The little known story of F = ma and beyond

1153RESONANCE � December 2009

GENERAL � ARTICLE

The Little Known Story of F =ma and Beyond

Amitabha Ghosh

KeywordsNewton's laws, Kepler's laws.

Amitabha Ghosh was aProfessor at IIT Kanpurtill his retirement in 2006.In between he spent fiveyears (1997 to 2002) at IIT

Kharagpur as theDirector. Currently he isworking as a SeniorScientist of the Indian

National Science Academy,New Delhi and is ssociatedwith both IIT Kanpur andBengal Engineering &

Science University, Sibpuras Honorary Distinguished

Professor.

F = m a is th e m o st fre q u e n tly u se d e q u a tio n

in b o th sc ie n c e a n d e n g in e e rin g . H o w e v e r, n o t

m a n y a re v e ry fa m ilia r w ith th e e v o lu tio n a ry p ro -

c e ss th ro u g h w h ic h th e e q u a tio n ¯ n a lly e m e rg e d .

T h is a rtic le p re se n ts a b rie f a c c o u n t o f th e o rig in

o f th e m o d e rn c o n c e p t o f fo rc e -m a tte r in te ra c -

tio n a n d th e tra n sito ry c o n c e p ts. T o w a rd s th e

e n d , th e a rtic le a lso ta k e s u p so m e still-to -b e -

re so lv e d m a tte rs a sso c ia te d w ith th e e q u a tio n .

It is d ou b tless th at th e m ost ex ten sively u sed q u a n ti-tative law of n atu ra l scien ce is F = m a . In a lm ost allb ran ch es of scien ce an d en gin eerin g th is law p lay s a p iv -

ota l role. F or th is reason all stu d en ts are in tro d u ced toth is law of m o tio n a t th e sch o ol level. H ow ever, it isstra n ge th a t very few u sers o f th is law a re fam ilia r w ithth e ev olu tio n ary p ro cess th ro u gh w h ich th e law h as b eend iscovered . M o st a re of th e u n d ersta n d in g th a t N ew -

ton d iscovered th is law an d w ro te it in h is fa m o u s b o okP rin cipia an d th ere is n o n eed to h ave a relo o k in to th eissu es in volved a s a ll asp ects are fu lly u n d ersto o d .

T h e p rim a ry o b jectiv e o f th is a rticle is to p resen t th ew on d erfu l ev olu tio n ary h istory b eh in d th e d evelop m en tof th e p rofou n d u n d erstan d in g req u ired to u n earth th em y stery of th e scien ce of m otio n . It is also im p o rtan t tok n ow a b o u t th e co n trib u tion s of g rea t in tellectu a ls of th e

p ast w h o se u n d ersta n d in g s a n d ¯ n d in g s u ltim ately ledN ew to n to sy n th esize th eir id ea s p ro p osin g th e law s ofd y n am ics. T h is m ay a lso h elp resea rch ers an d stu d en tsof scien ce to realize h ow actu al p rogress is m ad e in sci-en ti c d iscoveries th rou g h th e cu m u la tiv e accu m u lationof th e co n trib u tion s b y a large n u m b er of scien tists over

a lo n g p erio d of tim e.

1154 RESONANCE � December 2009

GENERAL � ARTICLE

O n ce h u m an s d iscov ered ag ricu ltu re, fo o d su p p ly b eca m e assu red an d th e w an -

d erin g n a tu re of th eir life ch an g ed to a m o re stab le ch a racter. T h u s, for th e¯ rst tim e, th ey b eca m e aw a re of th eir su rro u n d in gs an d n oticed th e n atu ral p h e-n om en a h ap p en in g a rou n d th em . M ost likely th e ¯ rst th in g th at attracted th eiratten tio n w a s th e m ovem en t of ob jects, i.e., m o tio n . B u t th e b ew ild erin g co m -p lex ity an d va riety o f a ll ty p es of m o tio n s m ad e th e u n d erstan d in g of th e scien ce

b eh in d th ese p h en o m en a d i± cu lt. A ristotle (38 4{3 22 B C ) w as th e ¯ rst p h ilo so-p h er w h o p rov id ed a com p reh en siv e th eory o f m otion . S in ce th e con cep t o f forceas a cau se of m otion w a s b ey on d co m p reh en sion d u rin g th ose d ay s, all m otion sw h ich to o k p lace w ith o u t th e d irect ob serva b le in terv en tio n o f an ag en t (likeh an d ) w ere term ed as n atu ral m o tio n s'. T h is w ay it w a s ex p lain ed th a t a h eav y

ob ject falls tow a rd s th e E arth as th e cen tre o f th e E arth w as th e n atu ral p la ce'of th ese o b jects. S im ilarly, ¯ re a n d ¯ ery ob jects m oved u p w ard s as th e h eaven 'w as th eir n atu ral p la ce. T h ose m o tio n s w h ich d e¯ ed th e n atu ra l' ex p ectation sw ere term ed as v io len t m otion s' (lik e a sto n e th row n u p w a rd s).

