erbmpfa dabmapr~ be pfa ynpqgpnbl pfgp pfa rgpnbGg"Cebr pfa mgrk ny awqgc pbGg"C ebr pfa obo, Pfa
magyqra be pfa uacbinp~ ny pfa nypglia prguacca` nl pfa enryp `ayialp, Ebr g dnual i~icbn`gc zal`qcqmpfa gri caldpf prguacca` `qrnld pfa enryp fgce be pfa enryp ytnld ny zrbzbrpnblgc pb pfa uacbinp~ gp pfa
obppbm be pfa zgpf &Dgqc` ?::0.,Ybma be pfa ugcqay ebr pfa lqmoar be byinccgpnbly nl ibcqml 1 be Pgoca 7 iblpgnl ergipnbly, Nl
gl agrc~ tbrknld zgzar qzbl tfnif pfny Yifbcnqm ny ogya` &Ibfal 7>47# z,7:9. Latpbl avzcgnla`
fbt fa grrnua` gp yqif ugcqay, Ebr avgmzca# nl igya 5 tfara pfa lqmoar be byinccgpnbly ny 9?> pfa
obo tbqc` oa gobua npy pgrdap be 15 nlifay gepar > ytnldy oqp oacbt pfa pgrdap gepar 7: ytnldy,
Latpbl lbpa` pfa gmbqlpy o~ tfnif pfa klbp tgy gobua br oacbt pfa pgrdap gl` qya` pfaya ugcqay
pb aypnmgpa pfa lqmoar be ytnldy tfal pfa pgrdap tgy ragifa`,Latpbly avzaipgpnbl pfgp pfa raynypglia tbqc` oa zrbzbrpnblgc pb pfa ywqgra be pfa uacbinp~
tgy iblenrma` ebr cgrdar uacbinpnay o~ `nun`nld pfa ugcqay ebrGg&tfnif gra pfa ogyni magyqray be
raynypglia giibr`nld pb awqgpnbly [7]# [?] gl` [9]. o~ pfa ywqgra be pfa gri prguacca` &tfnif ny pfa
ogyni magyqra be uacbinp~ gp pfa obppbm be pfa ytnld. nl pfa enryp `ayialp ebr igyay 7-5,
Nl fny mbra raenla` glgc~yny fa gyyqma` pfgp pfny raynypglia tgy mg`a qz be pfraa ibmzblalpy
zrbzbrpnblgc# rayzaipnuac~# pb U# U9"? gl` U
?, Fa dgua lb ragybl ebr pfny gl` np gzzagry pb oa g zqrac~
ebrmgc gyyndlmalp, Fny zraunbqy pfabrapnigc `nyiqyynbl fg` blc~ `agcp tnpf raynypgliay zrbzbrpnblgc
pb U:# Ugl` U
?, Qynld igyay ?# 0 gl` 5 gl` gl avzaipgpnbl pfgpGg = GU + OU
9"?+ IU
?Latpbl
`aparmnla` pfa ugcqay beG#O gl` Itfnif algoca` fnm pb igciqcgpa pfa racgpnua ynxay be pfaya pfraaibmzblalpy ebr `neearalp uacbinpnay, Nl br`ar pb `aparmnla pfa raynypglia erbm pfa gobua awqgpnbl
Latpbl qya` awqgpnbly [?] gl` [9] pb grrnua gp
Caldpf
IUOUGU
Tandfp
_aynypglia ?09?
9
7:4
774
++
=
Pfa egipbr 4"7: nl pfa yaibl` parm ny pfa magl be 4"77 gl` 9"0 ebql` nl awqgpnbly [?] gl` [9],
Latpbl zrayalpa` pfa `gpg ebr igyay 4-7? nl tfnif pfa gmzcnpq`a tgy ra`qia` o~ bla wqgrpar
be pfa nlnpngc `ayialp ypgpnld pfgp pfaya rayqcpy tbqc` zrbun`a mbra giiqrgpa nlebrmgpnbl pfgl igyay7-5 oqp fa pbbk pfanr glgc~yny lb eqrpfar, Fa trbpa8 N cagua pfa igciqcgpnbl pb yqif gy gra `nyzbya`
pb mgka np' Np ny nlparaypnld pfgp pfa ugcqay beGgebr pfa igyay 7-5 gra iblynypalpc~ cgrdar pfgl pfbya
ebr pfa ibrrayzbl`nld igyay 4-7?, Bla ragybl ebr pfa `neearalia ny pfgp# gy pfa zal`qcqm byinccgpay#npy uacbinp~ `airagyay# pfa raynypglia `airagyay gl` pfa ugcqa beGg`airagyay tnpf agif ytnld,
Pfqy# `nun`nld pfa `neearalia oaptaal pfa enryp `ayialp gl` pfa cgyp gyialp o~ pfa lqmoar be ytnldy
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8/3/2019 Colin F. Gauld- Newton's Use of the Pendulum to Investigate Fluid Resistance: A Case Study and Some Implications
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dnuay pfa guargda `airagya zar ytnld gl` lbp pfa airagya ebr pfa enryp ytnld, Oaigqya pfa ymgccar
`airagyay ebr pfa cgpar ytnldy nl igyay 4-7? gra bmnppa` nl igyay 7-5 pfa guargda ugcqay beGgebr
igyay 7-5 gra cgrdar pfgl ebr pfa ibrrayzbl`nld ugcqay ebr igyay 4-7?, Ne bla gyyqmay g zgrpniqcgr
`azal`alia bl U&br bl pfa gmzcnpq`a. ebr pfa ugcqa beGg nl agif ytnld np ny zbyynoca pb `arnua g
mbra giiqrgpa aypnmgpa beGgebr pfa enryp ytnld &qynld# ebr avgmzca# pfa Dbgc Yaakpbbc nl MY
Aviac., Pfa ugcqay ebrGgqynld pfny mapfb` gra mqif icbyar ebr pfa ptb yapy be rayqcpy,
@AZAL@ALIA BL PFA @NGMAPA_ BE PFA OBO &Igyay 7-?5# ?4.
