I

I

~Galileos Gravitational Units u

tr By Stillman Drake

Sti1lnuut ~ is emerilW professor ofhistory ofscience OJ the Uvenity ofTorol1lo He l has uanslaJed aD of Ga1ikos sciel1lific books and written research papers on JUs woricl lpoundu Galileo at Work ~riJlen a dicade ago will be suppkmel1led this year by GaliJelti Pioneer Scienlis~ wriuen particularly for physicists c_

~ bull 432 TIlE PHYSICS TEACHER SEPTEMBER 1989 ~ ~IQIl~rj 1

---------------------- 0 - ~~C~ ~t Jtulo II) ~~--1 t

gt II]J

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Galileo is remembered in physics chiefly for Ills discovshyeries of the pendulum law the law of fall and the parabolic trajectory of borizontally launched projectiles Those conshysequences of his early studies of motion were set forth in his last book Two New Sciences in 1638 There he derived the times-squared law for distances in fall mathematically from the defmition of uniformly accderated motion but Galileodid not explain how he had flfst discovered the law

The measurements and timings that led Galileo to the pendulum law and from tbat to the law of fall bave now been found in his working papers of 1604 from which Ills experimental procedures and mathematical reasoning can be reconstructed Until it was knoln (bat the pendulum law had been a necessary preliminary to the law of fall in the procedure followed by Galileo historians generally

I believed that he must somehow have followed the lead of

I certain 14th-century Titers on accelerated motion who hld devised the so-called Merton rule a mean-speed posshytulate But none of these earlier Titers bad ever applied that postulate to the fall of heavy bodies or carried out any

I

measurements of accelerated motions something obvishyously required in order to determine what law (if any) migbt govern free fall The introduction of careful meashysurements into physics was in a way Galileos principal contribution to the birth of modern physics But the units he devised are of even greater theoretical interest because

I they imply a universal constant of graitational accelerashytion that has been strangely neglected

In 1604 there were of course no standard units of length Every nation and in Italy each province had its OTI length-unit For time the astr0nomical second was

I

available but tbat was not a practical unit of measurement until Huygens inventedthe pendulum clock and fitted it vitb cycloidal cbecks The units GalileO used in bis discov eries of the laws of pendulum arid fall and again in 1608 when he identified the paths of projectiles to be parabolic in shape were of Ills own creation His punta 094 mm in length was entirely arbitrary but usefuLbecause the millimiddot meter is a precision measure that can also be quite accushyrately bisected with the naktd eye

Galileos tempo 1J92 s was not-arbitr~for it was gravitationally linked to the punto How Galileo managed shytbat and why it is still of interest today will ~ explained here It may seem that all units of length and time are necessarily arbitrary and that in principle no pair can have any advantage over another in physics though some ~tS are more convenient than otliers for making measure ments and calculations In modern pbysics for instance units for distance and time are related by the invariant speed of light For convenience in some calculations thiSshyspeed is given tbe arbitrary magnitude of one

Mathematically speaking space and time are incomshymensurable magnitudes But that does not mean tbat a measurement of lengtb cannot be numerically identical with the measurement of some related time and in fact shyfor graitational phenomena measured lengths are associ-~ ated with measured times What will here be called Galilean units or GU for short areA = 09422119 mm and T = 19188025 s (to seven significant figures) In relating these units we will use the quarter period of a pendulum the time for the bob to swing (in a vanishingly small arc) through any small amplitude to the verticaL ~~

~ ~ ~)

~ ~

Numerical Example of Galilean Units

(Edilors Addition)

fl a fall of 100 m (g = 980) starting from rest h = 100 mJ(0942 x 1O-3mJt) =1060t

tFJ~b = ~- = 0452 s = (0452 s)(9188 Tis) = 415 bull

~ 2 t falJz = 1-U T

t Fa12t = ~ 2Cr) = 0639 s

TI = 211 J~ ~ 4 g

0639 = pound~ L L = 162 m = 17lJ)t 2 980

Hence in GV the square of the time of fall through 1 m is numerically equal to the length of the pendulum whose quarter period is equal to the time of fall through 2 m

this lime Tl4 Then in GV the square of the time of free - fal~ startIng from rest through any height h is numerically

equal to the length of a pendulum whose quarter period is equal to the time taken for a fall from 2h (See example t in box) This relationship holds anywhere in the universe ~ where heavy bodies fall and pendulums oscillate It does

not depend on the strength g of the local gravitational field The rate of acceleration in GV is giT2

where g JtZ(23 = 123370055 This follows from the ordinary

equations for distance fallen from rest in a given time and fo the time of a quarter period of a pendulum

It is no mere coincidence that Galileos punlo and tempo were nearly t and T or that the implied rate of

accelera tion g )2 is a univ~rsal physical constant - unlike standard g = 980665 mJs which holds only at sea-level at 450 latitude (or the gravitational equivalent elsewhere) Galileo did not know enough physics to hit on a dynamic constant of acceleration varying ith the strength of gravshyitational field Modern physical constants arise from physshyinl equations and such constants would have cancelled (H in the older notations using the classical mathematics of proportion Galileo never used algebra even in his working papers the first modern mathematical physicist never wrote a physical equation in his life

What Galileo did use was the Euclidean theory of ratios and proportionality for mathematically continuous magnishytudes (such as lengths and times) set forth in Book V of

the Elements around the year 300 RC That was enough for Galileos physical discoveries But for his proof of the