T h e situ a tion rem a in ed m ore o r less u n ch an g ed for a b o u t tw o m illen n ia u n tilK ep ler (1 57 1{1 63 0) started in vestigatin g p lan etary m o tion s. T ill th at tim e allp h ilo sop h ers a n d scien tists con sid ered th e h eaven ly o b jects to b e m a d e u p of asu b stan ce called th e ` fth elem en t' (d i® eren t fro m th e fo u r b a sic elem en ts con sti-tu tin g all terrestria l ob jects) w h ose n atu ra l m otion w as u n iform circu la r m o tion

aro u n d th e cen tre of th e u n iv erse (till C o p ern icu s's tim e it w as n ea r th e cen treof th e E a rth ); later th is ch a n ged to th e th eory th a t a ll p lan eta ry m otio n s co u ldb e d escrib ed b y sy stem s of p erfect circles. H ow ev er th e cen tres of th e circlesd escrib in g th e p lan ets' m otion s w ere v oid p oin ts in sp ace a little aw ay fro m th ecen tre o f th e E a rth (a fter C o p ern icu s th ese w ere a little aw ay fro m th e S u n )!

U sin g th e very accu rately ob serv ed sy stem a tic an d ex ten sive d a ta, accu m u la tedb y h is em p loyer T y ch o B rah e (1 546 {1 601 ), K ep ler w as ab le to d em on strate th a tth e p la n ets m oved in tra jectories ¯ x ed in th e h elio-a stral sp a ce (ex cep t for th e ex -trem ely sm all p recessio n al m o tio n ). T h e co n cep t of a n orb it, i.e., a p a th ¯ x ed inth e h elio-a stral sp a ce th at is fo llow ed b y a p lan et, w as ¯ rst in tro d u ced b y K ep ler.

In itially h e a lso con sid ered (lik e all a stron o m ers b efore h im for a b o u t 20 00 years)th e o rb its to b e eccen tric circles w ith th e S u n a little aw ay from th eir cen tres assh ow n in F igure 1.

T h e lin e A P , th e d ia m eter p a ssin g th ro u gh th e cen tre O a n d th e S u n , is ca lledth e lin e of a p sid es'. F or th e ¯ rst tim e in th e h isto ry of a stron o m y, K ep ler p rov edth at th e orb ital p lan es of a ll p lan ets w ere tilted a t sm all an g les w ith th e eclip ticp lan e, i.e., th e orb ital p la n e o f th e E arth . K ep ler w as a cu tely u n com forta b le w ith

GENERAL � ARTICLE


Planet

Line ofapsides

SunP A

O

Figure 1. Eccentric circleas planetary orbit.

Figure 2. Concept ofequant.

th e id ea th a t a h u ge ob ject lik e T h e S u n ju st sits id le a little aw ay from th e

cen tres, aro u n d w h ich th e p lan ets m ove, w ith ou t d o in g an y th in g ! H e in tu itiv elyfelt th at th e S u n m u st b e b eh in d th e orb ital m o tion s o f a ll th e p lan ets. H issu sp icio n w a s in ten si ed w h en h e fo u n d th at th e o rb ital p la n es o f a ll p la n etscon tain on e co m m on ob ject { th e S u n an d th e a p sid al lin es o f th e orb its of allp lan ets p a ss th rou gh th e S u n ! T h u s, K ep ler a rrived a t th e va gu e con cep t of an

en tity lik e force' th at cau ses b o d ies to m ove. H is su sp icio n tra n sform ed in to a¯ rm con v ictio n th a t th e S u n d rives all th e p lan ets in th eir o rb its b y ex ten d in g aforce th rou gh sp ace to th ese h eaven ly b o d ies. T he con cept of force in the m odernsen se w as born .

It w a s k n ow n sin ce a n tiq u ity th at th e m otion s o f th e p lan ets a lo n g th eir resp ectivecircu la r orb its w ere n ot u n iform as seen from th e cen tre o f th e circle. P tolem yin tro d u ced th e con cep t of eq u a n t', a p o in t w ith resp ect to w h ich a p la n et d e-scrib es th e circu la r orb it w ith u n ifo rm a n gu lar sp eed . F igu re 2 sh ow s th e eq u a n tfor a p lan et. W h en a p la n et g o es from a p osition P 1 to P 2 in a tim e in terval ¢ t

an d d u rin g a n eq u al tim e in terva l th e p lan et go es fro m p ositio n P 3 to P 4 , so th a tth e ra te of ch a n ge o f a n gle μ ab ou t th e eq u a n t p oin t E is con sta n t. K ep ler, w h ow as ob sessed w ith th e id ea th at th e S u n forces th e p la n ets to m ove, cam e w ith ap h y sical reason b eh in d th e a b ov e featu re o f p lan etary m otion . H e reaso n ed th a tas a p lan et a t th e ap h elio n p o sitio n A is farth est from th e S u n , th e S u n 's force

on th e p la n et w ill b e m in im u m resu ltin g in m in im u m sp eed of th e p la n et an d

