The Moon's Rotation

Date: 
Tuesday, April 1, 1919
Volume: 
22
Pages: 
886-892
Archived Page: 
Author: 
Subject: 

The Moon's Rotation by Nikola Tesla
Publication: Electrical Experimenter
Date: April 1, 1919

from which I calculate its volume to be approximately 5,300,216,300 cubic miles. Since its mean density is 3.27, one cubic foot of material composing it weighs close on 205 lbs. Accordingly, the total weight of the satellite is about 79,969,000,000,000,000,000 and its mass 2,483,5000,000,000,000 terrestrial short tons. Assuming that the moon does physically rotate its axis, it performs on revolution in 27 days 7 hours, 43 minutes and 11 seconds, or 2,360,591 seconds. If, in conformity with mathematical principles we imagine the entire mass concentrated at a distance from the center equal to two-fiths of the radius, then the calculated rotational velocity is 3.04 feet pr second, at which the glove would contain 11,474,000000000000000 short foot tons of energy sufficient to run 1,000,000,000 horsepower for a period of 1,323 years. Now, I say, that there is not enough of that energy in the moon to run a delicate watch.

In astronomical treatises, usually the argument is advanced that "if the lunar globe did not turn upon its axis it would expose all parts to terrestrial view. As only a little over one half is visible it must rotate" But this inference is errenous, for it only admits of one alternative. There are an infinite number of axis besides its own in each of which the moon might turn and still exhibit the same pecularity.

I have stated in my article that the moon rotates about an axis passing thru the center of the earth, which is not strictly true, but it does not violate the conclusions I have drawn. It is well known, of course that the two bodies revolve around a common center of gravity, which is at a distance of a little over 2,8999 miles from the earth's center.

Another mistake in books of astronomy is made in considering this motion equivalent to that of a weight whirled in a string or in a sling. In the first place there is an essetnail difference between these two devices tho involving the same mechanical principle. If a metal ball, attached to a string, is whirled around and the latter breaks, an axial rotation of the missile results which is definitely related in magnitude and direction to the motion preceding.

By way of illustration - if the ball is whirled on the string clockwise ten times per second, then when it flies off it will rotate on it's axis ten times per second, likewise in the diretion of a clock. Quite different are the conditions when the ball is thrown from a sling. In this case a much more rapid rotation is imparted to it in the opposite sense. There is not true analogy to these in the motion of the moon. If the gravitation string, as it were, would snap, the satellite would go off in a tangent without the slighest swerving or rotation, for there si no moment about the axis and consequently no tendency whatever to spinnin motion.

Mr manieere is mistaken in his surmise as to what would happen if the earth were suddenly eliminated,. Let us suppose that this would occur at the instant when the moon is in opposition. Then it would continue on its eliptical path around the sun, presenting to it steadily the face which was always exposed to the earth. IF, on the other hand the latter woudl disappear at the moment of conjunction, the moon would gradually swing around thru 180 degrees and after a number of oscillations, revolve, again with the same face to the sun. In either case there would be no periodic changes but eternal day and night, respectively, on the sides turned towards and away from the luminary.

Some of the arguments advanced by the correspondents are ingenious and not a few comical None, however, are valid.

ONe of the writers imagines the earth in the center of a circular orbital plate, having fixedly attached to it's periperal portion a disk-shaped moon, in frictional or geared engagement with another disk of the same diameter and freely rotatable on a pivot projecting from an arm entirely independent of the planetary system. The arm being held continuously parallel to itself, the pivoted disk, of course is made to turn on ixs axis as the orbital plate is rotated. This is a well known drive, and the rotation of the pivoted disk is as palpable a fact as that of the orbital plate. But the moon in this model only revolves about the center of the system _WITHOUT THE SLIGHTEST angular displacement_ on its own axis. The same is true of a cart-wheel to which this writer refers. So long as it advances on the earth's surace it turns on the axle in the true physical sense;: when one of its spokes is always kept in a perpendicular position the wheel still revolves about the earths center, but axial rotation has ceased. Those who think that it then still exists are laboring under an illusion.

An obvious fallacy is involved in the following abstrat reasoning. The orbital plate is assumed to gradually shrink so that finally the centers of the earth and the satellite coincide when the latter revolves simultaneously about its own and the earths axis. We may reduc the earth to a mathematical point and the distance between the two planets to the radius of the moon without affecting the system in principle, but a further diminution of the distance is manifestly absurd and of no bearing on the question under consideration.

In all the communications I have received, tho different in the manner of presentation, the successive changes of position in space are mistaken for axial rotation. So, for instance, a positive refutation of my arguments is found in the observation that the moon exposes all sides fo other planets! It resolves, to be sure, but none of the evidence is a proof that it turns on its axis. Even the well-known experiment with the vouvault pendulum, altho exhibiting similar phenomena as on our globe, would merely demonstrate a motion of the satellite about _SOME AXIS_. The view I have advanced is NOT BASED ON A THEORY but on facts +DEMONSTRABLE by experiment_. It is not a matter of _definition_ as some would have it. A MASS REVOLVING ON ITS AXIS MUST BE POSEST OF MOMENTUM. If it has none, there is no axial rotation, all apperance to the contrary notwithstanding.