Nl igyay 79-7> Latpbl qya` g obo tnpf g ymgccar `ngmapar &? nlifay. gl` tandfp &?5,?1 bqliay.
pfgl ebr igyay 7-7? tfnca mgnlpgnlnld pfa ygma caldpf ebr pfa zal`qcqm, Ebr pfaya igyay fa ibqlpa`pfa lqmoar be ytnldy pb ornld gobqp g ra`qipnbl nl pfa gmzcnpq`a be bla andfpf, Ebr igyay ?:-?5 pfa
avzarnmalpy tara razagpa` ebr g ra`qipnbl nl gmzcnpq`a be bla wqgrpar oqp pfaya rayqcpy tara lbpglgc~ya` nl pfaZrnlinzng,
Latpbl glgc~ya` igyay 71# 74 gl` 7> qynld pfa ygma ebrm be racgpnblyfnz ebrGggy ebr igyay
?# 0 gl` 5 gl` ebql` pfgp pfa rgpnb be pfa uacbinp~ ywqgra` ibmzblalpy be raynypglia `arnua` erbm
pfa ugcqay beIebr igyay ?# 0 gl` 5 gl` igyay 71# 74 gl` 7> &ebr tfnif ibrrayzbl`nld uacbinpnaytara pfa ygma. tgy 4,99"7 tfnca pfa rgpnb be pfa ywqgray be pfa `ngmapary tgy 77,6"7, Fa grdqa`
pfgp ne bla pbbk nlpb giibqlp pfa raynypglia be pfa pfrag`# tfnif fa aypnmgpa` pb oa dragpar pfgl gpfnr` zgrp be pfa tfbca raynypglia be pfa cayyar zal`qcqm# pfa enryp rgpnb oaibmay &4,99 :,99."&7,::
:,99. br 7:,1"7 g ugcqa tfnif ny icbyar pb pfa rgpnb be ywqgray be pfa `ngmapary,
Latpbl ragcnya` pfgp pfa aeeaip be pfa pfrag` tbqc` oa racgpnuac~ cayy ebr dcboay tnpf cgrdar
`ngmapary gl` fa igrrna` bqp gl avzarnmalp &igya ?4. tnpf g dcboa be `ngmapar 76,41 nlifay gl`
mgyy ?:6 bqliay, Pfa zal`qcqm caldpf tgy 7??,1 nlifay, Pfa rayqcpy be pfny avzarnmalp tara
ibmzgra` pb pfbya erbm igya 5 gl`# ynlia pfa zal`qcqm caldpfy tara `neearalp# pfny laiayynpgpa` pfa
prglyebrmgpnbl be zal`qcqm zgrgmapary ebr igya ?4 pb pfbya be igya 5 oaebra g iblicqynbl ibqc` oa
ragifa`, Fa ebql` pfa rgpnb be raynypgliay pb oa 4"7 gl` pfa rgpnb be pfa ywqgra be pfa `ngmapary pb oa
4,0"7,
@AZAL@ALIA BL PFA @ALYNP^ BE PFA ECQN@ &Igyay ?6-09.
Latpbl igrrna` bqp g yarnay be avzarnmalpy nl tfnif pfa obo be g zal`qcqm mbua` pfrbqdf gnr
&igyay ?6-97G. gl` pfal pfrbqdf tgpar &igyay 9:T-95., Erbm igyay ?6 gl` 97G nl gnr Latpbl
`aparmnla` pfa ugcqay beG gl` Inl pfa avzrayynblGg= GU + IU?, Fa ragcnya` pfgp fa ibqc` lbp
ynmzc~ qya pfa gri caldpf gy g magyqra be pfa uacbinp~ gp pfa obppbm nl tgpar oaigqya pfa uacbinp~tgy mqif cayy nl tgpar pfgl nl gnr ebr pfa ygma zal`qcqm zgrgmapary, Fbtauar# fa grdqa` gy fa
`n` tfal ibmzgrnld pfa oafgunbqr be inriqcgr gl` i~icbn`gc zal`qcqmy, Fa ragybla` pfgp ne pfauacbinp~ be pfa zal`qcqm nl tgpar tara pb oa nliragya` yb np tgy pfa ygma gy pfgp ebr pfa zal`qcqm
nl gnr pfa pnma pfa obo yzalp prguaccnld pb pfa obppbm be npy zgpf tbqc` oa `airagya` nl pfa ygmargpnb, Pfqy# ynliaGg ny zrbzbrpnblgc pb raynypglia v &pnma.
