Analytical Proof of Newton's Force Laws


 Walter Sullivan
 5 years ago
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1 Analytical Poof of Newton s Foce Laws Page 1 1 Intouction Analytical Poof of Newton's Foce Laws Many stuents intuitively assume that Newton's inetial an gavitational foce laws, F = ma an Mm F = G, ae tue since they ae clea an ( istance M m) simple Howeve, thee is an analysis that ties the two equations togethe an emonstates that they must be tue The analysis povies answes to questions such as, "Is the inetial mass exactly the same as the gavitational mass? Why is the exponent of istance,, an not 199 o 01 o 1? Why is a constant equie in one law an not in the othe?" The ieal way to pove new theoetical laws is to foecast the outcome of an expeiment using the laws, pefom the expeiment, an fin that the esult is as foecast But natue ha aleay pefome the expeiment with planets in the sola system, an Keple ha etemine the esults So, Newton, in his 1669 pape, "Mathematical Pinciples of Philosophy", (now pat of the Geat Books Seies in local libaies), applie his foce laws to the sola system an obtaine the same esults that Keple ha state This confime Newton's ieas, put physics on a fim mathematical basis an answee the above questions Summay of Analytical Poof of Newton's Foce Laws In the 8 step poceue that follows, Newton's foce laws ae applie to the planet sun system, an the planet (eath) path aoun the sun is shown to be an ellipse This poceue below uses the mathematics foun in fist yea college texts an explains the mathematics within the eivations as they ae being evolve 1 Obsevations show that the planets follow a smooth cuve aoun the sun Sketch the planet at position P, using pola cooinates, an θ, within an othogonal cooinate system
2 Page Analytical Poof of Newton s Foce Laws Diffeentiate planet position functions to obtain planet velocities, v an v x y 3 Diffeentiate planet velocities to obtain planet acceleations, ax an ay 4 Equate Newton's inetial an gavitational foce laws as applie to the planet In this step we assume that inetial mass is ientical to gavitational mass an that the foce of gavity eceases as the squae of istance The acceleation equie in the inetial law is also assume to be the planet aial acceleation G must also be assume to make the equations consistent The en esult of this analytical poceue must show that all these assumptions ae coect, o Newton s equations woul not be tue 5 Convet acceleations, ax an ay, to assume planet aial, a R, an tansvese, a T, acceleations 6 Replace the time epenent tem, θ of a R, with a function of, in the expession 7 Replace the time epenent tem, of a R, with a function of an θ, in the expession 8 Solve the iffeential equation containing as a function of θ The solution is the pola equation of an ellipse This esult is the same as Keple's etemination fom astonomical ata an analytically poves Newton's foce equations 1 Planet Position in Pola Cooinates, an θ This analytical poof of Newton's foce laws begins with a planet P, moving along a smooth cuve in a pola cooinate system as shown in Figue 1 The planet is moving elative to the stationay sun
3 Analytical Poof of Newton s Foce Laws Page 3 y y position of P P Eath Sun 0 θ x position of P x Figue 1 Raius vecto,, is attache to planet, P, an vaies in length as P moves Also, angle θ an its ate of change vay as P moves Theefoe, the velocity an acceleation of P vay continuously as the planet moves along its path Recall that acceleation, velocity an foce have magnitue an iection Newton ha peviously pove that, as fa as the foce of gavity was concene, the entie mass of the planet an sun can be consiee to be at the cente of thei sphees The aius vecto stats at the cente of the sun an ens at the cente of the planet Detemine the x an y positions of P, as a function of an θ, by using the tigonometic functions that ae inicate by Figue 1 The x istance of P fom the oigin; P x = cos θ The y istance of P fom the oigin; P y = sin θ As time passes, P moves along its cuve, making an θ epenent upon time, t The positions of P as functions of time ae inicate as; P ( t ) ( t ) ( t ) P ( t ) ( t ) ( ) x = cos θ, y = sin θ t This completes step 1
4 Page 4 Analytical Poof of Newton s Foce Laws Planet Velocity in x an y Diections Figue inicates that the change in x an y istances is a function of both an θ as P moves in time along its path y velocity of P y y position of P P x velocity of P 0 θ x position of P x Figue The velocity of the planet, P, is the change of istance along the cuve pe the change in time O, change in istance change in time s = t The calculus expession fo velocity in the x iection, as P x t the change in time is mae vey small is; ( ) P Velocity in the y iection is; ( t ) y