APS O E

P2

P1

P3

P4

ee

r

��


GENERAL � ARTICLE

Figure 3. Kepler’s idea ofgravitational interactionbe-tween theSunandaplanet.

th e d istan ce trav elled in th e given tim e in terval, P 1 P 2 , w ill b e m in im u m . O n th e

oth er h a n d at th e p erih elio n p o sition P th e p lan et is n earest to th e S u n a n d w illm ove at th e fastest sp eed d escrib in g th e larg est arc P 3 P 4 in th e given tim e in terval¢ t. In itially K ep ler form u lated a d istan ce law ' acco rd in g to w h ich th e in ° u en ceof th e S u n , i.e., a n en tity lik e th e force ex erted , w as in versely p ro p o rtion al to th ed istan ce. K ep ler reason ed th at th e p u rp ose of th is force em an a tin g from th e S u n

w as to m ove th e p la n ets in th e orb ital p lan e an d th erefo re it w ill sp rea d a lo n g th ep lan e m a k in g th e in ten sity fall in v ersely w ith th e d istan ce. H e called th is fo rce asa n im a m o trix ' an d th e sch em e is sh ow n in F igure 3. T h erefo re, as p er K ep ler'sth eory

F ® 1= d

A s K ep ler erron eo u sly a ssu m ed sp eed to b e p rop ortion a l to force in h is sch em e,th e p lan et's sp eed w as to b e in versely p ro p ortion a l to th e d istan ce from th eS u n . T h ou gh h e w as th e ¯ rst to create a p h y sical m o d el o f th e p lan etary m o tionrep la cin g th e 2 000 -y ea r-old geom etric m o d el, b eca u se of w ro n g p h y sics h e co u ldn ot m a tch th e o b servation s w ith h is th eory.

K ep ler a lso p rop osed a ra te of rota tio n fo r th e S u n con sid erin g th e rota tio n al m o-tio n of th e p la n et u n d er th e in ° u en ce of th e S u n . G a lileo's telesco p ic ob serva tionof th e su n sp o ts a ctu a lly rev ea led th e rota tio n of th e S u n an d th e ra te o f rota tionw as fou n d to b e close to th at p red icted b y K ep ler. T h o u gh th is p rov id ed im m en se

sa tisfa ction to K ep ler it w a s n oth in g b u t a p u re coin cid en ce! W h ile stru g glin gto ex p lain th e o b servation w ith h is p h y sics', K ep ler assu m ed th e area law as anap p rox im a tio n for d escrib in g th e actu al m otio n . H e ¯ rst th ou gh t th is to b e anap p rox im a te a d h o c a ssu m p tion b u t later h e d iscovered th at th e area law w as anex a ct law . H is a n aly sis an d rea son in g w ere a s follow s:

F rom F igure 4 it is seen th a t

P 1 A =1

2¢ μ (r ¡ e);

Sun

Planet

Anima Motrix


GENERAL � ARTICLE

1The reader is advised to prove the area law for a general point on the orbit. This is valid for an elliptic orbit also!

Figure 4. Emergence of arealaw.

APS O E P1

P3��2

w h ere r is th e ra d iu s o f th e circle a n d e is th e d istan ce o f th e eq u a n t p o in t E from

th e cen tre O (it sh o u ld b e n oted th at th e d istan ce of th e S u n S fro m th e cen treO is a lso e). A g ain

P 3 P =1

2¢ μ (r + e):

W h en ¢ t is very sm a ll, th e sm a ll circu lar arcs m ay b e ap p rox im a ted as straig h tlin es. T h e area o f ¢ S P 1 A is eq u al to

1

2S A 1 :A P 1 =

1

2(r + e)

1

2¢ μ (r ¡ e) = 1

4(r 2 ¡ e 2 )¢ μ : (1 )

S im ila rly th e a rea of ¢ S P 3 P is eq u a l to

1

2S P 3 :P 3 P =

1

2(r ¡ e)1

2¢ μ (r + e) =

1

4(r 2 ¡ e 2 )¢ μ : (2 )

T h erefore, as ¢ μ is sam e for sam e tim e in tervals, th e p la n et's m o tio n d escrib eseq u a l a rea s a b o u t th e S u n in eq u al tim es a s fo u n d from (1 ) a n d (2).1 T h e rea d ercan ea sily n otice th at acco rd in g to K ep ler's d ista n ce law ' th e sp eed s a t A an d P ,V A a n d V P ca n b e ex p ressed a s follow s w ith C a s th e con sta n t:

V A = C = (r + e); V P = C = (r ¡ e):

S o, th e a rea s o f th e tria n gles S P 1 A an d S P 3 P for a given tim e in terva l ¢ t can b eex p ressed as fo llow s:

A rea o f ¢ S P 1 A =1

2S A 1 :¢ tV A =

1

2(r + e):C ¢ t= (r + e) =

1

2C ¢ t;


GENERAL � ARTICLE

an d

A rea of ¢ S P 3 P =1

2S P :¢ tV P =

1

2(r ¡ e):C ¢ t= (r ¡ e) = 1

2C ¢ t:

T h is eq u ality is N O T tru e in gen eral for o th er lo cation s in th e orb it; b u t th is ledK ep ler a lo n g a w ron g tra ck fo r q u ite som e tim e. H ow ev er, all th ese correct an din co rrect b u t ap p aren tly co rrect resu lts con v in ced K ep ler th at a fo rce from th eS u n is d riv in g th e p lan ets.

S u b seq u en tly, w h en K ep ler d iscovered th at all p la n ets m ove in ellip tic o rb its w ithth e S u n at th eir fo ci, th ere w a s n ot a sh red of d o u b t in h is m in d th at th e S u nd rives th e p la n ets. U n fo rtu n ately, n on e of th e con tem p orary scien tists like G a lileob elieved in a ction -at-a -d istan ce an d d id n ot a ccep t th e con cep t of g rav itation .

In ° u en ced b y G ilb ert's b o ok on m agn etism K ep ler b elieved in th e ex isten ce o fforce th a t can act a t a d istan ce. F u rth erm ore a s G ilb ert d em on stra ted th at th eE a rth is a b ig m ag n et, K ep ler to o k to th e id ea of an in teraction b etw een th e S u n

an d th e p lan ets sim ila r to th at fou n d b etw een m agn ets an d p rop o sed a th eo ryth at ca n b e said to b e th e foreru n n er of N ew ton 's th eory o f u n iversal grav itation .T h e follow in g p a ssag e from A stron om ia N ova sh ow s h ow clo se K ep ler cam e toth e con cep t of g rav itation .

\G rav ity is th e p rop en sity b etw een like b o d ies to u n ite or com e to-geth er ::: so th at th e E a rth d raw s th e ston e to it m u ch rath er th a n

th e sto n e seek s th e E a rth :::. If tw o ston es w ere to b e p la ced a n y -w h ere in th e w o rld o u tsid e th e ran ge o f in ° u en ce of a th ird sim ila rb o d y, th en each sto n e, lik e tw o m ag n etic b o d ies, w ou ld co m e to geth erat an in term ed ia te p oin t, ea ch sto n e trav ellin g tow a rd s th e o th er ad istan ce p rop ortion a l to th e b u lk of th e o th er" .

T h e area law , lu ck ily d iscovered b y K ep ler, p lay ed th e m ost cru cia l ro le in h is

d iscovery of th e ellip tic n atu re of th e p la n etary orb its. H ow ever, th e n atu re o faction b etw een force a n d m a tter, i.e., th rou g h a far m ore su b tle in teractio n in -vo lv in g in stan tan eou s a cceleration w a s co m p letely b ey on d h im . F u rth erm o re, inth e ab sen ce of a law o f in ertia of u n ifo rm rectilin ea r m o tio n an d th e seco n d law ,F = m a , it w a s im p o ssib le for h im to u n d erstan d th a t th e orb ital m otion w ascom p o sed of tw o m o tio n s { a u n ifo rm m otion d u e to in ertia an d a n accelera ted

m otion d u e to an a ttra ctive force' b y th e S u n . N o tw ith stan d in g th e ab ove sh ort-com in g , K ep ler's co n trib u tion s to th e early d ev elop m en t o f th e co n cep t of force


GENERAL � ARTICLE

2 It is truly strange that themost fundamental typeofmotion, i.e.,uniform rectilinearmotion, that plays the foundationalrole in the science of motion, is never observed in reality and the philosophers for two millennia were misguided bythe uniform circular motion of the heavenly bodies. According to scientists the discovery of the first lawwas, perhaps,the most difficult and important steps in the history of scientific revolution!

an d its ab ility to ca u se m o tio n of m atter a re p h en o m en al. K ep ler w as a lso th e

¯ rst to in tro d u ce th e term in ertia' in th e vo cab u lary of d y n am ics. T h is term w asu sed b y h im to d escrib e th e p rop erty of m a tter to rem ain at rest u n less oth erw iseim p ressed u p o n b y a fo rce. B u t h e d id n o t h ave th e u n d erstan d in g th at m atterten d s to rem a in in u n iform m otion also u n less acted u p o n b y a fo rce.