?gl` ne pfa raynypglia tara zrbzbrpnblgc
pb U?#Gg tbqc` oa qlifglda` gl` lb arrbr tbqc` oa nlprb`qia` o~ gyyqmnld pfgp pfa magyqra be
pfa uacbinp~ be pfa zal`qcqm obo prguaccnld pfrbqdf tgpar tgy pfa caldpf be pfa gri,
Ne pfny iblicqynbl tara prqa pfal# ebr igyay 97G gl` 97T#_&tgpar."_&gnr. =Gg&tgpar."Gg&gnr. = L&gnr."L&tgpar. = 191"7,?, Qynld fny ugcqay beG gl` Iebql` zraunbqyc~
Latpbl pfal grdqa` pfgp_&gnr[U?]."_&gnr. = 06,75"57,47 tfara_&gnr[U
?]. ny pfa ibmzblalp be pfa
raynypglia `qa pb pfa gnr tfnif ny zrbzbrpnblgc pb U?
erbm tfnif fa iblicq`a` pfgp_&tgpar."_&gnr
[U?]. = 147"7, Pgknld nlpb giibqlp pfa egip pfgp pfa pfrag` tgy pbpgcc~ nmmarya` nl gnr oqp lbp
pbpgcc~ nmmarya` nl tgpar# Latpbl# tnpfbqp gl~ eqrpfar `apgncy# grrnua` gp fny iblicqynbl pfgp
_&tgpar."_&gnr[U?]. tgy gobqp 61: pb 7# pfgp ny gy# pfa `alynp~ be pfa tgpar pb pfa `alynp~ be pfa gnr#
lagrc~,Latpbl igrrna` bqp eqrpfar avzarnmalpy ebr tfnif fa `bay lbp zrbun`a gcc pfa `apgncy, Fa
ragcnya` pfgp# nl ybma igyay# fny uayyac tgy pbb ymgcc ebr pfa ynxa be pfa obo gl` fa trbpa
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N nlpal`a` pb fgua razagpa` pfaya avzarnmalpy tnpf cgrdar uayyacy# gl` nl macpa` mapgcy# gl` bpfar cnwqbry
obpf fbp gl` ibc`; oqp N fg` lbp pfa canyqra pb pr~ gcc8 gl` oayn`ay# erbm tfgp ny gcrag`~ `ayirnoa`# np
gzzagry yqeeninalpc~ pfgp pfa raynypglia be ob`nay mbunld ytnepc~ ny lagrc~ zrbzbrpnblgc pb pfa `alynpnay be
pfa ecqn`y nl tfnif pfa~ mbua,
Pfa cgyp avzarnmalp fa `ayirnoa` nl pfny Yifbcnqm tgy bla pb `aparmnla pfa raynypglia `qa pb
pfa gapfar, Pfa gapfar tgy ibliazpqgcnya` gy g ecqn` tfnif lbp blc~ encca` amzp~ yzgia oqp gcyb
zarug`a` pfa nlparnbr be ybcn` bohaipy, Fa yqyzal`a` g tbb`al obv bl pfa al` be g pfrag` 77 eaap
cbld pfal `rat np gyn`a gobqp 5 eaap# gccbta` np pb byinccgpa gl` lbpa` pfa zbynpnbl be pfa obo gepar
bla# ptb gl` pfraa ytnldy, Pfa obv tgy pfal encca` tnpf mapgc, Tfal eqcc pfa tandfp be pfa obv
tgy 46 pnmay pfa ibmonla` tandfp be pfa amzp~ obv# pfa gnr nl np gl` pfa pfrag` grbql` np, Tfalgccbta` pb byinccgpa# pfa encca` obv pbbk 44 ytnldy pb ragif pfa enryp mgrk, Latpbl tgy nlparaypa` pb
yaa ne pfny `neearalia tgy `qa pb pfa aeeaip be pfa gapfar bl yqregiay nlyn`a pfa obv gy tacc gy bl pfabqpyn`a yqregia, Qynld pfa racgpnblyfnz [7]# [?] br [9]# lgmac~#
T
_
C
LgyialpEnlgc`ayialpNlnpngck
C
kGg=
'
=
".&
Latpbl iblicq`a` pfgp pfa tfbca raynypglia be pfa obv tfal eqcc# fg` lbp g dragpar zrbzbrpnbl
pb pfa raynypglia be pfa obv# tfal amzp~# pfgl 46 pb 44, Yazgrgpnld pfaya raynypgliay nlpb ptb
ibmzblalpy pfgp `qa pb pfa bqpar yqregiay gl` pfgp `qa pb pfa nllar yqregiay - fa iblicq`a` pfgp
pfa raynypglia be pfa amzp~ obv nl npy nlparlgc zgrpy tncc oa gobua 1::: pnmay cayy pfgl pfa
raynypglia bl pfa avparlgc yqzareninay [yqregiay] pfgp ny# pfa raynypglia be pfa gapfar# ne gl~# tgy uar~
ymgcc,
Latpbl zbnlpa` bqp pfgp pfny avzarnmalp ny racgpa` o~ mambr~# pfa zgzar oanld cbyp nl tfnif Nfg` `ayirnoa` np; yb pfgp N fgua oaal bocnda` pb bmnp ybma ergipnblgc zgrpy# tfnif gra ycnzp bqp be m~
mambr~; gl` N fgua lb canyqra pb pr~ np gdgnl,
@NYIQYYNBL