Iw of fall he needed something that the ancient Greek nathematicians did not supply Euclid excluded the infishynite from mathematics by his axiom that -the whole is greater than the part Galileo needed to prove that there are as many different speeds as there are instants of time during any fall from rest for that he introduced the

concept of one-to-one correspondence betveen members of an infinite set and its half That concept was no small contribution to pure mathematics and it was quite essenshytial to the birth of rigorous mathematical physics

About the beginning of 1604 Galileo set out to find if he could a rule for increases of speed during natural descent as any spontaneous motion of a heavy body was called That would be hard to do if he had had to rely on direct measurements of free fall But the motion of a ball rolling down an inclined plane was just as surely a natucal descent as was straight falL and rolls could be made slow enough to time by musical beats_ On folio 107 of his working papers (as they are now numbered) Galileo reshycorded the results of some measurements that led him on to the discovery of the law that distances in fall from rest are as the squares of the elapsed times For about ten years I believed this page to have been the discovery document for Galileos law of fall

The calculations down the middle show how Galileo arrived at the eight figures he tabulated at upper left A number was multiplied by 60 and a number less than 60 was added Clearly Galileo had a ruler accurately divided into 60 equal parts which he used when making measureshyments of lengths In fact he used the same ruler for drawing and measuring several diagrams on other pages of these Dotes Those diisions were 094 mm each or a punto as Galileo called this hat he had measured were places of a ball at each ofeight successive equal times when it rolled from rest down a plane grooved to guide it The plane was something over 2100 punti long raised 60 punti ~ from the horizontal at one end to an angle of about 17 full roll would take about four seconds permitting eight half-second marks Calculation shoM that Galileo was accurate within 164 s for every mark except the last when the ball was moving about 1000 punti per second That

SEPTEMBER 1989 ruE PHYSICS TEACHER 433

I

Numerical Example of Galilean Units

(Edilors Addition)

fl a fall of 100 m (g = 980) starting from rest h = 100 mJ(0942 x 1O-3mJt) =1060t

tFJ~b = ~- = 0452 s = (0452 s)(9188 Tis) = 415 bull

~ 2 t falJz = 1-U T

t Fa12t = ~ 2Cr) = 0639 s

TI = 211 J~ ~ 4 g

0639 = pound~ L L = 162 m = 17lJ)t 2 980

Hence in GV the square of the time of fall through 1 m is numerically equal to the length of the pendulum whose quarter period is equal to the time of fall through 2 m

this lime Tl4 Then in GV the square of the time of free - fal~ startIng from rest through any height h is numerically

equal to the length of a pendulum whose quarter period is equal to the time taken for a fall from 2h (See example t in box) This relationship holds anywhere in the universe ~ where heavy bodies fall and pendulums oscillate It does

not depend on the strength g of the local gravitational field The rate of acceleration in GV is giT2

where g JtZ(23 = 123370055 This follows from the ordinary

equations for distance fallen from rest in a given time and fo the time of a quarter period of a pendulum

It is no mere coincidence that Galileos punlo and tempo were nearly t and T or that the implied rate of

accelera tion g )2 is a univ~rsal physical constant - unlike standard g = 980665 mJs which holds only at sea-level at 450 latitude (or the gravitational equivalent elsewhere) Galileo did not know enough physics to hit on a dynamic constant of acceleration varying ith the strength of gravshyitational field Modern physical constants arise from physshyinl equations and such constants would have cancelled (H in the older notations using the classical mathematics of proportion Galileo never used algebra even in his working papers the first modern mathematical physicist never wrote a physical equation in his life

What Galileo did use was the Euclidean theory of ratios and proportionality for mathematically continuous magnishytudes (such as lengths and times) set forth in Book V of

the Elements around the year 300 RC That was enough for Galileos physical discoveries But for his proof of the

Iw of fall he needed something that the ancient Greek nathematicians did not supply Euclid excluded the infishynite from mathematics by his axiom that -the whole is greater than the part Galileo needed to prove that there are as many different speeds as there are instants of time during any fall from rest for that he introduced the

concept of one-to-one correspondence betveen members of an infinite set and its half That concept was no small contribution to pure mathematics and it was quite essenshytial to the birth of rigorous mathematical physics

About the beginning of 1604 Galileo set out to find if he could a rule for increases of speed during natural descent as any spontaneous motion of a heavy body was called That would be hard to do if he had had to rely on direct measurements of free fall But the motion of a ball rolling down an inclined plane was just as surely a natucal descent as was straight falL and rolls could be made slow enough to time by musical beats_ On folio 107 of his working papers (as they are now numbered) Galileo reshycorded the results of some measurements that led him on to the discovery of the law that distances in fall from rest are as the squares of the elapsed times For about ten years I believed this page to have been the discovery document for Galileos law of fall

The calculations down the middle show how Galileo arrived at the eight figures he tabulated at upper left A number was multiplied by 60 and a number less than 60 was added Clearly Galileo had a ruler accurately divided into 60 equal parts which he used when making measureshyments of lengths In fact he used the same ruler for drawing and measuring several diagrams on other pages of these Dotes Those diisions were 094 mm each or a punto as Galileo called this hat he had measured were places of a ball at each ofeight successive equal times when it rolled from rest down a plane grooved to guide it The plane was something over 2100 punti long raised 60 punti ~ from the horizontal at one end to an angle of about 17 full roll would take about four seconds permitting eight half-second marks Calculation shoM that Galileo was accurate within 164 s for every mark except the last when the ball was moving about 1000 punti per second That