A s accelera tio n p lay s th e cen tral role in d y n am ics a n y fu rth er p rog ress co u ld b ep o ssib le on ly w ith a p rop er u n d erstan d in g o f a ccelerated m otion . S om e earlystu d ies on u n iform ly accelera ted ch an ge w ere con d u cted b y W illiam H ey tesb u ry(13 13{ 137 3) at M erto n S ch o ol, O x ford . M ore or less con tem p orarily, fu rth er

p rog ress w a s m a d e at P aris sch o ol u n d er th e lea d ersh ip of J ea n B u rid an (129 5{13 56 ) fo llow ed b y N ich o las O resm e (132 3{1 38 2). H ow ev er, th e w orld h ad tow ait till G a lileo G a lilei (15 64{ 16 42) co n d u cted ex ten sive stu d ies on u n ifo rm lyaccelerated m otion an d free fa ll. H e d em on stra ted th at th e d istan ce coveredu n d er u n iform a cceleration in creases w ith th e secon d p ow er o f tim e; h e d eriv ed

th e rela tion

s =1

2a t2 : (3 )

G a lileo m ad e tw o oth er critica lly im p orta n t con trib u tion s th a t led to th e d evel-op m en t of th e scien ce of m otion a n d d iscoverin g th e law F = m a . H e w as th e¯ rst to p ro p ose th e law of in ertia of m otion in a so m ew h a t ru d im en tary form .A ccord in g to h is u n d ersta n d in g if an ob ject is g iv en a m otion o n th e su rfa ce o f

th e E arth it w ill co n tin u e to m ove aro u n d th e E arth if n ot sto p p ed . T h u s it w asso m eth in g like a circu la r' in ertial m otion . T h e o th er m a jor b rea k th rou gh b y h imw as th e con cep t of resolv in g m o tio n s in to co m p on en ts. C o n versely, h e sh ow ed th ecom p o sition o f m o tion a n d w as th e ¯ rst to d em on strate th at a p ro jectile m o tionw as a com b in a tio n of a n in ertial m otio n (an ap p rox im a tely u n ifo rm rectilin ea r

m otion , in th e sm all scale a ccord in g to G a lileo) a n d a u n ifo rm ly accelera ted freefall. O n ly th e co n cep t of th e com p o sition o f m otion s cou ld u ltim a tely lead to th eth eory of orb ital m o tio n of th e p la n ets.

C losely follow in g G alileo, R en e D escartes (15 96{ 16 50) to ok u p th e stu d y o f d y -

n am ics an d w as th e ¯ rst to correctly en u m erate th e ¯ rst law o f m o tio n , i.e., ab o d y co n tin u es in a u n iform rectilin ea r m o tio n 2 w h en n o t o b stru cted . T h is cam e


GENERAL � ARTICLE

Figure 5. Logical thoughtexperimentbyHuygens todiscover the proportion-ality of force with accel-eration.

T T´ T

(a) (b) (c)

from th e ob serva tio n o f th e m o tion o f th e o b jects ° y in g o u t o f slin gs. M o tio n o f

w ater d ro p lets th row n aw ay from th e p erip h ery of sp in n in g w h eels also led to th erealization of ¯ rst law . T h is k in d of stu d y w as v ery p op u lar d u rin g th at p erio dto u n d erstan d th e situ ation relatin g to a sp in n in g E arth ! A s R en e D escartes'p h y sics a n d co sm olo gy in v olv ed co llision o f p articles, th e con cep t o f m om en tu m 'as a q u an tita tiv e m easu re of m otion w as p rop o sed b y h im an d h e attem p ted to

so lv e collision p rob lem s u sin g th is co n cep t w ith on ly p a rtia l su ccess. A ccord in gto h im m om en tu m w as th e p ro d u ct of sp eed an d b u lk o f m atter.

H is frien d an d d iscip le, C h ristian H u y gen s (162 9{ 169 5) to ok u p th e stu d y o f co l-

lision p rob lem s a n d so lv ed th e p ro b lem for ela stic collision s an d fra m ed th e con -cep t of th e law of con servation of lin ear m om en tu m correctly. H ow ev er, H u y gen s'm ost revo lu tion a ry con trib u tio n to d y n am ics w a s h is d iscov ery of th e rela tionb etw een im p ressed force a n d th e resu ltin g acceleratio n o f a b o d y. H is rea son in gan d th ou g h t p ro cess th at led to th is p rofou n d d iscov ery are p resen ted b elow .