Nl mgl~ rayzaipy Latpbly pragpmalp be pfa zrbocam be ecqn` raynypglia ny yndlneniglpc~ `neearalp
erbm pfa tg~ pfny zrbocam tbqc` oa pgikca` pb`g~ ayzaingcc~ nl a`qigpnblgc yappnldy, G lqmoar be
eagpqray be Latpbly tbrk nl pfny grag gra `nyiqyya` nl tfgp ebccbty,
Pfa dabmaprnygpnbl be lgpqra
Pfa pfabr~ zrayalpa` gobua nccqyprgpay pfa gzzrbgif pb ql`arypgl`nld zf~ynigc zfalbmalg tfnif
Latpbl g`bzpa` gl gzzrbgif `ayirnoa` o~ Kb~r gy pfa dabmaprnygpnbl be lgpqra &Kb~r 7>51#
zz,5-4; Taypegcc 7>>:., Zf~ynigc racgpnblyfnzy ibqc` oa mgzza` o~ gl` glgc~ya` pfrbqdf
dabmaprnigc racgpnblyfnzy, Pb`g~ pfa amzfgyny nl yinalia pagifnld ny bl gcdaorgni rgpfar pfgl
dabmaprnigc mb`acy, Bla eagpqra be Latpbly pragpmalp be Zrbzbynpnbl 9: ny pfa icbya racgpnblyfnz
oaptaal pfa dabmaprnigc `ngdrgm gl` pfa zf~ynigc ynpqgpnbl, Pfa ogya cnla#OG nl Endqra?#razrayalpy pfa zgpf be pfa zal`qcqm obo gl` pfa uacbinp~ be pfa obo ny razrayalpa` o~ uarpnigc cnlay
ypgrpnld erbm pfa zgrpniqcgr zbynpnbl be pfa obo bl pfgp cnla, Pfa raynypglia gp@ ny razrayalpa` o~pfa cnla@K, Nl agif igya pfara tgy g `nraip racgpnblyfnz oaptaal g `~lgmnigc br knlamgpni ugrngoca
gl` g cnla bl pfa `ngdrgm gl` pfa ibllaipnbl oaptaal `ngdrgm gl` ynpqgpnbl ny lbp `neeniqcp pb drgyz,Pfa raynypglia ny `nraipc~ racgpa` pb pfa `airagya nl pfa gmzcnpq`a be pfa zal`qcqm gy np mbuay
pfrbqdf pfa ecqn`,
-- ENDQ_A 5 --
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8/3/2019 Colin F. Gauld- Newton's Use of the Pendulum to Investigate Fluid Resistance: A Case Study and Some Implications
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Fbtauar# pfny ny lbp gctg~y yb nl pfaZrnlinzng gl`# nl ybma igyay# tfnca Latpbl zrbun`a` gdabmaprnigc ybcqpnbl pb g zrbocam ybmapnmay pfa cnlk oaptaal pfa `ngdrgm gl` pfa ynpqgpnbl tgy lbp
agy~ pb ibliazpqgcnya, Fny pragpmalp be Zrbocam 0 &Zrbzbynpnbl ?>. nccqyprgpay pfny zbnlp tacc, Pfny
zrbocam ny Yqzzbynld pfgp g ob`~ byinccgpnld nl g i~icbn` ny raynypa` nl g `qzcnigpa rgpnb be pfa
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tfnca# gy g rayqcp be raynypglia# np mbua` erbmO pb g, Iny pfa cbtayp zbnlp gl`@ ny gl~ zbnlp blpfny zgpf, Latpbl yap qz pfa f~zarobcg nl Endqra 5&o. nl br`ar pb enl` g tg~ be `aparmnlnld pfa
raynypglia avzarnalia` o~ pfa obo gy np mbua`, Fa ypnzqcgpa` pfa ebccbtnld nl br`ar pb oadnl fny
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np gzzcna` pb g i~icbn`gc zal`qcqm gl` tfnif ny bqpcnla` gobua gcpfbqdf pfa gzzrbvnmgpnbly mg`a
nl pfny ybcqpnbl ral`ara` np cayy rndbrbqy pfgl pfa Aqicn`agl mb`ac tfnif Latpbl gppamzpa` pb
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8/3/2019 Colin F. Gauld- Newton's Use of the Pendulum to Investigate Fluid Resistance: A Case Study and Some Implications
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glgc~yny, Ynmncgrc~# fa icgnma` pfgp# nl Endqra 0# pfa iqruaOehg tgy g yamninrica br gp cagyp
lagrc~ g yamninrica g zcgqynoca gyyqmzpnbl ebr tfnif fa zrbun`a` ybma hqypnenigpnbl,
Tfal fny zal`qcqm oboy mbua` pfrbqdf tgpar# Latpbl ragcnya` pfgp pfa uacbinp~ gp pfa cbtayp
zbnlp tbqc` oa yndlneniglpc~ cayy pfgp pfgp dnual o~ pfa racgpnblyfnz oaptaal gri &br ifbr`. caldpf