A few simple reflections based on well establish mechanical principles will make this clear. Consider first the case of two equal weights w and w1, in fig 1, whirled about the center O on a string s as shown. Assuming the latter to break at "a" both weights will fly off on tangents to their circles of gyration, and, being animated with different velocities, they will rotate around their common center of gravity o. If the weights are whirled n times per second then the speed of the outer and the inner one will be respectively, V= 2 ( R + r) n and V1= 2pi (R -r)n and, the difference V - V1 = 4pi r n , will be the length of the circular path of the outer weight., In asmuch, however, as there will be equalization of the speeds until the mean value is attained, we shall have V - v1 / 2 = 2 pi r n = 2 pi r N ; N being the number of revolutions per second of the weights around their center of gravity. Evidently then, the weights continue to rotate at the original rate and in the same direction. I know this ot be a fact from actual experiments. It also follows that a ball, as that shown in the figure , will behave in a similar manner for the two half-spherical masses can be concentrated at their centers of gravity m and m1, respectiely which will be at a distance from 0 equal to (obscured in the text reference).

This being understood, imagine a number o fballs, M carried by as many spokes S radiating from a hub G, as illustrated by a bike (ED.L) and in fig 2, and let this system be rotated w times per second around O on frictionless bearings. A Certain amount of work will be required to bring the structure to this speed, and itw ill be found that it equals exactly half the product of the masses with the square of the tangential velocity. Now if it be true that the moon rotates in reality on its axis this must also hold good for each of the balls as it performs the same kind of movement. Therefore, in imparting to the system a given velocity , energy must have been used up in the axial rotation of hte balls. Let M be the mass of one of those and R the radius of gyration then the rotational energy will be E= 1/4 M (2pi Rn)^2 since for one complete turn of the wheel every ball makes one revolution on its axis, according to the prevailing theory, the energy of axial rotation of each ball will be e= 1/2M (2 pi r1 n)^2, r1, being the radius of gyration about the axis and equal to 0.6325r. We can use as large balls as we like, and so make e a considerable percentage of E and yet, it is positively established by experiment that each of the rotating balls containing only the energy E, no power whatever being consumed in the supposed axial rotation, which is consequently, wholly illusionary. Something even more interesting may, however be stated. As I have shown before, a ball flying off will rotate at the rate of the wheel and in the same direction. But this whirling motion, unlike that of a projectile neither adds to , nor detracts from the energy of the translatory movement which is exactly equal to the work consumed in giving to the mass the observed velocity.

From the foregoing it will be seen that in order to make one physical revolution on it's axis the moon should have twice its present angular velocity, and then it would contain a quantity of stored energy as given in my above letter to the New York Tribune, on the assumption that the radius of gyration is 2/5 that of figure. This, of course, is uncertain as the distribution of density in the interior is unknown. But from the character of motion of the satellite it may be concluded with certitude that it is devoid of momentum about its axis. If it be bisected by a plane tangential w the orbit, the masses of the two halves are inversely as; the distances of their centers of gravity from the ce the earth's center and therefore, if the latter were to disappear suddenly no axial rotation, as in the case of a weight thorn off, would ensue.

Continued on page 3 out of 7 and at :