SEPTEMBER 1989 ruE PHYSICS TEACHER 433

I

permit anyone to adjust the the limit of accuracy of his oWn rh~thm

The rule Galileo found Was speeds during successive equal increase as do the odd numbers 1 7bull as seen from the diagrams he vith the page turned sideways was the kind of rule he sought put this page aside Later in diffe ink and a bit smeared he squeezed the leftmiddothand margin the first e square numbers which is why I 107l (Fig 1) to be the discovery ument for the law of falL But it me why in that case he had de entering so important a result mathematician knew that adding odd numbers lOgether gives the

] numbers Yet there was no point adding speeds together and was not thinking of these lengths distances but only as measures of cessive speeds in a series of equ times It had not ret occurred to

I

that some ruk might exist that the distance directly to the time of scent

vnat did occur to him before he f107 aside was that if a simple rule increase of speed in descent could found by merely equalizing time more might be found by] times of motions At the bottom of

1 page he sketched a kind of water lUtrlt___

watch In r o Sew Sciences he d scribed his dece as simply a bucket water lth a lUbe through its bott the tlow of water during a motion collected and weighed on a sensi

fig f 107 Vol 72 Galilean manuscripts reproduced by permission oC tbe National Central Li brary of FlorenceJ was the only length that Galileo charged later though he put a positive or negative sign on four or Lie other marks] Galileos father and brother were professional musimiddot cians and he was a good amateur perforIiler on the lute Al beats of half a second anyone can detect an error of1 1J25 or 130 of a second A good arraeur musician can do twiee as well and a professional eomiddot~j ceteer an error of 1100 s or less I assume that Galile0 tied strings around

-] his grooved plane as frets were tied around the neck of his lute His bronze ball would then make a bumping sound as it struck the plane after passing over a fret Galileo hadfinlY to adjust the strings until every bump agreed with a

beal of a crisply sung march and measure distances from t bull resting point to the lower side of each string That

kes patience to do but the apparatus described would f

i) 434 mE PHYSICS TEACHER SEPTiBER 1989

balance the weights being taken measures of times His sketch suggests a more pgtigt(rr

delce Analysis of the data recorded on two other shows that water flowed at very nearly three fluid per second Galileo recorded his timings in grains weight each equal to 11480 ounce of flow

Galileo measured fall time of 4(X)) punti = 376 m 1337 grains of flow (about 1C() s too high) He found pendulum taking half thal lime or 668 12 grains of to be 870 punti long He then doubled that pendulum 1740 punti and found its swing 10 take 942 grains of while reaching the vertical through a small arc In all work Galileo timed pendulum swings to the vertical not the full period He used the quarter-period lJCaLI)

the only swing he could time with precision was from instant of releasing the bob to its sound of impact with

at

I

that was fLXed in advance ag3inst a side of the bob banging plumb

froro the pendulum measurements of lengths and a table of the kind shovro below could be compiled

Galileo may not have actually compiled a table will serve to show the way in wruch he arrived at his

des simply by applng the general theory of ratio proportionality set forth in Euclids Elements Book

The lable ~as been extended far enough to show the of a very important number found in two of

surviving working papers on one of wruch he was when he first recognized the law of fall in timesshy

form from its mathematicaUy equivalent meanshy~-___r~t n form

TIme to the vertical (See text belOW)

i pumi in grains flow V(2L) T16

870 6685 417 418 1740 942 59 589 3480 1337 834 836 6960 1884 118 1178

13920 2674 1669 1671 27840 3768 236 2355

The third figure in the second columnl337 grains was mel5ure of time Galileo had found for free fall of 4000

punt The fust column above was formed by successive doublings and the second by alternate doublings Except for one slight discrepancy each of the fust two columns taken separately is in continued proportion Galileos new

bullbull o time unit the tempo came into being when he related the two columns horizontally so to speak His original meashy

middot SUfe of time in grains of now through a particular dece haing been completely arbitrary he v as free to alter it in any ratio he pleased Taking each time to be the mean proportional be tvmiddote en two and the lelgth of pendulum the

middot data ~ each column become related line bv line The same numbers also result almost exactly fromdiision of each

time in grains by 16 and 16 grains (ll 1 cc) became 1 tempo - the new unit adopted by Galileo as a result of

middotthese investigations relating the times of pendulums to ttheir lengths - Because division by 16 of the times in grains weight of

Gow did not exactly produce that mean-proportional rei ashymiddot tion between the two columns Galileo made a calculation

~~ tbat resulted in his changing 27~J as sholl above to 278-t That work was done on one of the working papers

fwrJch only a part survives On the blank side he wrote In 1609 a note on another topic cut it out and pasted it

--~~T [90 That was lifted at my request and on the hidden ~de [ saw the number 27834 twice with enough words to lentify it as a diameter Galileos diagram and calculashytions Were thrown away with the part of the page he cut off but the work was done before Galileo discovered the law

tz of fall because 27834 played a crucial part in the calculashy

tion on f 189v1 (Fig 2) which put that discovery into Galileos h3nds

With his adoption of the tempo as the unit of time Galileo had the pendulum law in a restricted form that is for any set of pendulums successively doubled in length It would have been a difficult task to test it for successively tripled pendulums let alone for any other integral multishyples and quite impossible to establish it by induction in its complete generality For that purpose Galileo next calcushylated the mean proportional of 118 and 167 - the times to the vertical in temp~ for the two pendulums of lengtbs 6960 and 13920 punti-and got 140 tempi The mean proportional of those two lengtlts is 9843 so if his reshystricted pendulum law were perfectly general a pendulum 98-13 punti long would sing to the vertical in 140 tempi