gl` uacbinp~ qya` ebr g zal`qcqm nl gnr, Fbtauar# fa grdqa` &tnpfbqp `apgnca` glgc~yny oqp tnpf
raearalia pb eagpqray be fny pfabr~ zrayalpa` agrcnar nl pfny zgzar. pfgp ne pfny ygma racgpnblyfnz taragyyqma` lb yndlneniglp arrbr tbqc` oa nlprb`qia` nlpb fny iblicqynbly,
Pfaya avgmzcay gra nl`nigpnbly pfgp Latpbl# tbrknld tnpf g pfabr~ pfgp `n` lbp eqcc~ zarebrm
pfa eqlipnbl tfnif fa nlpal`a` np pb# lauarpfacayy zrbiaa`a` bl pfa ogyny be tfgp mndfp ragyblgoc~
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paiflnwqa gy fa zrbdrayya` nl fny nluaypndgpnbly, Pgoca 7 iblpgnly nlebrmgpnbl gobqp 95 igyay ebrtfnif Latpbl zrbun`a` yqeeninalp `apgnc ebr qy pb klbt mbyp be pfa racauglp zgrgmapary gl` yb
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ebccbtnld Zrbzbynpnbl 97 pfa `apgnc ny nlyqeeninalp ebr pfny pb oa `bla gl` ta fgua pb pgka Latpbly
tbr` ebr fny iblicqynbly, Nl fny `nyiqyynbl be pfa nmzcnigpnbly be fny `gpg Latpbl glgc~ya` blc~ 7:
be pfbya igyay cnypa` nl Pgoca 7 gl` lb malpnbl ny mg`a be zbyynoca arrbry nl pfa `gpg, Erbm igyay 7-
5 fa qya` igyay ?# 0 gl` 5 pb `aparmnla pfa ibaeeninalpyG#O gl` Inl pfa racgpnblGg= GU + OU9"?
+ IU? gl` ndlbra` igyay 7# 9 gl` 1, Igyay 4-7? tara caep pb pfa rag`ar pb glgc~ya,
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iblypglp# zrbzbrpnblgc pb U# br zrbzbrpnblgc pb U?
fa dnuay lb ragybl ebr nlprb`qinld fara g
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`bmnlgpa,
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raynypglia bl pfa ywqgra be pfa `ngmapar be pfa obo br pfa `alynp~ be pfa ecqn` tgy egr erbm avgip, Nlgcc pfaya ynpqgpnbly Latpbl ybqdfp g ragybl ebr pfa `neearalia oaptaal fny avzarnmalpy gl` fny
avzaipgpnbly gl` pfa raiblincngpnbl ny lbp gctg~y iblunlinld, Ebr avgmzca# fa nlprb`qia` g rbqdf
aypnmgpa be pfa raynypglia be pfa pfrag` &tnpf lb giibqlp be fbt fa bopgnla` pfny. pb raiblinca
avzarnmalpgc rayqcpy nl igyay 7-5 gl` 79-7> tnpf pfa rgpnb be `ngmapary tfnif tgy 77,9"7, Pfa rgpnb
be raynypgliay oaebra mb`nenigpnbl tgy 4,99"7 tfnca gepar mb`nenigpnbl tgy 7:,1"7,
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oqp pfa pfrag` tgy lbp nmmarya` nl pfa tgpar yb gcc pfa nrracauglp ugrngocay tara lbp iblprbcca`
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8/3/2019 Colin F. Gauld- Newton's Use of the Pendulum to Investigate Fluid Resistance: A Case Study and Some Implications
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g`awqgpac~, Fa ypgpa` tnpfbqp yqzzbrp pfgp ne pfa pfrag` tara eqcc~ nmmarya` nl tgpar pfa rgpnb
tbqc` oa 61:"7# tfnif ny lagrc~ pfa rgpnb be pfa `alynpnay be tgpar gl` gnr, &Gp ?::I pfa rgpnb ny
gobqp 69:"7., Fbtauar# ne pfa nlebrmgpnbl fa dgua gobqp pfa raynypglia be pfa pfrag` tfal
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515"7, Np ny lbp icagr fbt fa grrnua` gp pfa rgpnb be 61:"7, Nl pfa pfnr` a`npnbl be pfaZrnlinzng
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Yifbcnqm gepar Zrbzbynpnbl 0: tfal fa nluaypndgpa` pfa raynypglia pb egccnld ob`nay# gl` 64: nl pfaYifbcnqm gepar Zrbzbynpnbl 1: nl tfnif fa nluaypndgpa` pfa yzaa` be ybql` nl gnr, Taypegcc &7>49.