ELECTRICAL EXPERIMENTER April, I9 1 9 he Qonis otation By NHKQLA TESLA INCE the ag earange of my article entitled the amous Scientific Illu- sions" in your ebrua issue, I have received a number of riétters criticiz- ing the views I exprest regarding the moon's “axial rotation.” These been partly answered by my state- ment to the New Y orlz Trsbune of Feb- ruar{ 23, which allow me to gusto: n Your issue of February , Mr. Char es' E. Manierre, commenting ?on my article in the Electrical :perimentcr for February which appeared in the Tribune of Janu- ary 26, suggests that I give a deh- nition of axial rotation. I intended to be explicit on this point as may be judged from the ollowing quotation: “The unfail- ing test of the spinning of a mass is, however, the existence of energy of motion. The moon is not posest of such ui: viva." By this I meant that “axial rotation” is not simply “rotation upon an axis nonchalantly defined in dictionaries, but is a cir- cular motion in the true physical sense-that is, one in which half the product of the mass with'the square of velocity is a definite and positive tkiantitff. The moon is a nearly sp erica body, of a radius of about 1,087.5 miles, from which have conclusions I have drawn. It is well known, of course, that the two bodies revolve around a common center of ¥ravity, which is at a distance of a ittle over 2,899 miles from the earth’s center. /-x ; , :_ -5 M, sling. ln this case a murh mare va/aid rotation is imparted to it in the oppo- site sense. There is no true analogy to these in the motion of the moon. If the gravitational string, at it wcrc, would map, the .mtellilz would go ol? in n tangent without the .slightest _rwnn/ing or rotation, for there is na moment about the axis und, cumequmtly, na tendency 'whatever to Ictainning motion. T r. Manierre is mistaken in his ,> ,. surmise as to what would happen if . s _/' <~ f the earth were suddenly eliminated. Let us suppose that this would oc- cur at the instant when the moon is in opposition. Then it would continue on its eliptical path around the sun, presenting to it steadily the face which was always exposed to the earth. lf, on the other hand, the latter would disappear at the moment of ronfunclian, the moon would gradually swin around thru 180° and, after a number of oscilla~ tions, revolve, again with the same face to the sun. ln either case there would be no periodic changes but eternal day and night, respec- tively, on the sides turned towards, and away from, the luminary. Some of the arguments advancuu by the correspondents arc ingenious 'i if . “ ¢ *sf-ev* 5 s ' 5 'e.` H E 5 li* | ,xi rn' 0 . - if .fig e ,_t, a” Mig: Ill Thlnk That the Moon Rotate: on It: Axle' I calculate its volume to be approx- imately 5,300,2l6,300 cubic miles, Since its mean density is 3.27, one cubic foot of material composing it weighs close on 205 lbs. Accord- ingly, the total weight of the satel- lite is about 79,969,000,000.000.000,000, and its mass Z,483,500,000,000,000,000 terrestrial short tons. Assuming that the moon does physically rotate upon its axis, it performs one revolution in 27 days, 7 hours, 43 minutes and 11 sec- onds, or 2,360,591 seconds. Ii, in con- formity with mathematical principles, we imagine the entire mass concen- trated at a distance from the center equal to two-hiths of the radius, then the calculated rota- tional velocity is 3.04 feet per second, at which the globe would contain ll,- 474,(X)0,000,000,000,00C short foot tons of energy sufficient to run 1,000,000,000 horsepower for a Rleriod of l,3Z3 years. ow, l say, that there is not enough of that energy in the moon to run a delicate watch. In astronomical treatisies usually the argument is advanced that "if the lunar globe did not turn upon its axis it would expose all parts to terrestrial view. As only a little over one- half is visible it must rotate." But this in- ference is erroneous. for it only admits of one alternative. There are an infinite number of axis besides its own in each of which the moon, might turn and still exhihit the same peculiarity. ' I have stated in; my article that the moon rotates about-an axis passing thru the center of the earth, whiQh_ is not -""""' true but it does not """"' the Look at Thl| Dlmgram and Follow Closely the Succelllve Polltlonl Taken y One at the Balls M Whlle It ls Ro- tatud by a Spoke of the Wheel. Suhltltuts Gravlty for the Spoke and the Anala Solve! the Moon Rotation Rh;/d le. Another mistake in books on astron~ omy is made in considering this motion equivalent to that of a wei%ht whirled on a string or in a sling. n the first place there is an essential dillerence be- tween these two devices tho involving the same mechanical principle. If _a metal ball, attached to xi string, is whirled around and the latter breaks, an axial rotation of the missile results which is definitely related in magnitude E believe the accompanying illustration and its explanation will dispel all doubts as to whether the moon rotates on its axis or not. Each of the balls, us M, delfricts a diyerent iositilm af, and rotates exactly like, the moon eeping a ways t e same fiwe turned towards the center 0, representing the earth. But as you study this diagram, can au conceive that any of the balls tum on theiruxis? Plainly tliilis is rendered physically impossible by the spokes. But i you are still unconvinced, Mr. Tesla’s experimental proof wi l surely satisfy you. A body rotating on its axis must contain rotational energy. Now it is u fact, as Mr. Tesla shows, that no such energy is imparted to the ball us, for instance, to a projectile discharged from a gun. I I5 is therefore evident that the moan, in which the gravitational ut- traction is substituted for u spoke, cannot rotate on its axis or, in other words, contain rotational energy. I f the earth’.s attraction would suddenly cease and cause it to fly of in, a tangent, the moon would have no other energy exceit that of translatory movement, and it would not spin like the ull.-Editor. and direction to the motion preceding. B wa of illustration-if the ball is whirlecl, on the string clocltwise ten times per second, then yvhen it_fl\es off, it will rotate on its axis ten times per_ second, likewise in the direction of a clock. Quite ditterent are the cond\~ tions when the ball is thrown from a and not a few comical. None, how- ever, are valid. One of the writers imagines the earth in the center of a circular nr- bital plate, having fixedl atisrclicil to its periperal oprtion a disk-shammi moon, in frictional or geared engagement with another disk of the same diameter and freely rotatable on a pivot projecting from an arm entirely independent of the plane- tary system. The arm being held continu- ously parallel to itself, the pivoted disk. of course, is made to tum on its axis as the orbital plate is rotated. This is a well- known drive, and the rotation of the pivoted disk is as palpable a fact as that of the orbital plate. But, the moon in this model only revolves about the center of the system u-itltout the .slighlrst angular dir- placemenr on its own axis. The same is true of a cart-wheel to which this writer re- fers. So long as it ad- vances on the ean.b's surface it turns on the axle in the true physi- cal sense: when one of its spokes is always kept _in a perpendicular position the wheel still revolves about the earth`s center, but axial rotation has reasrd. Those who think that it then still exists are laboring tm- der an illusion. _ An obvious fallacy is involved in the fol- lowing abstract reason- ing. The orbital plate is assumed to gmdually shrin .so that nally the centers of the earth :md the satellite coin- cide when the latter revolves simultaneously alwiut its own and the earth`s axis. \\’e may reduce the earth to a mathematical point and the distance between the two plriut-ts to the radius of the moon without atiectiiig ill* system in principle, but a further rlimimivton of the distance is mani- ((`u/ll. vu [2 Slljl