On the other side of the same page (f154) Galileo wrote the note mo br 16- the string is 16 braccia long From two lines dr3wn and labeled by Galileo at Padua the braccio that he used in 1604 was about 620 punti At 615 punti per braccio length of that pendulum would be 9840 punti or about 30 feet Such a pendulum could be hung from a window over the courtyard of the University of Padua and timed protected from wind Its time to tbe vertical at Padua would be 141 tempi by calculation

In the foregoing table Galileos timing of fall through -4000 punti 1331 grains of flow (or 835625 tempi) was sholl as the time for the pendulum of length 3480 pUllt~ whereas that fall is in fact timed by the pendulum of 4000g

324228 punti The error arose from his haing timed the faU as 1130 s too long The correct time was 80527 tempi not 835625 wruch Galileo adjusted to 80 23 immeshydiately upon rus discoery of the law of fall

That discovery occurred when Galileo calculated on f189vl (Fig 2) the distance a body would fall vertically in time 2SO tempi double the time he had just measured for s-ing to the vertial by his 3O-fool pendulum When he made that calcuacicn he supposed some distance of fall to be needed and for that he used fall of 4000 punti In fact it did not matlter what fall he used because that factor was cancelled by its presence as the numerator of one ratio and as the denominator of another The result he got was very near to the modern result for the latitude of Padua his 48143 punti as compared with our 48317 a difference of 174 punti or 15 em in 451)4 meters (about 036 low)

As you see Galileo at once took the mean proportional of rus 48143 punti and the assumed base 4000 punti saw this 13863 to be nearly the half of 27834 and thus realized t hat his roundaoout cakulation ofa distance of fall through an intermectiate penduium relation had been unnecessary the distance depended only upon the time At lower left he drew the conventional proportionality diagram for magnitudes middotcommensurable in the square which he used always thereafter for relating distances and times in fall

Strictly speaking Galileo never made use of any middotconshystant of gravitation in the modern sense The acceleration implied by ~ units of measurement was of course very nearly g = 1t 8 valid anywhere that pendulums oscillate

SEPTEMBER 1989 THE PHYSICS 435

that was fLXed in advance ag3inst a side of the bob banging plumb

froro the pendulum measurements of lengths and a table of the kind shovro below could be compiled

Galileo may not have actually compiled a table will serve to show the way in wruch he arrived at his

des simply by applng the general theory of ratio proportionality set forth in Euclids Elements Book

The lable ~as been extended far enough to show the of a very important number found in two of

surviving working papers on one of wruch he was when he first recognized the law of fall in timesshy

form from its mathematicaUy equivalent meanshy~-___r~t n form

TIme to the vertical (See text belOW)

i pumi in grains flow V(2L) T16

870 6685 417 418 1740 942 59 589 3480 1337 834 836 6960 1884 118 1178

13920 2674 1669 1671 27840 3768 236 2355

The third figure in the second columnl337 grains was mel5ure of time Galileo had found for free fall of 4000

punt The fust column above was formed by successive doublings and the second by alternate doublings Except for one slight discrepancy each of the fust two columns taken separately is in continued proportion Galileos new

bullbull o time unit the tempo came into being when he related the two columns horizontally so to speak His original meashy

middot SUfe of time in grains of now through a particular dece haing been completely arbitrary he v as free to alter it in any ratio he pleased Taking each time to be the mean proportional be tvmiddote en two and the lelgth of pendulum the

middot data ~ each column become related line bv line The same numbers also result almost exactly fromdiision of each

time in grains by 16 and 16 grains (ll 1 cc) became 1 tempo - the new unit adopted by Galileo as a result of

middotthese investigations relating the times of pendulums to ttheir lengths - Because division by 16 of the times in grains weight of

Gow did not exactly produce that mean-proportional rei ashymiddot tion between the two columns Galileo made a calculation

~~ tbat resulted in his changing 27~J as sholl above to 278-t That work was done on one of the working papers

fwrJch only a part survives On the blank side he wrote In 1609 a note on another topic cut it out and pasted it

--~~T [90 That was lifted at my request and on the hidden ~de [ saw the number 27834 twice with enough words to lentify it as a diameter Galileos diagram and calculashytions Were thrown away with the part of the page he cut off but the work was done before Galileo discovered the law

tz of fall because 27834 played a crucial part in the calculashy

tion on f 189v1 (Fig 2) which put that discovery into Galileos h3nds

With his adoption of the tempo as the unit of time Galileo had the pendulum law in a restricted form that is for any set of pendulums successively doubled in length It would have been a difficult task to test it for successively tripled pendulums let alone for any other integral multishyples and quite impossible to establish it by induction in its complete generality For that purpose Galileo next calcushylated the mean proportional of 118 and 167 - the times to the vertical in temp~ for the two pendulums of lengtbs 6960 and 13920 punti-and got 140 tempi The mean proportional of those two lengtlts is 9843 so if his reshystricted pendulum law were perfectly general a pendulum 98-13 punti long would sing to the vertical in 140 tempi

On the other side of the same page (f154) Galileo wrote the note mo br 16- the string is 16 braccia long From two lines dr3wn and labeled by Galileo at Padua the braccio that he used in 1604 was about 620 punti At 615 punti per braccio length of that pendulum would be 9840 punti or about 30 feet Such a pendulum could be hung from a window over the courtyard of the University of Padua and timed protected from wind Its time to tbe vertical at Padua would be 141 tempi by calculation