zbnlpa` bqp pfgp# nl pfa ugrnbqy `rgepy be pfa enryp a`npnbl be pfaZrnlinzng nl tfnif fa `ayirnoa` fny
nluaypndgpnbly nlpb pfa yzaa` be ybql`# Latpbl pamzbrgrnc~ g`bzpa` >:: gl` >1: oaebra yappcnld
bl 64: ebr pfny rgpnb# gzzgralpc~ pb enl` pfa bla tfnif oayp yqnpa` fny `gpg, Pfaya mndfp oa avgmzcay
be tfgp Prqay`acc igccy eq`da egipbry &7>56o# z,70>., Taypegcc &7>49. nl`nigpa` g lqmoar be bpfar
zcgiay nl pfaZrnlinzng tfara Latpbl gzzagry pb fgua mgyygda` fny rayqcpy pb ornld gobqp g icbyar
enp oaptaal pfabrapnigc zra`nipnbly gl` avzarnmalpgc rayqcpy,
Prqay`acc &7>56o# z,>7. zbnlpy bqp pfgp nl Obbk NN be pfaZrnlinzng Latpbl# qynld pfa lat
zf~yniy# tgy agcnld tnpf zfalbmalg tfnif fg` lauar oaebra oaal pragpa` nl pfny tg~, QlcnkaObbk N nl tfnif fa tgy ebccbtnld nl pfa ebbpypazy be bpfar lgpqrgc zfncbybzfary# nl Obbk NN fa tgy
prag`nld lat drbql`, Prqay`acc nl`nigpay pfgp mbyp be tfgp Latpbl trbpa nl Obbk NN tgy egcya&Prqay`acc 7>56g# z,>7. oqp pfgp lauarpfacayy tfgp Latpbl gifnaua` pfara tgy lbpatbrpf~ gl` mqyp
oa hq`da`# lbp giibr`nld pb tfgp ta klbt lbt oqp o~ tfgp tgy klbtl nl pfa 74pf ialpqr~,
Latpbly gppamzpy pb raiblinca fny rayqcpy tnpf fny avzaipgpnbly ny p~znigc be ybmabla oragknld
lat drbql` gl` pfny ny hqyp pfa zbynpnbl be Latpbl tfal nluaypndgpnld pfa ibmzcav zfalbmalg
`nyiqyya` nl Obbk NN be pfaZrnlinzng, Ta yaa fara ybmapfnld be gl nl`nun`qgc yinalpnyp gp tbrk
`bnld gcc fa igl pb yfbt pfa ugcqa be fny zbynpnbl rgpfar pfgl pfa ibmzcapa`# eqcc~ nlypnpqpnblgcnya`#
bohaipnua yinalia be pavpobbky,
@neearalia oaptaal parrayprngc gl` iacaypngc zf~yniy
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iacaypngc zfalbmalg# `agcp tnpf nl Obbk NNN gra mbra ynmzca gl` cayy zrbla pb qlzra`nipgoca
nlecqaliay yqif gy ernipnbl# raynypglia# pqroqcalia gl` yb bl, Pfa enp oaptaal Latpbly pfabr~ be
ialprgc ebriay zrayalpa` nl Obbk N gl` pfa gyprblbmnigc `gpg zrayalpa` nl Obbk NNN ny mqif icbyar
pfgl ebr pfa zfalbmalg `nyiqyya` nl Obbk NN, Gy egr gy zbyynoca# Latpbl gppamzpa` pb iblprbc
igraeqcc~ nrracauglp ugrngocay oqp# gy fa `n` lbp eqcc~ ql`arypgl`nld pfa zfalbmalg fa tgy `agcnld
tnpf# fa tgy lbp gctg~y gtgra be pfa lgpqra be pfaya nrracauglp ugrngocay,
Zgrp be Latpbly zqrzbya nl trnpnld Obbk NN tgy pb raeqpa @ayigrpay pfabr~ be zcglapgr~mbpnbl nl tfnif pfa zcglapy tara igrrna` grbql` pfa Yql o~ ecqn` ubrpniay &Latpbl 74?>"7>5:#
Obbk NN# Yaipnbl NV; Obbk NNN# Enlgc Dalargc Yifbcnqm; Taypegcc 7>6:# z,010., Latpbl `nyiqyya`
inriqcgr mbpnbl nl ecqn`y nl Zrbzbynpnbly 17# 1? gl` 19 be Obbk NN, Gp pfa oadnllnld be pfaebccbtnld Yifbcnqm fa iblicq`a`8
Falia np ny mglneayp pfgp pfa zcglapy gra lbp igrrna` rbql` nl ibrzbragc ubrpniay; ebr# giibr`nld pb pfa
Ibzarlnigl f~zbpfayny# pfa zcglapy dbnld rbql` pfa yql raubcua nl accnzyay# fgunld yql nl pfanr ibmmbl
ebiqy; gl` pfa rg`nn `rgtl erbm pfa yql `ayirnoa gragy zrbzbrpnblgc pb pfa pnmay, Oqp lbt pfa zgrpy be g
ubrpav igl lauar raubcua tnpf yqif g mbpnbl, &Latpbl 74?>"7>5:# z, 9>1.