5 Analytical Poof of Newton s Foce Laws Page 5 Theefoe, the velocity of P in the x iection is; v x = ( ( t ) ( t ) ) cosθ An the velocity of P in the y iection is; v y = ( ( t ) ( ) ) sinθ t The calculus ule fo obtaining the eivative of the pouct of two vaiables is to multiply the fist tem times the eivative of the secon tem plus the secon times the eivative of the fist Also, the eivative of the sine of an angle is the cosine of the angle times its eivative, an the eivative of the cosine of an angle is minus the sine of the angle times its eivative Using these iffeentiation ules; The expession fo the velocity of P in the x iection is, v x = θ ( cosθ ) = cos θ sinθ The expession fo the velocity of P in the y iection, following the same ules, is: v y = θ ( sinθ ) = sin θ + cosθ This completes step
6 Page 6 Analytical Poof of Newton s Foce Laws 3 Planet Acceleation in x an y Diections The next step is to obtain expessions fo the planet acceleations in the x an y iections inicate in Figue 3 y Acceleation in y iection P Acceleation in x iection θ 0 x Figue 3 Recall that acceleation is the ate of change of velocity Let the acceleation of the planet in the x iection be a x Let the acceleation of the planet in the y iection be a y Then; a x = v x, an a y = v y Fining acceleation causes us to take the eivative, with espect to time, of velocity Velocity, in tun, is the eivative of istance with espect to time Theefoe, acceleation is the secon eivative of istance with espect to time The eivative of a eivative is calle the secon eivative The symbol fo the secon eivative, in this case is; ( vaiable θ o )
7 Analytical Poof of Newton s Foce Laws Page 7 Replace v x an v y with thei eive expessions liste in Section Follow the same iffeentiation ules as given above an obtain: a x = θ θ θ cos θ sin θ + a y = θ θ θ sinθ + sin θ + This completes step 3 Acceleations a x an a y must be use to obtain expessions fo the aial an tansvese acceleations of the planet in Step 5 4 Equate Gavitational Foce to Planet Inetial Foce Newton's foce of gavity law as applie to eath mass, m, an sun mass, M, is; F Gavitational = G Mm Whee is the aius vecto, the vaying istance fom eath to sun, an G is the gavitational constant F Gavitational is the amount of foce that acts in a staight line between the planet an sun This foce woul place the planet in fee fall towa the sun if it wee not fo the counteacting planet inetial foce What is the inetial foce on the planet? Newton s inetial foce law states that the inetial foce is equal to the acceleation of the planet times the mass of the planet F Inetial = ma Raial
8 Page 8 Analytical Poof of Newton s Foce Laws The inetial an gavitational foces must be equal to each othe in magnitue but opposite in iection, o else the planet woul leave its obit With unequal foces, the planet woul fall into the sun, o attain a iffeent obit in a new equilibium path, o go spinning off into space Since the planet oes maintain its obit, the sum of the two foces must be zeo F Gavitational + F = 0 Inetial G Mm + ma Raial = 0 Divie though by m an obtain;g M + a = 0 Raial Then; G M a Raial = This is an impotant place in the poof whee the inetial mass is assume to be ientical to gavity mass an the aial acceleation is shown opposite to the attaction of gavity We must continue to be skeptical of these assumptions, incluing istance to the secon powe, until we eive the elliptical path of the planet aoun the sun This equation of the aial acceleation shows that a Raial is popotional to the invese of istance squae By applying some mathematics we will moify the equation to obtain a Raial as a function of an θ This aial acceleation equation is the basic equation that will evolve into the equation showing that the eath obit is an ellipse
9 Analytical Poof of Newton s Foce Laws Page 9 Notice also that Newton's inetial foce law can be consiee simply as the efinition of the unit of foce Once the stanas of kilogam, mete an secon ae agee upon, the unit of inetial foce is establishe We nee a constant, (G), to make the gavitational units of foce have the same imensions an the same magnitue as inetial foce units But we have no eason (as yet), to believe that the inetial foce, base on anom but agee upon stanas, is iectly popotional to the gavitational foce We just assume the equivalence when we cancele "m" in the above eivation If the path of the eath aoun the sun is analytically etemine to be an ellipse, then the assumption is coect This completes step 4 The next pat of this poof to fin expessions fo aial an tansvese planet acceleations in step 5 of the poceue
