Name
P e r i o d-
Date
O KEPTER,S
LAw oF EQUALAREAS
IohannesKepler discoveredthat the planetsfotlow eltipticalpaths
a.ound the sun. He also discoveredthai an inraSinaryline from the sun
to a planet sweepsout equal areas in equal t:ne intefvals.The
photographsin Figlre 14- 2 on page60 sere madeby i]luminatinga white
pendulumbob and photographin8it in jts rest pos ion. The bob was then
madeto swingso that ii follou.edan
ellipticalpath.Bv meansofa motof,
ized stroboscope.
the bob was then photog.aphed
in seve.alpositjons
all on the samefilm exposurethat was usedto photograph
ihe restposiiion of the bob.The notion of the bob closelyduplicaresrhe '1otionol
a planetas it orbits the sur.
Camera
F r p p l el a n k l i g h t
E a l lp a i n t e dw h i t e
Blackpaper
F i q ! r e 1 4 - 1 . T h e p h o r o q r a p h si . F i c ! r e 1 4 - 2 w e r e t a k e n w i t h a p p a r a i u ss r i r i t a rl o
l h i s . A w h i l e b a l l i s s u s p e n d e da b o v ea d a r k b a c k q r o ! . d T h e c e n t r a lp o s i | o . o f r h e
b a l l l s r e c o r d e do n t h e t i r m .O n t h e s a m e ' m e x p o s ! r e ,t h e s w n g t . g b a i s p h o r o .
9 f a p h e du s n g l h e s l f o b o s c o p er o o b l a i n e ! p o s u r e sa t l r e q ! e n r i . t e d a t s a t o . g r h e
p a l h o l t h e b 3 l l I s i m p o d a n tr o c t o s e l h e c a m e r as h u l l e r a f i e r t h e b a h a s c o m .
p r e t e d o n er e v o r uor n . A s e c o n dm e r h o do t o b t a . i n g r h e s a m e e i t e c rs r o u s e a s r r o b e
l i g h l a s a s o u r c eo l l r ! h i n a t i o n a s r h e p h o l o g r a p hr s r a k e n .
\-,/oleclrves
During this investigation
vou will
study Kepler'slaw oi equalareas.
notethat suchIaas are not conlinedto specialcases,but afe tfue iol
all circular o. elliDlicalmotion.
Equipment
rulef and pencil
thin tracingpaper
t4
s
E
E
F Q U r e - 1 4 - 2l a
. ) T h e e l t i p t i c a tp a l h o ' a s u s p e n d e db a L E a c h p o s i l i o n
s separated
r r o o r h e n e x r b y e x a c l r yr h e s a m e r i m e i n t e d a . ( b )
A n e . t a r g e d p o r l i o no r ; ; t .
60
Name
Period _
Date
Procedure
{
1. Examinethe photographs
in Figure14 2. Assumethat the iniervals
betweenconsecutive
imag!sof the bob are equal_
The bob therefore
movedffom any position shown to iis next position in equal time intervals- To determinethe area s$,eptout by an imaginary lhe from
the bob to tle central imagein equal times,placea pieceof thin paper
over Figure 14-2.
2. Placea dot ovefthecenterofthecentralimageiOl.Locatethe centers
oi the imagesmarled A, B. C, and D. place a dot over the centers
of each of theseimages.
3, Removethe paperand constructtwo tfiangles(ABOand CDO)using
lhe five points mafked on the paper. With a ruler. carefully measure
the bases and heights of each triangle.Record in Table 14-1.
Calculatethe area of each.
4. RepeatSteps2 and 3 usingdifle.ent positionsof the images.you may
useeitherFigur!14-2{a)orFiSureta-2(b).Do not try to usethe ex,
treme light of the pholographs
$.hefe the imagesrun together.
Doto ond Colcuiotions
Calculations
iriangle
Base
AOB
coD
Interpretotion
1. Ivere the afeas of the trianglesin closeagreement?
How do you
accountior slightdiscrepancies?
6t
2 . \ \ h " n l h " o a b f o l o q - a n c l j r p t i d, l p " r r . i r s . , m a r m p s
L.oiej.o tro
. p n r r a l p o . r ' r o nl h " r . t i s
".orhpr..mas.Il an inagin"rI Jn- fron
r h l eb o b o r h e / p n . r . l i m o s e ! $ e p p ( o u r
in "qu"il.n ""
$na 7o- lo- on, Jd"
u le "br o" ubi$ h p n
l" ot ur h
I rsL.o,er
".b6onupr"r,h" c s p p F , o
l o l h p L e n l r d li n a B e
1 \ \ . . t hr h p . p e p o o l r r e o o b h h . n
"
.
ii is larlher awa! irom the cenirat image?
i
3 . The"ar rh fotlouran "Jipri,aJ,p"rhdroL.nd
.h. .1.n.The
ear,n:r . toser
r o r n es u nD \ a b o u r6 4 , 1 0 b I mi n l n e $ . i n t e lrh a n i r
isrnthe,umm"L Comparethespepdol rhe-d h aronSils orbrrur rhp
u.nr"r q. rh
r n Fs p a p do l h F F a r , ha o n sr r - o r b r t: n l h L . - m m e r .
Extensions
1
Although lhe orbil of a planet around the sun
is curved, we can ap,
p r o \ : m . t p s p s m p n ro
s f l h p o . b j r a s s t r . i S b tl i n e - r l s e ,
n o o . cc o n
s p . u l t r e p o < r l . o n 5o l l h e p l " n e l r e r \ . l o s F r o g e l h F r
As r"n be seen
r4_2.thFsh.peoi hp1.pa,BFpro r. br an inagrn.rl
i:T,il:il:
l l r e 0 F | l r e e t] h ep t " n e t
d n d . n es u n . : d l r : d n S . pT.h e h e . r h t ; t t b F
triangleis approximalely
r. and iis baseis equalro the a;sjancethe
plan!thastraveledin the lime inrerlal i. Show
rhai if the p)anerhls
a speedvr at distancer, lrom lhe sun and speea,,
ar aisrrrc".r,
f1v1 = r2v2.
2 . Sinc!ihe massof lhe planei is consiant.!\!
can !|.rjt!
m-rrtr : mr2r,
Ihe qu"ntrr\,1.rr .."lted ',
" "nsul"r mompnr-ffot tnep.a-e,.i- ors u r P rm o T e n t u m
o l r h ep l " 1 - r o n - e f f " d ' r r p l " i n
62
t-