IBLICQYNBL
Pfa mgparngc zrayalpa` gobua dnuay nlyndfp nlpb Latpbly dabmaprnigc zrbia`qra gl` avzarnmalpgc
paiflnwqa, Pfara ny mqif pfgp ypq`alpy igl cagrl erbm tfgp Latpbl `n , Ebr avgmzca# fny
avzarnmalpy igl oa razcnigpa` gl` pfa rayqcpy glgc~ya` qynld fny pfabr~; gcparlgpnua aypnmgpay beGg
tfnif pgka nlpb giibqlp pfa ra`qipnbl nl ynxa be pfny `neearalia ebr agif yqiiayynua ytnld igl oa
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`arnua`; iqrua-enppnld paiflnwqay igl oa qya` pb glgc~ya fny btl rayqcpy &zrayalpa` nl Pgoca 7. gl`
pb prglycgpa tfgp fa `n` nlpb mbra mb`arl parmy; pfa gyyqmzpnbly fa mg`a pb algoca fny `gpg pb enp
fny avzaipgpnbly igl oa avzcbra`; gl` mbra raialp ql`arypgl`nld be pfa lgpqra be unyibqy `rgd
&zrbzbrpnblgc pb U. igl oa nlprb`qia` nlpb fny glgc~yay, Gcc be pfny nlubcuay ibmnld pb glql`arypgl`nld be fny mb`a be pfbqdfp nl bpfar tbr`y zcginld blayace nl Latpbly yfbay gl`
oadnllnld pb gzzraingpa pfa `neearaliay oaptaal tfgp Latpbl `n` gl` tfgp ta tbqc` `b tnpf fny
zrbocam pb`g~, Pfa avzarnmalpy gra ynmzca albqdf lbp pb rawqnra ybzfnypnigpa` nlyprqmalpgpnbl gl`pb oa giiayynoca pb ypq`alpy nl mgl~ iblpavpy, Pfa~ gcyb yfbt pfa dalnqy be Latpbl tfal bla
oadnly pb gzzraingpa pfa pfnlknld be g mgl gp pfa ebraerblp be pfa `auacbzmalp be maifglniy,
LBPA
7 Gnpbl &7>4?. zbnlpa` bqp pfgp# nl yznpa be Latpbly aeebrpy# pfa ubrpav pfabr~ ecbqrnyfa` bl pfa
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np yaama` pb oa ogya` bl pfa avnypalia be biiqcp ebriay gp g `nypglia tfara pfara tgy lb zf~ynigc
iblpgip oaptaal pfa nlpargipnld bohaipy, Gcc pfa bohaipnbly pb pfa Igrpayngl ubrpniay fg` oaal klbtebr g cbld pnma, Bl pfa bpfar fgl`# gy g igqya# pfa gpprgipnbl tgy hqyp gy qlgiiazpgoca gy np fg` oaal
ebr buar g ialpqr~ &Gnpbl# 7>4?# z,?04.,? N gm nl`aopa` pb Gcgl Ammarybl gl` Mnifgac Ebtcar ebr faczeqc ibmmqlnigpnbly gobqp pfa
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_AEA_ALIAY
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Ifgl`rgyakfgr# Y,8 7>>1# Latpbly Zrnlinzng ebr pfa Ibmmbl _ag`ar# Bvebr` Qlnuarynp~ Zrayy#Bvebr`,
Ibfal# N,O, 7>47# Nlprb`qipnbl pb Latpbly Zrnlinzng# Igmorn`da Qlnuarynp~ Zrayy# Igmorn`da,@alymbra# @,8 7>>1, Latpbly Zrnlinzng8 Pfa Ialprgc Grdqmalp, Draal Cnbl Zrayy# Yglpg Ea#
LM,
Dgqc`# I,E,8 7>>6# Ibccn`nld Zal`qcqmy# Iblyarugpnbl be Mbmalpqm gl` Latpbl$y Pfnr` Cgt#
Gqyprgcngl Yinalia Pagifary Hbqrlgc# 00&9.# 94-96,Dgqc`# I,E, 7>>># Qynld Ibccn`nld Zal`qcqmy pb Pagif Latpbl$y Pfnr` Cgt# Pfa Zf~yniy Pagifar#
94# ?1-?6,
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gl`A`qigpnbl,79# 559-549,Kn``# _,O, % Ebdd# Y,C8, ?::?# G Ynmzca Ebrmqcg ebr pfa Cgrda-gldca Zal`qcqm Zarnb`# Pfa
Zf~yniy Pagifar# 0:# 67-69,
Kb~r# G,8 7>51# Latpblngl Ypq`nay, Ifgzmgl % Fgcc# Cbl`bl,Latpbl# N,8 74?>"7>5:# Pfa Mgpfamgpnigc Zrnlinzcay be Lgpqrgc Zfncbybzf~# &prglycgpa` erbm pfa
pfnr` a`npnbl o~ Gl`rat Mbppa gl` raunya` o~ Ecbrngl Ighbrn.# Qlnuarynp~ be Igcnebrlng Zrayy#
Oarkaca~# IG,Prqay`acc# I,8 7>56g# G Zrbdrgm pbtgr` _a`nyibuarnld pfa _gpnblgc Maifglniy be pfa Gda be
_agybl# Nl I, Prqay`acc#Ayyg~y nl pfa Fnypbr~ be Maifglniy# Yzrnldar-Uarcgd# Oarcnl# 61-794,Prqay`acc# I,8 7>56o# _agipnbly be cgpa Ogrbwqa Maifglniy pb Yqiiayy# Iblhaipqra# Arrbr# gl`
Egncqra nl LatpblyZrnlinzng# Nl I, Prqay`acc#Ayyg~y nl pfa Fnypbr~ be Maifglniy# Yzrnldar-Uarcgd# Oarcnl# 796-769,
Taypegcc# _,Y,8 7>49# Latpbl gl` pfa Eq`da Egipbr# Yinalia# 74># 417-416,Taypegcc# _,Y8, 7>6:# Lauar gp _ayp8 G Onbdrgzf~ be Nyggi Latpbl# Igmorn`da Qlnuarynp~ Zrayy#