H u y g en s ex a m in ed th e m a tter o f free fall u n d er th ree d i® eren t situ ation s a s in d i-cated in F igure 5 . H e rea son ed th a t th e tu g ' (ten sion in m o d ern la n gu ag e) T onth e strin g from w h ich an o b ject is h a n gin g p reven ts it from acceleratin g d ow n -w ard s d u e to free fa ll. S im ila rly, w h en th e o b ject is o n an in clin e (F igu re 5 b )th e tu g T 0is less an d is p rop ortion a l to th e accelera tion th e o b ject acq u ires on ceth e tu g va n ish es (i.e., th e strin g b reak s o r th e ob ject is released ). T h e situ a tionin d icated in F igure 5 c d em on strates th e fa ct th at th e tu g d ep en d s on ly on th eacceleratio n of th e ob ject im m ediately after th e relea se a n d n ot on th e su b seq u en tm otion . T h u s, H u y gen s reason ed th a t th e tu g th at p rev en ts th e b o d y from b ein g

accelerated is p ro p ortion a l to th e a cceleration it p reven ts. In versely, a tu g (o rforce in m o d ern la n gu a ge) ca n cau se a b o d y to accelerate an d th e m ag n itu d e o fth e a cceleration w ill b e p rop ortion al to th e tu g. T h u s, th e m o st im p orta n t (an d


GENERAL � ARTICLE

Figure 6. Study of circularmotion by Huygens.

d i± cu lt to o) q u an titativ e relation sh ip in d y n a m ics

T ® a (4 )

w as u n raveled b y H u y gen s. H e w a s also th e ¯ rst scien tist to arriv e at th e correct

m agn itu d e of cen trip etal a cceleration o f an ob ject m ov in g w ith sp eed V in acircu la r p ath o f ra d iu s r as V 2 = r. H e stu d ied th e p rob lem in a m an n er sim ila rto w h a t is p resen ted b elow . H e con sid ered a p oin t on a circle ex ecu tin g u n iformrota tio n . If th e p article is freed from th e circle at th e p o in t A it w ill m ove in astra igh t lin e acco rd in g to th e law of in ertial m otion a n d rea ch a p o in t P 0a fter ap erio d o f tim e t. H ad it rem ain ed a ttach ed to th e circle it w o u ld h av e reach eda p o in t P after th e sam e p erio d of tim e. H u y g en s ¯ rst p roved th a t th e p a th o frela tive m o tio n o f th e free p article w ith resp ect to th e p a rticle th a t is attach edto th e circle at P is rad ial at p oin t P (F igu re 6). W ith th is h e d em on stra tedth at th e ten d en cy of a free p a rticle is to ° y ou t rad ially ou tw ard s from th e p oin t

of release. S o, a p o in t/p a rticle th a t is ex ecu tin g u n iform circu la r m o tion h as arela tive m otion w ith resp ect to a free (i.e., u n a ccelerated ) p a rticle th a t is ra d ia llyin w a rd s at th e p o in t of th eir sep a ration , i.e., p oin t A . If th e circu m feren tial sp eedb e u an d th e free p article rea ch es th e p oin t B 00after a tim e t (w h en th e p articlein th e circu la r m otion reach es th e p o in t B ) th en A B 00= u t. F u rth erm ore, w h en tis very sm all th e cu rve B B 00w ill b e in d istin gu ish ab le fro m th e lin e B B 0(ex ten sionof th e ra d ial lin e O B ). N ow ,

B B 0=p(O A 2 + A B 02 ) ¡ O B :

B ecau se O A = O B = r an d A B 0¼ A B 00= u t,

B B 0 ¼ O Ap[1 + (A B 0= O A )2 ] ¡ O A

¼ 1

2(u 2 = r)t2 : (5 )

A AP'

P

O O

C

C' C''

B

B B''


GENERAL � ARTICLE

Figure 7. An experiment oncentrifugal force.

S im ila rly, after a p erio d of tim e 2t, th e free p article reach es th e p oin t C 00 (veryn ear to C 0) an d th e p o in t u n d er circu lar m o tio n go es to C . T h e d istan ce is

C C 0¼ (u 2 = r)(2 t)2 : (6 )

T h u s, th e p article u n d ergoin g circu la r m otion falls rad ially in w ard s w ith resp ectto th e u n a ccelerated free p a rticle (a t th e p oin t of sep aration ) w ith a u n iformacceleratio n as th e d istan ce fallen in creases w ith th e seco n d p ow er of tim e (asob serv ed in free falls b y G alileo). T h e m a gn itu d e of th is a cceleration is given b yu 2 = r.

H u y g en s con d u cted m an y ex p erim en ts w ith slin gs an d th rou g h ex p erim en ts onsta b ilizin g w h irlin g o b jects on a sm o oth tab le b y h a n gin g w eigh ts (as sh ow n inF igure 7 ) a n d com p a red th e tu g ex erted b y th e w eig h t w ith th e th eoretically esti-m ated cen trip etal accelera tion of th e ob ject. T h ese resu lts con ¯ rm ed h is in tu itived iscovery

F ® a : (7 )

T h is w as th e m o st im p ortan t d iscovery in th e scien ce o f m o tio n an d w a s p er-fected b y Isa ac N ew ton (1 64 2{1 727 ) in th e form of h is seco n d law of m otion .