In the foregoing table Galileos timing of fall through -4000 punti 1331 grains of flow (or 835625 tempi) was sholl as the time for the pendulum of length 3480 pUllt~ whereas that fall is in fact timed by the pendulum of 4000g

324228 punti The error arose from his haing timed the faU as 1130 s too long The correct time was 80527 tempi not 835625 wruch Galileo adjusted to 80 23 immeshydiately upon rus discoery of the law of fall

That discovery occurred when Galileo calculated on f189vl (Fig 2) the distance a body would fall vertically in time 2SO tempi double the time he had just measured for s-ing to the vertial by his 3O-fool pendulum When he made that calcuacicn he supposed some distance of fall to be needed and for that he used fall of 4000 punti In fact it did not matlter what fall he used because that factor was cancelled by its presence as the numerator of one ratio and as the denominator of another The result he got was very near to the modern result for the latitude of Padua his 48143 punti as compared with our 48317 a difference of 174 punti or 15 em in 451)4 meters (about 036 low)

As you see Galileo at once took the mean proportional of rus 48143 punti and the assumed base 4000 punti saw this 13863 to be nearly the half of 27834 and thus realized t hat his roundaoout cakulation ofa distance of fall through an intermectiate penduium relation had been unnecessary the distance depended only upon the time At lower left he drew the conventional proportionality diagram for magnitudes middotcommensurable in the square which he used always thereafter for relating distances and times in fall

Strictly speaking Galileo never made use of any middotconshystant of gravitation in the modern sense The acceleration implied by ~ units of measurement was of course very nearly g = 1t 8 valid anywhere that pendulums oscillate

SEPTEMBER 1989 THE PHYSICS 435

l

the ratio 9421850 1108 his rati time to tbe vertical for a pendu1~~ ~ length 1140 punti to time of fall 11

~~ ~ punti from rest He did not know 1t had anything to do ith the matter ~ relied entirely on the most accu

measurements he could make and rigorous Euclidean theory of nr_~ tionality for mathematically magnitudes

GU provide a method of ~UC mental measurement ofre that unllltupo

students should be inited to test in laboratory No better way showing them the nature of length and time based not on the tance from pole to equator Paris and 1186400 of one axial of Earth but on actual VUU11U~iJ(lUllli vertical fall and of horizontal spontaneously produced by gr That physical relationship was built Galileos units as accurately as be co measure it and of course we can b into G U the exactre lationship middotnWlv1if

ing a transcendental number re cannot be exactly measured in any tual phenomena of nature

In conclusion I add that in day there existed no measurements planetary distances n terrestrial such as millions of miles There now and it is easy to convert those GU Anyone who does lhat will some very interesting relations arno the places and perioiic times of plan over and above these deducible Keplers wellmiddotkno -n planetary lawsbullif~

Teachers and students interested ~

a

fig 2 f 189vl At cenkr calculation ofdistance fallen in 280 tempi At top acd not required for this calculation are timings or pendulum 0 r length 1740 punti through a large and a small arc

and heavy bodies spontaneously fall when released from restraint But the coostant Galileo used was about Ii -c(22) ane bat is what I call Galileos conmiddot stant In theory it is UlO720735 whereas Galileo used

this subject will find more about GU ~~ planetary data in books by Stillman Drake History ofFre(f Fall (Wall amp Thompson Toronto 1989) and Galileo1 PioneerScientisr (Univ ofToronro in press) middott~

shy

- -- shy

436 ruE PHYSICS 1EACHER

II DETERMINACION DE LAS MASAS DE LA TIERRA Y DEL SOL

I Prime-ra Parte

I Por Francisco S Ramirez Avila

El problema que nos contie-rne abora es sobre la determinacion de las

I masas de la Tierra y del Sol Para ree-velver satisfacroriamenre dicha tarea detgtemos ~tableuro(er de antemano la relacion que existe entre la fuerza gravitadonal de dos masas y el Pfso de una con res~to a la otra Hare-mooI esto romando el caso de 1a Tierra y despues e1 caso del Sol

I Masa de la Tierra

I Si decimos que nuestro planeta tiene una masa M entonces para un objeto de Olasa m somelido ala influenda gravitatoria rorrestre

I

GMmr2 =mg ( 1)I Aqui G representa la constan~ universal de la gravitacion (=667110-11

Nm2kg2) g es 1a aceleracton gravitactonal de la Tierra i r ~ la distanda entre el centro de 1a Tierra y el centro del objeto Si asumimos que el obieto

I

repltgtsa sobre la superficte de la Tierra y si este es ademas pequeno comparado con las dimensiones geometricas de nuestro planeta enronces la ec(O se reduce a la siguiente forma

GMJR2 = g donde R representa el radio terrestre Resolviendo ~ta Ultima ecuacion para M tenernos que

M = gR2G (2)I Podriamos determinar 1a masa de la Tierra consuHando cualqujer Hbro sobre astronomia elemental y viendo en et los valores ofidalmente

I aceptados para R y g Esto serra 10 mas comodo de hacer Sin embargo una manera mas practica y comprensib1e de conocer M seria midi~ndo R y g por cuenta propia mediante dos experimentos relativamente senci11os comoI expUacamos a con tinuadon

I a) Determinacion del radio Terrestre La meaidon del radio de nuestro planeta debe hacerse partiendo

del h~o de que 1a Tierra es reuroltlonda as como 10 asumiera en laI antiguedad e1 sabio filosoio y matematico alejandrino EratOstenes (280-192 aCJ -la historia oHcia1 10 cons1dera como el primero en