cmzcrnicar. exrcnrmenrsn April, im The M 0on’s Rotation tGontimi:d from page 866) festly absuriajnd Ui* no'l>caxi11€ 0" tht question und r consideration. _ In ull the communications I have received, tho different in the manner of presentation, the successive changes of position in space are mistaken for axial rotation. So, for in- stance, a positive reiutation of my argu- ments is found in the observation that the moon exposes all sides to other planets! It revolves, to be sure, but none of the evidenca is a proof that it tums on its axis. ven the well-known experiment with the Foucault pendulum, altho exhibf iting similar phenomena as on our globe, would merely deinonstrate a motion of the satellite about .tome axis. The view I have advanced is NOT BASED ON A THE- ORY but on facts demonstrable b Mperi- menl. It is not a matter of dejhilian as some would have it. A MASS REVOLV- ING ON \ITS AXIS MUST BE POSEST OF MOMENTUM. If it has none, there is no axial rotation, all appearances to the contrary notwithstanding. A few simple reflections based on well establisht mechanical grincigles will make this clear. Consider rst e case of two equal weights 'w and wi, in Fig. 1, whirled about the center O on a strings as shown. Assuming _the latter to brea at a both' weights will fly off on tangents to their circ es of gyration, and, being animated with different velocities, they will rotate around their common center of gravity o. If the weights are whirled n times per sec- ond then the speed of thc outer and the inner one will be,`respectively, V = 2 (R -l- r) n and Vi = 2'If(R - r) n, and the difference l’ - Vi = 4’lIrn, will be the length of the circular path of the outer weight. Inasmuch, however, as there will be equalization of the speeds untillthe ,mean - i value is attained, we shall have -T = Zfrn = ZTTPN, N lacing the number of revolutions pcr second of the weights around their center oi gravity. Evidently then. the weights continue to rotate at the original rate a|\d in the same direction. I know this to he a fact from actual ex- periments. It also follows that a ball, as that shown in the figure, will behave in a similar manner for the two half-spherical masses can he concentrated at their centers of gravity and in and mi, respectively, which will bc at a distance from a equal to 5g r". 'l`his being understood, imagine a numbcr of halls ,ll carried by as man spokes .Y radiating from a hub H, as iliiistratetl in Fig, Z, and let this system be rotated n- tinics per second around center O on fric- tiunlcss hearings. A certain amount of work will bc required to bring the struc- ture to this speed, and it will he found that it equals exactly half the product of the masses with the square of the tangential velocity. Now if it he true that the moon rotates in rcality on its axis thi: must also hold gnnd /nr EACH nf I/ir ball: as il por- jtvrm: lln' sonic kind of mo:'rnmiI. There- fore, in imparting to the system a given velocity, energy must have been used up in the axial rotation of the halls. Let M be the mass of one of these and If the radius ol gyration_ then the rotational energy will he IE = MM t2'lfRn)'. Since for one com- plete turn of the wheel every hall makes one revolution on its axis, according to the prevailing theory, the energy of axial rotation of each ball \vill be c = MM (2‘lf r\n)’, ri being the radius of gyration about the axis and equal to 0,6325 r. We can use as large balls as we like, and so make c a considerable percentage of E and yct. it is positively established hy experi- ment that each of the rotating halls contain only the energy E, no power whatever being 'consumed' in the supposed axial rotation, which is, consequently, wholly illusionary. Something even more interesting may, how- ever, be stated. As I have shown before, a ball Hying 0,5 will rotate at the rate of the wheel and in the same direction. But this whirling motion, unlike that of a pro- jectiletaneither aldds to, nor detracts from; the energy of the translatory movement which is exactly equal to 'the work con- sumed in giving to the mass the observed velocity. From the foregoing it will be secn that in order to rmake one 'physical 'revolution on its axis the _moon should have twice its pnesent angular velocity, and then it would contain a quantity of stored energy as iven in my above letter to the New York $3411- nne, on the assumption that the radius of gyration is 2/5 that of figure. This, of course, is uncertain, as .the Jclistsibution of density in the interior is unknown, But from the character of\motioix of tha' satel- lite it may be ,coricludefl with cer-titude that it is det/aid of monuzntum abouriits-a.z~is. If it be bisected by a planqtangential to the W T, , )\ il / \\ R r I / : 5 / . / 1 / 1 1 / i / > / , / :__ u F ~ f, ° ,af 4 wi Dlagram lllustratlng the Flotatlon, of Weights Thwwn Off By Centrlfugal Force. orlmt, the masses of the two halves are inversely as;the distances of their centers of gravity from the carth’s center and, therefore, if the latter were to disappear suddenly, no axial rotation, as inzthe case of a weight thrown 05, would ensue.,