N ew to n in tro d u ced th e co n cep t of m ass a s a m ea su re o f q u an tity o f m atter an dth e p rop ortion a lity w a s rep la ced b y eq u ality in th e follow in g m a n n er

F = m a : (8 )

T h e o rig in a l fo rm of th e seco n d law , a s p rop o sed b y N ew to n , w a s, o f cou rse,in term s of th e ra te o f ch an g e of m om en tu m as w e a ll k n ow . W h en m a ss iscon sid ered in va rian t, th e rate o f ch a n ge o f m om en tu m is n oth in g b u t th e m asstim es th e acceleratio n . T h e rea son b eh in d N ew ton 's ex p ressin g seco n d law interm s o f m om en tu m lies in h is w ork on orb its of p lan ets d u e to th e grav ita tio n al

attra ction o f th e S u n a s in d icated in F igure 8 . H e ex p la in ed th e orb ital m o tionas fo llow s. A p lan et at lo cation A go es in a sm a ll tim e in terval ¢ t to p oin t B .


GENERAL � ARTICLE

Figure 8. Demonstration ofplanetary orbit by Newton.

O A

B

B'

b

C

c

C'

D

S u b seq u en tly, it w ou ld h ave reach ed p o sition B 0 h ad th ere b een n o fo rce a ctin gon it d u rin g a n eq u al in terval of tim e ¢ t in su ch a w ay th a t A B = B B 0. B u tth e p lan et is su b jected to a n attractive fo rce tow ard s th e S u n at O a n d its e® ectd u rin g th e tim e in terval ¢ t can b e rep resen ted b y an im p u lse on th e p lan et a t Bactin g a lo n g th e d irectio n B O . T h is im p u lse in tro d u ces a corresp on d in g ch an gein th e m om en tu m of th e p la n et resu ltin g in a ch an g e in th e v elo city of th e p la n et

given b y a vecto r B b. T h is ca u ses th e p lan et to arrive at lo cation C d u rin g atim e in terva l ¢ t d u e to th e a ltered resu lta n t velo city. In a sim ilar w ay th e p la n etreach es a p oin t D in stead of th e p oin t C ' after a tim e in terva l ¢ t u n d er th ein ° u en ce o f th e a ttractiv e cen tral force d u e to g rav itation . T h is a p p roa ch , u sin gth e ch an g e o f m o m en tu m d u e to a n im p u lse d u rin g a sh ort tim e in terval, p erh ap s

led N ew to n to form u late h is secon d law in term s of force an d th e rate of ch an geof m om en tu m . H ow ever, so lon g as th e m ass m is trea ted as a n in varian t th is lawis sa m e a s F = m a .

C o n tra ry to com m o n b elief th e sto ry o f th e secon d law of m o tio n d o es n ot en dh ere. S eriou s q u estion s on th e n atu re of th e o rig in of th e in ertia ' p rop erty of m at-ter w ere raised b y th e co n tem p o rary p h ilo sop h ers. T h e m ost p rom in en t am on gth em w as G eorg e B erkeley (1 68 5{1 75 3). H e p rop osed th at in ertia is n ot a n in trin -sic p rop erty of m a tter a s su ggested b y N ew ton , b u t it arises o u t of th e in teractionof a b o d y w ith th e m atter p resen t in th e rest of th e u n iv erse w h en th ere is an

attem p t to accelera te th e b o d y. A n o th er m a jor m y stery co u ld n ot b e resolved sat-isfactorily. T h o u gh th e p h en om en on o f grav itatio n al in teraction b etw een b o d iesis com p letely d i® eren t fro m th e o ccu rren ce of a resistive force w h ile acceleratin g ab o d y, th e grav itatio n al m ass o f a b o d y is alw ay s fou n d to b e eq u iva len t to an o th erp rop erty of th e b o d y { its in ertia l m ass. In th e 18 th cen tu ry su ch d eep ro o ted

p h ilo sop h ical q u estion s d id n o t h av e m u ch im p act b eca u se o f th e trem en d ou ssu ccess o f m ech an ics, as sy n th esized in th e ¯ n al form b y N ew to n , in ex p lain in gall m otion s an d m otion -related p h en o m en a . T h e p oin t w a s aga in raised in th esecon d h a lf of th e 19 th cen tu ry b y th e n oted A u stria n p h iloso p h er E rn st M ach


GENERAL � ARTICLE

Figure 9. Illustration ofMach’s Principle.

3 In fact it is not that the faraway matter instantaneously act on the accelerating body to produce the resistance asnothing can move with a speed more than that of light. What is presumed is that all the matter present in the universegenerate their influence at the location of the body being accelerated; and this local field interacts with the acceleratingbody to produce the resistance we term as inertia.