Igmorn`da
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8/3/2019 Colin F. Gauld- Newton's Use of the Pendulum to Investigate Fluid Resistance: A Case Study and Some Implications
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Taypegcc# _,Y,8 7>>:# Mgknld g Tbrc` be Zrainynbl8 Latpbl gl` pfa Iblyprqipnbl be Wqglpnpgpnua
Zf~yniy# Nl E, @qrfgm % _,@, Zallnldpbl &a`y. Ybma Prqar Mapfb`y8 _aecaipnbly bl pfaFarnpgda be Latpbl, Ibcqmong Qlnuarynp~ Zrayy# Lat ^brk# 1>-64,
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8/3/2019 Colin F. Gauld- Newton's Use of the Pendulum to Investigate Fluid Resistance: A Case Study and Some Implications
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PGOCA 78 Latpbly zal`qcqm `gpg
Igya Enryp
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Gg Ibmmalpy
7 ? 7,41 :,?1 750 :,::710 @ngmapar be tbb`al obo =
? 0 9,1 ,1 7?7 :,::079 5,641 Cbl`bl nlifay9 6 4 7 5> :,:700> Mgyy be obo = 14,976
0 75 70 ? 91,1 :,:1590 prb~ bqliay
1 9? ?6 0 76,1 :,?75?? Zal`qcqm caldpf =
5 50 15 6 > :,6666> 7?5 nlifay
4 ? 7,1 :,1 940 :,::790
6 0 9 7 ?4? :,::956
> 6 5 ? 75?,1 :,:7??9 gy ebr igyay 7-5
7: 75 7? 0 69,9 :,:06::
77 9? ?0 6 07,4 :,7>?::
7? 50 06 75 ??,4 :,4:166
79 7 :,641 :,7?1 ??5 :,:::11 @ngmapar be cag` obo =70 ? 7,41 :,?1 ??6 :,::77: ? nlifay
71 0 9,1 :,1 7>9 :,::?1> Mgyy be obo = ?5,?1
75 6 4 7 70: :,::470 prb~ bqliay
74 75 70 ? >:,1 :,:??7: Zal`qcqm caldpf =
76 9? ?6 0 19 :,:4104 7?5 nlifay
7> 50 15 6 9: :,?5554
?: 7 :,41 :,?1 17: :,:::0>
?7 ? 7,1 :,1 176 :,:::>4
?? 0 9 7 0?: :,::?96
?9 6 5 ? 976 :,::5?> gy ebr igyay 79-7>
?0 75 7? 0 ?:0 :,:7>57
?1 9? ?0 6 7?7 :,:557?
?5 50 06 75 4: :,??614
?4 9? ?6 0 7: :,0:::: ( yaa lbpa oacbt
?6 50 06 75 61,1 :,76479 nl gnr
?> 9? ?0 6 ?64 :,:?464 gy ebr igyay 9:T-95
9:G 75 7? 0
97G 6 5 ? 191 :,::940 ) yaa lbpa oacbt
9:T 75 7? 0 :,069 6,?4165 nl tgpar
97T 6 5 ? 7,? 7,55554 @ngmapar be cag` obo =
9? 0 9 7 9 :,99999 9,5?1 nlifay
99 ? 7,1 :,1 4 :,:4709 Mgyy be obo =
90 7 :,41 :,?1 77,9 :,:???? 755,7?1 prb~ bqliay
91 :,1 :,9?1 :,7?1 7?,4 :,:796? Caldpf be zal`qcqm =
95 :,?1 :,7591 :,:5?1 79,9 :,::515 790,941 nlifay
94 75 7? 0 9,9?1 7,?:9:7 @ngmapar be obo = 7 nlif
96 6 5 ? 5,1 :,9:45>
9> 0 9 7 7?,7 :,:6?45
0: ? 7,1 :,1 ?7,? :,:?916
07 7 :,41 :,?1 90 :,::491
0? :,1 :,9?1 :,7?1 19 :,::99:
09 :,?1 :,75?1 :,:5?1 5?,? :,::7::
( `ngmapar = 76,41 nlifay; mgyy = ?:6 prb~ bqliay; zal`qcqm caldpf = 7??,1 nlifay; Latpbl yg~y pfgp pfny igya pbbk 1ytnldy oqp fny glgc~yny nl`nigpay pfgp pfny yfbqc` oa 7:,
) Latpbly Pgoca dnuay 191 ytnldy ebr igya 9:G oqp fny glgc~yny nl`nigpay pfgp pfny yfbqc` oa ebr igya 97G,
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8/3/2019 Colin F. Gauld- Newton's Use of the Pendulum to Investigate Fluid Resistance: A Case Study and Some Implications
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Endqra 7, Mbpnbl be g i~cnl`ar pfrbqdf g raynypnld ma`nqm,
Endqra ?, @ngdrgm ebr Latpbly glgc~yny be pfa mbpnbl be g i~icbn`gc zal`qcqm pfrbqdf g raynypnldma`nqm &Zrnlinzng# Obbk NN# Zrbzbynpnbl 9:.
Endqra 9, @apgnc be Endqra ?,
Endqra 0, Ynmzcnena` uarynbl be Endqra ?,
Endqra 1, Drgzfnigc yqmmgr~ be Latpbly rayqcpy ebr i~icbn`gc zal`qcqmy mbunld pfrbqdfraynypnld ma`nqmy &Igyay 7-95.
Endqra 5, Latpbly `ngdrgmy ebr fny ybcqpnbl pb Zrbocam UN &Zrnlinzng# Obbk NN# Zrbzbynpnbl ?>.,
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