|32 The ln llii: nrliclc l)r Tesla prnzcr conclusir/c , - l tional and that the mmm carilnirtr abrolulely N revising my article on "The Moon's Rotation’ , which appeared in the April issue of the ELECTRICAL Exccmmsnrcn, I appended a few remarks to the orig- ` inal text in further support and eluci- ' 1 . , ny 1 W il. ll // /l n I' °' / 4 'l' Vg 0 ll ll ll M | Flq. 1. In Determlrllng the Klnetlc Energy of B Rntillnn Mass, T ls Figure Shows the SeleC!|0t'\ of a Number of Folnts Taken Wlth- ln the Stralrht Rod or Mau M, at Successlve Dlstances rom the Axls of Knowlng These Values and the tatlon the Klnetlc Energy of Readlly Computed. Rotailorl O, Speed of Ro- the Mass Is dation of the theory advanced. Due to the and, in con- printer's error these were lost sequence, I found it necessary to forward another communication which, unfortunate- ly, was received too late for embodiment in the May number. Meanwhile many letters have reached me in which certain phe- nomena presented b rotatin bodies, as the moon’s librations oglongituiie, are cited as evidences of energy due to spinning motion, i. e., proofs of axial rotation of the satellite in the true physical sense. I trust that the following amplilied statement will mcct all nt' the trltjt-t-tilnm mist-rl null <~.»»w<»t-t tu my views those who are still nut-tntvinced. The kinetic energy of a rotating mass can he determined in four ways which are illustrated in diagrams, Figs. 1, 2, 3 and 4 and may be found more or less suitable. Referring to Fig. 1, the method consists in selecting judiciously a ntnnber of points as ot, oi, os, etc., witlnn the straight rod or mass M, respectively at distances rt, rg, ri, etc., from the axis of rotation O and cal- culating the square root of the mean square of these distances. Its value being R., de- noted radius of gyration, the effective ve- locity of the mass at 1| revolutions per sec- ond will be V. I 2'lTR,.n and its kinetic energy E : % M V..' = M M (2 '7fR,n)". In Fig. 2 the mass M, rotating n times per second about an axis 0 at right angles to the plane of the paper, is divided into numerous elements or small parts, most conveniently very thin concentric laxninae, as lt, li, lt, etc., at distances rt, rs, rs, etc., from 0. Since the kinetic energy of each part is equal to half the product of its mass and the square of the velocity, the sum of all these elemental energies E = % E in V’ : % n1tVt’+}AmrV¢ + %m.V|’-l- ...... : % mi (2 1frin)' + 56 mi (2 1rr1n)' -1- % ms (Z 1I'rsn)' + ...... _ A different form of expression for the cncrgy of a rotating body may he obtained by determining its moment of inertia. For this purpose the mass M (in Fig. 3), ro- tating 1i times per second about an axis 0, is separated into minute parts, as in., nu, nu, etc., respectively at distances rt, ri. rt, etc., from the same. The sum of the products of all these small masses and the squares ELECTRICAL EXPERIMENTER June, l9l9 ®©n°s Rotation 3' NHKOLA TESLA of their distances is the moment of inertia I, and then E = % I /=~", 'H = 2 fr n being the angular velocity. It is obvious that in all these instances many points or elements will be required for great accuracy but, as a rule, very few are sufficient in practice. Still another way to compute the kinetic energy is illustrated in Fig. 4, in which C356 the quantity I is given in terms of the mo- mcnt of inertia Ir about another axis paral- lel to 0 and passing thru the center of gravity C of mass M. In conformity with this the energy of motion E = % M V' + % I= Lv” in which equation V is the velocity ol' the center of gravity. The |n-t-cvtliug is tler-med indispensable as l note that the eurrcspoiitlriits, even those who seem thoroly familiar with mechanical principles, fail to make a distinction be- tween lheoretical and physical truths which is essential to my argument. In estimating the kinetic energy of a ro- tating mass in any of the ways indicated we arrive, thru suitable conceptions and methods of approximation, at expressions which may be made quantitatively precise to any desired degree, but do not truly de- ny: __ Flg. 3. Another Form of Expreaslon for the Energy of a Rotating Body May Bo Obtalned by Determlnlng Its Moment of Inertia. Here the Mass M ls Subdlvlded Into Minute Part: m,, my m,, .... etc. The Sum of the Prod- uct: of These Maisel and the Sguarea of Thelr Dlstances ls the Moment o lnertla, Whlch wlth the Angular Speed, Glvel the Klnetle Energy E. Fine the actual condition of the body. To illustrate, when proceeding according to the plan of Fig. 1, \ve find a certain hypothetical velocity with \vhich the entire mass should fgl Flu. 4. ln thlo Cane the Motlon ll Resolved Into Two Separate Components-One Tran:- Iatlonal About 0 and the Other Rotntlonal About C. The Total Klnetlc Ensr of the Mau Equals the Sum of These Two Ignerglu. y by lltanry and m'/wrirnmil that all the kineli: energy of a rotating mar.: ir purely translu- no rotational energy, nz other -wvrdr, doe: not folate an its axu.-Eorron. move in order to contain the same energy, a state wholly imaginary and irreconcilable with the actual. Only, when all particles of the body have the same velocity, does the /'" "` 'W “ _A /., . WL 1 U ”2 / /'@.z __ Flql: 2. ln Thls Case the Mano M, Rotntlng n Imeo Per Second, About An Axls 0, In Dlvlded Into Numoroul Elements or Small Parte at Vnrloul Radll from 0. Knowlna the Klnetlc Enorgy of Each Part. tho Whole Klnotle Energly of the Mau ls Easlly Deter- mlned by Ta Ing a Summatlon of the Indl- vldual Guantltles. product % M V’ specify a physical fact and is numerically and descriptively accurate. Still more remote from palpable truth is the equation of motion obtained in the manner indicated in Fig. 4, in which the first term represents the kinetic energy of translation of the body as a whole and the second that of its axial rotation. The for- mer would demand a movement of the mass in a definite path and direction, all particles having the same velocity, the latter its simul- taneous motinn in nnothernpath and direc- tion, the parlit-|r~u having di crcnt velocities. This abstract itlca ol' angular motion is chiefly responsible for the illusion of the moon's axial rotation, which I shall en- deavor to dispel by additional evidences. With this object attention is called to Fig. 5 showing a system composed of eight balls M, which are carried on spokes S, radiating from a hub H, rotatable around a central axis`0 in bearings supposed to be frictionless_ It is an arrangement similar to that before illustrated with the exception that the balls, instead of forming parts of the spokes, are supported in screw pivots J, \vhich are normally loose but can be tight- ened so as to permit both free turnin and rigid lixing as may be desired. To gacili- tate observation the spokes are provided with radial marks and the lower sides of the balls are shaded. Assume, first, that the drawing depicts the state of rest, the balls being rotatable \vitl\out friction. and let an angular velocity Lv = 2 'II' n be im- parted to the system in the clockwise direc- tion as indicated by the long solid arrow. Viewing a ball as M, its successive positions 1, 2, 3-8 in space, and also relatively to the spoke, will be just as drawn, and it is evident from an inspection of the dia- gram_ that while movin with the an ular velom? f-I about 0, in the clockwise direc- tion, t e ball turns, with respect to its axis, at the same angular velocity but in the op- position direction, that of the dotted ar- row. The combined result of these two motions is a translatory movement of the ball such that all particles are animated with the same velocity V, which is that of its center of gravity. In this case, granted that there is absolutely no friction the