I determinar el tamafio del p1aneta- Nosotros emplearernos el mismo

I fmHodo que utiHzara Erat-)stenes en didla t)Jea

I

1 Q AHa -uID1D1S VoL 11 I~ NUffi 20 N07iembre de 1939

t SI )tlT77x-r=r=c ItC-- I 5 fAI

deg

Como se iIustra en la figura la el radio de la Tierra se obtiene1 partiendo de la reladon

R = S0 (3)11 El angulo 0 senala la separacion angular y S senala la separadon lineal entre los puntos A y B

I Si asumirnos que -s- representa la geodesica que une las latitudes de la Cd de middotfexico y de Cd Juarez (1370 Km aproximadamente) para determinar R wnemos entonces que ca1cular ~ de manera directa esI dedr de manera experimental

Vemos pues que si exwndemos las lineas OA y OB mas alla del

I circulo como se aprecia en la fig lb encontraremos una relacion muy obvia entre los angulos 0 1) Y 0 at proyecru una serie de tres lineas horizontlles tales que pasen caltia una por los punlos A B Y O EstaI relation es la siguiente

3-5=0 (tnI A1 medir 3 y 5 podemos ca1cular automaticamente II y Rpor sustitucion

en las ecuaciones (middotn y (3) respectivamenle

I A ----------------~~~

B -------i~3 Omiddot------t-+--Iawo

I ~

(a) (b)

]

J IIGURA 1 (a) rltscion entre las Tariables S Y if con respecentto e1 rejio terrestre

R =Oa =OB (1)) Relacion entre los engulos 13 () Y iI Las Eneas AA BB Y 00 son paralel~ entre st

Tanto B~omo 0 S~ obtienen si asumimos que las lineas AA Y BB] repres~ntan rayos solareuro-s incidenteuros sobre la Tierra Entonces a1

adoptar las extensiones de las lineas OA y OB como varil1as sobresaliendo la su~rficie terrestre podemos medir el largo de sus

bull I sombras sobr~ 1a Tierra a una hora espociiica dol aia con una es(ala

conven~ional - utiliando las reglas bitsica~ d~ 1a trigon~)metri3 -St0S -shy ~raul(lt c ~ --~-- 1lt1 r dlmiddotfrt -J ImiddotI)C1jtC iof bull flLJ-~o v rrr ~ 1 -)

bull~J1Jmnt 1 unttri0lt (J~l rPJint( sS-rngtJIoS de fr3-par~lt~middotna dd mmiddottl-

I

II DETERMINACION DE LAS MASAS DE LA TIERRA Y DEL SOL

I Prime-ra Parte

I Por Francisco S Ramirez Avila

El problema que nos contie-rne abora es sobre la determinacion de las

I masas de la Tierra y del Sol Para ree-velver satisfacroriamenre dicha tarea detgtemos ~tableuro(er de antemano la relacion que existe entre la fuerza gravitadonal de dos masas y el Pfso de una con res~to a la otra Hare-mooI esto romando el caso de 1a Tierra y despues e1 caso del Sol

I Masa de la Tierra

I Si decimos que nuestro planeta tiene una masa M entonces para un objeto de Olasa m somelido ala influenda gravitatoria rorrestre

I

GMmr2 =mg ( 1)I Aqui G representa la constan~ universal de la gravitacion (=667110-11

Nm2kg2) g es 1a aceleracton gravitactonal de la Tierra i r ~ la distanda entre el centro de 1a Tierra y el centro del objeto Si asumimos que el obieto

I

repltgtsa sobre la superficte de la Tierra y si este es ademas pequeno comparado con las dimensiones geometricas de nuestro planeta enronces la ec(O se reduce a la siguiente forma

GMJR2 = g donde R representa el radio terrestre Resolviendo ~ta Ultima ecuacion para M tenernos que

M = gR2G (2)I Podriamos determinar 1a masa de la Tierra consuHando cualqujer Hbro sobre astronomia elemental y viendo en et los valores ofidalmente

I aceptados para R y g Esto serra 10 mas comodo de hacer Sin embargo una manera mas practica y comprensib1e de conocer M seria midi~ndo R y g por cuenta propia mediante dos experimentos relativamente senci11os comoI expUacamos a con tinuadon

I a) Determinacion del radio Terrestre La meaidon del radio de nuestro planeta debe hacerse partiendo

del h~o de que 1a Tierra es reuroltlonda as como 10 asumiera en laI antiguedad e1 sabio filosoio y matematico alejandrino EratOstenes (280-192 aCJ -la historia oHcia1 10 cons1dera como el primero en

I determinar el tamafio del p1aneta- Nosotros emplearernos el mismo

I fmHodo que utiHzara Erat-)stenes en didla t)Jea

I

1 Q AHa -uID1D1S VoL 11 I~ NUffi 20 N07iembre de 1939

t SI )tlT77x-r=r=c ItC-- I 5 fAI

deg

Como se iIustra en la figura la el radio de la Tierra se obtiene1 partiendo de la reladon

R = S0 (3)11 El angulo 0 senala la separacion angular y S senala la separadon lineal entre los puntos A y B

I Si asumirnos que -s- representa la geodesica que une las latitudes de la Cd de middotfexico y de Cd Juarez (1370 Km aproximadamente) para determinar R wnemos entonces que ca1cular ~ de manera directa esI dedr de manera experimental