(18 38{ 19 16), an d th is tim e w ith b etter su ccess. H e su gg ested th a t w h en a b o d y

is a ccelerated a ll th e m a tter p resen t in th e rest of th e u n iverse o® ers a resistan ceto th e a cceleration (F igure 9 ), an d , th a t is th e origin of in ertial force3 . S in ceth is p h ilo sop h y received a lo t of su p p o rt from a few scien tists (E in stein b ein gon e of th em ) th e id ea ga in ed a certain d eg ree of legitim acy a n d b ecam e k n ow nas M ach 's P rin cip le'. O f co u rse, it rem ain ed ju st a p h iloso p h ica l statem en t an d

for th e ¯ rst tim e a q u an tita tiv e m o d elin g of M ach 's P rin cip le a s attem p ted b yD W S cia m a (192 6{ 200 1) tow ard s th e b eg in n in g o f th e secon d h a lf of th e 20 thcen tu ry. H e p ro p o sed a m o d el of d y n am ic grav ita tio n al in tera ction (term ed asin ertial in duction b y h im ) acco rd in g to w h ich th e grav ita tio n al force b etw eentw o b o d ies d ep en d s n ot o n ly o n th eir m u tu a l sep a ratio n b u t a lso on th eir relative

acceleratio n . In a v ery cru d e fo rm th e g rav itation a l fo rce on th e b o d y A (F igure10 ) d u e to th e b o d y B ca n b e w ritten a s fo llow s:

f = ¡ (G m g A m g B = r2 ) ¡ (G m g A m g B = c

2 r )a ; (9 )

w h ere r is th e d istan ce b etw een th e tw o b o d ies, a is th e accelera tio n o f b o d y Aw ith resp ect to b o d y B , c is th e sp eed of ligh t, m g A an d m g B a re th e grav ita tio n alm asses of th e tw o b o d ies an d G rep resen ts th e g rav itation a l con stan t. W h enan ob ject o f grav ita tio n al m a ss m a ccelerates a t th e rate a w ith resp ect to th e

m atter p resen t in th e rest of th e u n iv erse th e to tal resistan ce ex p erien ced b y th eb o d y can b e ob tain ed b y su m m in g u p th e in ertial in d u ction w ith resp ect to allth e m atter in th e u n iv erse. T h u s,

F = § f = ¡ § (G m g B = c2 r )m a : (10 )

MACH’S PRINCIPLEag

F

a

Pmg


GENERAL � ARTICLE

Figure 10. Model of inertialinduction.

Suggested Reading

[1] D W Sciama, The Unity of the Universe, Doubleday Anchor Publication, 1959.[2] Bernard Cohen, The Birth of a New Physics, Anchor Books, Garden City, NY, 1960.[3] Ernst Mach, Science of Mechanics, Open Court Publishing House, 1988.[4] Julian Barbour, Discovery of Dynamics, Vol.1 of Absolute or Relative Motion?, Cambridge University Press,

1989.[5] Galileo Galilei, Dialogues Concerning Two New Sciences, Prometheus Books, 1991.[6] Thomas S Kuhn, The Copernican Revolution, Harvard University Press, 1992.[7] Amitabha Ghosh, Origin of Inertia, Apeiron, Montreal, 2000; East West Press, New Delhi, 2002.

AB

r

f a

T h e su m of th e ¯ rst term in th e R H S of (1 0) b eco m es zero d u e to th e iso trop y

an d h o m o gen eity o f th e m atter p resen t in th e u n iv erse w h en con sid ered in th elarge scale. T h e valu e of th e term w ith in th e b rackets in th e R H S of (11 ) b ecom esap p rox im a tely eq u al to u n ity resu ltin g in th e fo llow in g relatio n :

F ¼ ¡ m a :

T h is sh ow s w h y th e g rav itation a l m ass a p p ears as th e in ertia l m ass in th e forcelaw . It a lso d em on strates th at th e seco n d law o f m otion is n ot a law ' b u t ad eriv ed rela tion from d y n a m ic g rav itation a l in teractio n o f an acceleratin g b o d y

w ith th e m atter p resen t in th e rest o f th e u n iv erse! H ow ev er, th e m y stery stillp ersists a s th e eq u ivalen ce is ex act an d n o t ap p rox im ate. H ow su ch a ¯ n e tu n in gis p ossib le of th e variou s p a ram eters of th e u n iverse to m a ke th e eq u iva len ceex a ct an d th e force law to b e F = m a h as b een a lso w ork ed ou t b y in tro d u cin ga velo city -d ep en d en t term in th e m o d el of in ertial in d u ctio n g iv en b y (10 ), b u t

th at is an o th er story.

Address for Correspondence: Amitabha Ghosh, INSA Senior Scientist, Bengal Engineering and Science Univer-sity, Shibpur, P.O. Botanic Garden, Howrah 711 103, India. Email: [email protected], [email protected]

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The little known story of F = ma and beyond