_lune, |9l9 kinetic energy of each hall will be g-iven by the product of M M V’ not approximately, bitt with mathematical rigor. If no\v the pivots are screwed tight and the balls fixt rigidly to the spokes, this angular motion relatively to their axes becomes physically im- possible and then it is found that the kinetic energy of each ball is in- creased, the increment being exactly the energy of rotation of tlte ball on its ax-is. This fact, which is borne out both by theory and experiment, is the foundation of the general notion that a gyrating body-in this instance face towards the center of motion, nrtually rotates upon fits axis in the same sense, as indicated by the short full arrow. Un! il doc: nut lho to the eye i»'.rer1us so. The fallacy will become manifest on further inquiry. To begin with, observe that when a mass, say the armature of an elec- tric motor, rotating with the angular velocity U, is reversed, its speed is - 1-1 and the difference w - (- w) :_ Z H. Now, in fixing the ball to the spoke, the change of angular velocity is only v; therefore, an additional velocity w would have to be imparted ball M-presenting always the same /l .I ELECTRICAL EXPERIMENTER 3-lo. M I 5' -M, A 9 ,W , My ,\ s / I yu . /,, .~ - .fs.,_~- f--L-,~~@ /, pe/ WM ny. 5 ' /M 133 about the parallel axis passing thru C and equal to 3/§ M i’ so that E. : %M (21fRrl)’+%Mr’<21fn)’- Neither of these two expressions for E describes the actual state of the body but the first is certainly prefer- able conveying, as it does, the idea of a single motion instead of two, one of which moreover 'is devoid of ex- istence. I shall first undertake to demonstrate that there is no torque or rotary effort about center C and that the kinetic energy of the sup- posed axial rotation of the ball is matltematically equal to zero. This makes it necessary to consider the two halves separated by the tangen- tial plane pp wholly independent from one another. Let ci :\|\d C- be their centers of gravity, then Cc, : Cc; I M r. ln order to ascertain the kinetic energy of the hemispheres we have to find their radii of gyration which can be done by determining the moments of inertia Ic. and lc, about parallel axes passing thru ci and cz. Complex calculation will be avoided by remembering that the moment of inertia of either one of the half spheres about an axis thru C is IC V ' i / 4* 4 / // /7 // , \ I/ ( Y // lf / | Fl . 5. Thll Dlnnrnnt Hspraltmtl tl Syuttsni Compound of to it in urrler tu cause at clockwise rotation of the ball on its axis in tlte truc significance of tlte word. The kinetic energy would then he equal to the sum of the energies of the translatory and axial motions, not merely in the ab- stract mathematical meaning, but as a phy- sical fact. I am well aware that, according to the prevailing opinion, when the ball is free on the pivots it does not turn on its axis at all and only rotates with the angu- lar velocity of the frame \vhen rigidly at- f i . ,- f P , ‘ F V We ,"' if I l Flg. 6. Diagram Showing a Ball Havlnq Milt M, 0| Radlul r, Flotatlnq About Center 0, and Used In the Theoretlcal Analyull of the Moon's Motion. tached to the same, but tlte truth will ap- pear upon a closer examination of this kind of movement. Let the system be rotated as first assumed and illustrated, the balls being perfectly free on the pivots, and imagine the latter to be gradu- ally tightened to cause friction slowly reducing and iinally pre- venting the slip. :\t the ntttset all particles of each ball have been moving with the speed of its center of gravity, but as tlte bearing resistance asserts itself more and more the translatory velocity of tlte part-icles nearer to the axis 0 will be diriiiltixliintg, while that of tlte diametrically opposite ones will be iurrcasing, until the maxima of tltese I gill! M, Clrrled on Spoke! S and Rntntlng Around center o. 'rm Batt. Ars Freely notatatns on Plvotn Which Can Be Tlghiellsd. Wlih hll Model the Fillatiy of the Moo|t'| Rotation on Its Axle Is Demonatrahle. those parts of the masses which are nearer to the center of motion, of some kinetic en- ergy of lrun.rlr|li01i while adding to the en- ergy of those which are farther and, obvi- ously, the gain \vas greater than the loss so that the :Iertivc velocity of each ball as a whole was increased. Only .ru have \ve aug- mented the kinetic energy of tlte system, not by causing axial rotation of the balls. The energy E of each of these is .solely that of translatory movement with an eflective ve- locity V. as above defined such that E = % M V.’. The axial rotations of the ball in either direction are but apparent; lln-_v Inn/c nn rrnlily tnliulctfrr aml call for no mechani- cal eflort. lt is merely when an extraneous force acts independently to turn the whirl- ing bod on its axis that energy comes into play_ lliicidentally it should be pointed out that in trite axial rotation of a rigid and homogenous mass all symmetrically situated particles contribute equally to the momen- tum which is not the case here. That there exists not even the slightest tendency to such motion can, however, be readily estab- lished. lfnr this purpose l wonhl refer tn Fig. (1 slmwin); at ball l\l of rntlitis r, tlte center C of which is at a distance R front axis 0 and which is bisectcd by a tangential plane [fp as indicated, the lower half sphere being shaded for distinction. The kinetic energy of the ball when whirled n times per second about 0 is according to the first form of expression E = % M Vt.” = M l\l (2 WR., n)“, M being the mass and R, the radius of gyration. But, as explained in connection with Fig. ~l, we have also E = M l\1V“+ yé l-- I-f”,V : 2 1r It n being the velocity of tlte center of gravity C and l~ thc moment of inertia of the ball, ;= "' »_-_._- > fs-fr. 'LF' -EE K- hy a 4; M >< Ks M r", : Ke M r', and since M = 2 in, lg : if, m r“. This can be exprest in terms of the mo- ments lci and Icz; namely, Ig : Iv, -lv- ln (Mi r)’2 Iva -l- in (Mg r)‘. Hence lci=Ic¢= lg-in(9/§r)’:$@n1i’- 9/64 m r' = 83/320 m r’. Following the same rule the moments of inertia of the ltalf spheres about the axis passing thru the center of motion 0 can be fotmd. Designating thc moments for the ttpper and lower halves of the ball, respectively, Igi and Ig, we have lui = m (R -1- yt r)’ + In = m (R -l- M r)‘ -l>- 83/320 m r’ and lot = m (R _ _T i J 7 _ FEL 7. Here Two Mane: m-rn, Are Con- sl ered al Condensed Into Polnts, Attached to Welqhtleas Strlrlgl of Dlfferertt Radll. If Both Strings Are Cut, and the Mane: Con- sldered an Joined, Then There Wlll Be No Rctatlon About the Common Center of Gravity. - My rl' + lu, : m (R - Mg r)‘ -l- 83/320 in ri. Thus for the upper half sphere the gyration RH, : radius of IO' = t;\tinn ht-ing itil will bv exact- " W/ l l Il FI . s. Tn MRKG U18 PI‘Dhl8|\l Sl\0w|\ lll Fl(|.~ 7 Clt!i\I`» lIl\I\ ||I\tJ TWU clntngcs are attained when tlte balls are lirutly held. ln llnsrnp- eration we have thus deprived Rl%¢ Birrsll Parallel to Eatllt 0K|\t|l'. If Two B.\ll|t M-M llkro Flrutl Slnltllttldlbtlllé. Jnlllrd by n T|\ut\t'|l\lt.n| Unltd, They Wlll Rnvnlvn Abotll Tlldlr nlltllmll Cultlul' ol tlritvlly, I‘|nvl|\\| ltnl thu Muull Puuuueii Only Kinetic Energy of Trzinltatlun. ly t-‘phil In tht- until tt»i|u~ti<~ ctr rrpy ul tht- l».tl| ns tt tnnt. 'l`ltt1 (t,`unIin|n'tl un [tug/e l5(>)