Vemos pues que si exwndemos las lineas OA y OB mas alla del

I circulo como se aprecia en la fig lb encontraremos una relacion muy obvia entre los angulos 0 1) Y 0 at proyecru una serie de tres lineas horizontlles tales que pasen caltia una por los punlos A B Y O EstaI relation es la siguiente

3-5=0 (tnI A1 medir 3 y 5 podemos ca1cular automaticamente II y Rpor sustitucion

en las ecuaciones (middotn y (3) respectivamenle

I A ----------------~~~

B -------i~3 Omiddot------t-+--Iawo

I ~

(a) (b)

]

J IIGURA 1 (a) rltscion entre las Tariables S Y if con respecentto e1 rejio terrestre

R =Oa =OB (1)) Relacion entre los engulos 13 () Y iI Las Eneas AA BB Y 00 son paralel~ entre st

Tanto B~omo 0 S~ obtienen si asumimos que las lineas AA Y BB] repres~ntan rayos solareuro-s incidenteuros sobre la Tierra Entonces a1

adoptar las extensiones de las lineas OA y OB como varil1as sobresaliendo la su~rficie terrestre podemos medir el largo de sus

bull I sombras sobr~ 1a Tierra a una hora espociiica dol aia con una es(ala

conven~ional - utiliando las reglas bitsica~ d~ 1a trigon~)metri3 -St0S -shy ~raul(lt c ~ --~-- 1lt1 r dlmiddotfrt -J ImiddotI)C1jtC iof bull flLJ-~o v rrr ~ 1 -)

bull~J1Jmnt 1 unttri0lt (J~l rPJint( sS-rngtJIoS de fr3-par~lt~middotna dd mmiddottl-

I

1 Q AHa -uID1D1S VoL 11 I~ NUffi 20 N07iembre de 1939

t SI )tlT77x-r=r=c ItC-- I 5 fAI

deg

Como se iIustra en la figura la el radio de la Tierra se obtiene1 partiendo de la reladon

R = S0 (3)11 El angulo 0 senala la separacion angular y S senala la separadon lineal entre los puntos A y B

I Si asumirnos que -s- representa la geodesica que une las latitudes de la Cd de middotfexico y de Cd Juarez (1370 Km aproximadamente) para determinar R wnemos entonces que ca1cular ~ de manera directa esI dedr de manera experimental

Vemos pues que si exwndemos las lineas OA y OB mas alla del

I circulo como se aprecia en la fig lb encontraremos una relacion muy obvia entre los angulos 0 1) Y 0 at proyecru una serie de tres lineas horizontlles tales que pasen caltia una por los punlos A B Y O EstaI relation es la siguiente

3-5=0 (tnI A1 medir 3 y 5 podemos ca1cular automaticamente II y Rpor sustitucion

en las ecuaciones (middotn y (3) respectivamenle

I A ----------------~~~

B -------i~3 Omiddot------t-+--Iawo

I ~

(a) (b)

]

J IIGURA 1 (a) rltscion entre las Tariables S Y if con respecentto e1 rejio terrestre

R =Oa =OB (1)) Relacion entre los engulos 13 () Y iI Las Eneas AA BB Y 00 son paralel~ entre st

Tanto B~omo 0 S~ obtienen si asumimos que las lineas AA Y BB] repres~ntan rayos solareuro-s incidenteuros sobre la Tierra Entonces a1

adoptar las extensiones de las lineas OA y OB como varil1as sobresaliendo la su~rficie terrestre podemos medir el largo de sus

bull I sombras sobr~ 1a Tierra a una hora espociiica dol aia con una es(ala

conven~ional - utiliando las reglas bitsica~ d~ 1a trigon~)metri3 -St0S -shy ~raul(lt c ~ --~-- 1lt1 r dlmiddotfrt -J ImiddotI)C1jtC iof bull flLJ-~o v rrr ~ 1 -)

bull~J1Jmnt 1 unttri0lt (J~l rPJint( sS-rngtJIoS de fr3-par~lt~middotna dd mmiddottl-

I

1 - r 8 == iI

tuto Tecnol6gico y de Estudios Superiores de Monterrey (IT ESMJ

I de los campus de Cd Juarez y del Edo de Mexico se prestaron para realizar la meuroltiicion de dichos Mooulos Ambos campus nos pusimos de acuerdo para colltXaf cada uno una plomada de un metro de atoo sobe ~ una superficte totalmente horiZontal (Ia borizontalidad del suelo se determino utilizando nive1es de burbujas de aire) justo el dra 25 de

I Octubre de este ailo Sin embargo~ como la Cd de M~xico esta ubicada a una Iongitud de 7305ifJ (eqUiValenre a 0483 brs) at este de nosotros los alumnos del campus de aquella ciudad midieron la sombra de suI plomada exactanente a las 1300 ru-s mientras que nosotros 1a tuvimos que bacer a las 1329 tJs con ello asegurabamos que el Sol en Cd Juarez

I estaria en una ~icton en el ctelo casi identica a la que tuvo en la Cd de MexiCO a la bora en que se tom6 1a medici6n all~L

I Plomada

I I

1m

~ Sombra de

I la Plomada (a)

I

Plomada

1m

Sombra de la Plomada

(b)

rIGURA 2 Pora wnocer los engulos ~ y B lIleOi81te IfiS reglas lgt6siltamps de 18 lligonometria eud~1ea re miltien las som)r-es que hacente1l dos p10r0aOOs de 1 m ltle alto CWa una sotgtre un soolo terrestre perfectamente horizontal (a) LaI medici6n que re hizo en e1 Cempus LT Isl~ de Cd Juarez el 25 de Octutgtre de este ano a ias 13 brs revelo una sombre ~e 0982) metros je longitud con 10 cuu e obtuvo 1 engulo 0 igual a 144910 (07765 raj) (b) Una me-Hci6n