ELECTRICAL EXPERIMENTER june, ww The Moon’s Rotation By Nikola Tesla (Cauimurd /ram pm/4' 133) significance of this will he understood by reference to Fig. 7 in which the two masses, condensed into mints, are repre- sented as attached to independent weight- less strings of lengths Ru and R.: which are purposely shown as displaced but should be imagined as coincident. It will be readily seen that if both strings are cut in the same instant the masses will ily off in tangents to their circular orbits, the angular movement becoming rectilinear without any transformation of energy occurring. Let us now inquire what will happen if the two masses are rigidly joined, tie connection being assumed imponderable. Hen' 'wr come to the real bu in the querliou under discussion. Evidcntlaly, so long as the whirl- ing motion continues, and both the masses have precisely the same angular velocity, this connecting link will be of no eliect whatever, not the slightest turning effort ahout the eonuuon center ul' gravity of the masses or tendency of equzilization of ener- gy between them will exist. The moment the strings are broken and they are thrown oli they will begin to rotate but, as pointed out before, this motion neither adds to or detracts from tl\e energy stored. The ro- tation is, however, not due to an exclu- sive virtue of angular motion, but to the fact that the tangential velocities of the masses or parts of the body thrown ott' are diiierent. To make this clear and to investigate the etiects produced, imagine two rifle barrels, as shown in Fig. 8, placed parallel to each other with their axes separated by a dis- tance Ru - R.. and assume that two balls of same diameter, each having mass m, are discharged with muzzle velocities V, and Vi, respectively equal to Z fl' n Ru and 21' n Rn as in the case just considered. If it be further supposed that at the instant of lea\'~ ing the barrels the balls are joined by a rigid but weightless link the will rotate about their common center oi' gravity and in accordance with the statement in my previous article above mentioned, the rela- Vi - Vs tion will exist -T = 1r n (Rn - Rn) n bein the number of revolutions per sec- ond. %`he equalization of the speeds and kin- etic energies of the halls will be, under these circumstances, very rapid but in two heav- enly bodies linked b gravitational attrac- tion, the rocess miglit require ages. Now, this whirling movement is real and requires energy which, obviously, must be derived from that originally imparted and, conse- quently, must reduce the velocity of the balls in the direction of flight by an amount which can be easily calculated. At the moment of discharge the total kinetic ener- gy was E = % m Vi' + % n1 Vi’ which is evidently equal to n1 V¢’, V., being the ef- fective velocity of the common center of gravity, from which follows that Vi = V 1' + Vt’ 17-. The speed of revolution of Y. - V: the masses is, of course. »---- and the 2 rotational cnergy`of both lmlls, wdiich must .-Vt ' he considered as points, is e : ru(-A Z The kinetic energy nf translation in the direction of flight is then % in Vi’ + % in Vi-V1 ‘ Vi +V, ' Vi' - m -_-» =m _~ : m 2 Z V- 'I \"¢ V¢', V. : T-- heing the speed of the