I I similar hecha en e IT Isl~ Campus del fdo ltie Mexico e1 mismo ltfia pero a

18$ 1300 hIS revelo una rombra de 06792 metros de largo con 10 cual se logro ceterminRr un anguJo8 igUii1 R)2]]80 (0)6] 6 rsd)

Durante la medid6n nuestra plomada describio una sombra de 09325

I metros de largo mientras que la plomada del campus Edo de Mexico

I trazo unasombra de 06292 metros Con estos datos y auXiliandonos de tanto de las figs 2a y 2b como de la 1a trigonometria vemos que

r I

C tlImiddot ts I tl Xr=r=rl

y re-emplazando ~te (tltimo angulo y Ia g0cltlf~ica s= 1370 (m en la ee(3) concluimos que el radio de la Tierra es equivalente a

)

I t

R =041005 km (5) La ltlivergencia de este ltlata lton el l81or oficialmente aceptaoo (6378 km) fue de poco menos de n km un error muy pequeno en comparacion ltOn el tamano de nuenro planetB-

I b Determinadon de la Aceleracion Gravitacional -g- La aceleracton gravitactonal g se obtiene directamenoo recurriendo

I al concepto de caida libre Una de las ecuactones basicas de la cinematica que relactona esta constanoo g con la altura 11 a la que se lioora un objeto y con el tiempo -t- que este tarda en caer at suelo esI h = vot+ (l2)gt2

I donde va representa la velocidad inicial del objeto Sin embargo como esoo desciende en caida libre su veloctdad inidal es cero Siendo a51 la eltuadon anterior se reduce a la siguiente forma

I g = 2ht2 (5)

I

I Cronometrando el tiempo que tardan diversos objetos de distintos

pesos y tamanos en caer al suelo desde una altura predeterminada podemos determinar g recurriendo a la elt(S) En una serie de pruebas que se hideron en dias pasados en el 1TESM campus Cd Juarez se

L liberaron una bOla de madera una esferita de acero y una pelota de beisbOl diez veces ltada una desde una altura h de 054 metros y S~ procedio a tomar el tiempo promedio de vuelo para cada una de elias (ver tabla de abajo)

I Objeto Literado

I Bola de Mooera Esfera de acero Pe10ta de Beislgtol

I

liempoPromedio ltle C8i6a Livre (reg)

1179 1155 1164

Alturaen C81cula de g en metres metros lseo 2

0

654 654 6gt1

940 980 96j

Promeltlio 961 mreg2

E1 promedio de g sali6 menor que el valor ofictalmente aceptado deI 98mseg2 porque la reuroSistencia del aire influy6 considerablemente en las medictones

I Ahora st como ya oonernos los valor~ oe R1 g los sustituimos en la ec(2)

I y venlOS finalmente que la masa de la Tierra es equivalente a

M = 592X 1024 Kg Dejemos -de tasea a11ector que confirme este -da~o en algim 111gt10 tgterico -de BStronomia

1 () ~

I

C tlImiddot ts I tl Xr=r=rl

y re-emplazando ~te (tltimo angulo y Ia g0cltlf~ica s= 1370 (m en la ee(3) concluimos que el radio de la Tierra es equivalente a

)

I t

R =041005 km (5) La ltlivergencia de este ltlata lton el l81or oficialmente aceptaoo (6378 km) fue de poco menos de n km un error muy pequeno en comparacion ltOn el tamano de nuenro planetB-

I b Determinadon de la Aceleracion Gravitacional -g- La aceleracton gravitactonal g se obtiene directamenoo recurriendo

I al concepto de caida libre Una de las ecuactones basicas de la cinematica que relactona esta constanoo g con la altura 11 a la que se lioora un objeto y con el tiempo -t- que este tarda en caer at suelo esI h = vot+ (l2)gt2

I donde va representa la velocidad inicial del objeto Sin embargo como esoo desciende en caida libre su veloctdad inidal es cero Siendo a51 la eltuadon anterior se reduce a la siguiente forma

I g = 2ht2 (5)

I

I Cronometrando el tiempo que tardan diversos objetos de distintos

pesos y tamanos en caer al suelo desde una altura predeterminada podemos determinar g recurriendo a la elt(S) En una serie de pruebas que se hideron en dias pasados en el 1TESM campus Cd Juarez se

L liberaron una bOla de madera una esferita de acero y una pelota de beisbOl diez veces ltada una desde una altura h de 054 metros y S~ procedio a tomar el tiempo promedio de vuelo para cada una de elias (ver tabla de abajo)

I Objeto Literado

I Bola de Mooera Esfera de acero Pe10ta de Beislgtol

I

liempoPromedio ltle C8i6a Livre (reg)

1179 1155 1164

Alturaen C81cula de g en metres metros lseo 2

0

654 654 6gt1

940 980 96j

Promeltlio 961 mreg2

E1 promedio de g sali6 menor que el valor ofictalmente aceptado deI 98mseg2 porque la reuroSistencia del aire influy6 considerablemente en las medictones

I Ahora st como ya oonernos los valor~ oe R1 g los sustituimos en la ec(2)

I y venlOS finalmente que la masa de la Tierra es equivalente a

M = 592X 1024 Kg Dejemos -de tasea a11ector que confirme este -da~o en algim 111gt10 tgterico -de BStronomia

1 () ~

I