une. l9l9 ELECTRICAL EXPERIMENTER t'u|n|n\||t ccntcr of grztvity, sn that V. - Vi is the hiss of velocity ill thu tlircctiun ul' tligbt owing to the rotation of the two mass points. If instead of these we would deal with the balls as they are, their rotational energy V. + \'; 2 e\=e+iw“=m if ‘i‘i(27f|\)z i being the moment of inertia of each ball about its axis. As will he seen. we arrive at precisely the same results whether the movement is recti- linear or in a circle. In both cases the total kinetic energy can he divided into two parts, respectively of the same numerical values, but there is an essential dijerenre. ln angular motion the axial rotation is noth- ing more than an abslrad conception; in rectilinear movement it is a posilit-c t'1't'nt, Virtually all satellites rotate in like inan- ner and the probability, that thc accelera- tion or retardation of their axial motions- if they evcr existed--should come to a stop precisely at a definite angular velocity, is in- finitesimal while it is almost absolutely eer- tain that all movement of this kind would ultimately cease. The most plausible view is that no true moon has ever rotated on its axis, for at the time of its birth there must have been some deformation and displace- ment of its center of gravity thru the at- tractive force uf the tnuther planet so as tu make its peculiar position in space, relative to the latter, in which it persists irrespective of distance, more or less stable, In ex- plunation of this, suppose that one of the balls as M in Fig 5 is unt of homogenous material and that it is similarly supported but on an axis passing thru its center of gravity ‘instead of form. Then, no matter in what position the ball is fixed on the pivots, its kinetic energy antl centrifugal pull will be the same. Nevertheless a tli- rective tendency w-ill exist as the two cen- ters do not coincide and there is, conse- quently. Ito tiynumic balance. When per- mitted to turn freely on the axis of gravit the body, of whatever shape it may he, wiii tend to place itself so that the line joining the two centers points to O and thcre may he twu positions of stability but, generally, if the center of gravity is not greatly t|i~- placed, the heavier side will swing: out- wardly, Such condition may obtain in the moon if it had solidified before receding from the earth to great distance, when the ((`fml‘inm'4l un [fugr l60)

ELECTRICAL EXPERIMENTER une I9 I9 The Moomfs Rotation By Nikola Tesla (Crvziliiiurd from pugr 157) nrrnngcntcnt of the masses i|\ its interior became subject to gravitational forces of its own, vastly greater than the terrestrial. It has been suggested that the planet is egg-shaped or ellipsoidal but the departure from spherical forln must he inconsider- ahle. It may even he a perfect sphere with the centers of grav-ity and symmetry coin- ciding' and still rotate as it docs. \Vhatever he its origin and past history. the fact is, that at present all its parts have the same angular velocity as though it were rig~idly connected with the earth. This state must endure forever unless forces from without the lnna-terrestrial system brin about dif- ferent conditions and thus the hope of the star-gazers that its other s/ide may become visible some day must be indefinitely de- lcrrrtl. !\ inolion of this character, as I have shown. precludes the possibility of axial rotation. The easiest \vay to free ourselves of this illnsion is to conceive the satellite snlxdivided into minute and entirely inde- pendent parts, as dust particles, which have different orbital. but rigorously the same angular, velocities. One must at once recognize that the kinetic energy of such an agglomeration is solely translational, there licing alisolntcly no tendency to axial ro- lalinn. This makes it also perfectly clear why the moon. provided its distance does not greatly increase, must always turn the same face to ns even without any ifllzvrnll dir1‘cIi'z'1' Irnduziry nor so much as the slighlrxl rifnrt from llzr rurtli. Referring to the librations of longitude, I do \\ot see that they have any bearing on this question. ln astronomical treatises the axial rotation of the moon is accepted as a material fact and it is thought that its angu- lar velocity is constant while that of the orbital movement is not. this resulting in an apparent oscillation revealing more of its surface to our vie\v. To a degree this may he true. Init I hold that the mere change of orliital velocity, as will be evident from what has been stated before could not produce these pltenonirnzt, for no matter li-iw fast -.r slim ilu- gyrniion, the posi- tion of the hotly relative to thu: center of attraction remains the same. The real cause of these axial displacements is the changing distance of the moon from the earth owing to which the tangential com- ponents nl velocity of its parts are varied. ln <1/fogrr. \vhen the planet recedcs. the radial component of velocity decreases while the tangential increases but, as the decrement of the former is the same for all parts. this is more pronounced in the re- gions facing the earth than 'in those turned away from it. the consequence being an axial displacement exposing more of the cnsterirside. In /rrriqrr, on the contrary, the radial component increases and the ef- fect is just the opposite with the result that more of the western side is seen. The moon actually swings on the axis passing tl\r\\ its center of gravity on which it is supported like a ball on a string. The forces involved in these peudular move- ments are iucomparably smaller than those required to effect changes in orbital ve- locity. If we estimate the radius of gyra- tlon of the satellite at 600 miles and its mean distance from the earth at 240,000 miles, lhen the energy necessary to rotate it once in a month would be only (00 2 l -~» ~ - - ul tl-u kinetic energy 2~l(),(lfltl ll'»0,0tll) of the orbital xnovenient.

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