10 Page 10 Analytical Poof of Newton s Foce Laws 5 Planet Raial an Tansvese Acceleations Figue 4 shows the geometic constuction to etemine the aial an tansvese planet acceleations y a y cosθ a y a x cosθ a θ a R a x sinθ a T P θ a y sinθ θ 0 a x x Figue 4 A vecto epesenting an assume total planet acceleation, a, is awn at some angle to the aius vecto, In Figue 4, it is convenient to aw a upwa an away fom the iection of the aial acceleation, a R, (which is opposite to the line of attaction between eath an sun)
11 Analytical Poof of Newton s Foce Laws Page 11 One component of assume planet acceleation, a, must be in line with (but in the opposite iection to) the foce of gavity between the sun an planet This acceleation component, a R, is the aial acceleation The othe component of assume planet acceleation, a, place pepenicula to the aial acceleation, is the tansvese acceleation, a T (This tansvese acceleation, is not to be confuse with the tangential, ie tangent to the path, acceleation Tangential acceleation, not elevant in this poof, is iscusse in Sections 71 an 7) Of couse, we know that if the planet actually has a tansvese acceleation, a tansvese foce must be applie But if a tansvese foce is applie, the planet will be pushe out of its obit So the tansvese foce must be zeo an the tansvese acceleation must also be zeo This concept of an assume tansvese acceleation, will povie one equation neee fo this poof If the assume acceleation, a, ha been place in line with the aius vecto, it woul have been ientical to a R an no new infomation coul be gaine fom the geomety Place as it is though, acceleation a is compose of two vectos, a R an a T The aial acceleation, a R is awn in line with the aius vecto, The tansvese acceleation, a T, is awn pepenicula to the aial acceleation It is seen in Figue 4 that acceleation a is equal to two iffeent sets of component vectos that povie the infomation neee to continue the poof One set of these components is a x an a y The secon set of a components is a R an a T The geometic constuction in Figue 4 shows how a X, a y an θ ae use to etemine a R an a T
12 Page 1 Analytical Poof of Newton s Foce Laws The constuction shows that; a = a cos θ + a sin θ R x y a = a cos θ a sin θ T y x The algebaic expessions fo a X an a Y wee eive in Section 3 Theefoe, eplace a X in a R an a T with θ θ θ cos θ sin θ + Also, eplace a Y in a R an a T with θ θ sinθ cos + + Make the substitutions an obtain; a R = θ a T = + θ This completes step 5
13 Analytical Poof of Newton s Foce Laws Page 13 θ 6 Replace Time Depenent Tem,, in a R Fom Section 4, G M a Raial = Replacing a Raial with its equivalent fom above; GM = θ A tem that is not a function of time must be evelope to eplace θ θ, because is a facto in a R Recall that we have aleay etemine that a T is zeo Fom above; a T = + θ Let s take the eivative of θ an see what esults θ θ θ = + Now multiply both sies by 1 an obtain; 1 θ θ θ = +,which is a T θ
14 Page 14 Analytical Poof of Newton s Foce Laws But a T is zeo So, 1 Since 1 θ is also zeo cannot be zeo, the eivative of θ must be zeo The ate of change of a constant is zeo Theefoe, must be a constant That constant is esignate K θ Substitute K Let K = θ fo θ K, an = θ in the equation of aial acceleation, G M = θ Then, G M = K 0, G M = K 3 This completes step 6 Now must be change to a tem containing an θ, an be time inepenent
15 Analytical Poof of Newton s Foce Laws Page 15 7 Replace Time Depenent Tem,, in a R The facto to eplace poceue is foun though the following Let = 1 Diffeentiate 1 with espect to time by n n following the calculus ule fo iffeentiating a vaiable to a powe The ule is: Place the exponent in font of the vaiable, subtact one fom the exponent an iffeentiate the vaiable The esult is, 1 1 n = n n Then, = n 1 n Multiply 1 n n by 1 in the fom of θ θ = n 1 n θ θ In Section 6, we foun that θ is equal to K Substitute an obtain; 1 n K n = = K n θ θ We want to eplace the secon eivative of with espect to t, theefoe we must iffeentiate the fist eivative,, once moe to obtain the secon eivative This secon eivative is, = K n θ θ
16 Page 16 Analytical Poof of Newton s Foce Laws We again substitute K fo θ, an obtain; = K n n θ The esult fom Step 6 above is G M = K 3 Substituting fo we obtain; GM = K n n θ K 3 Since n = 1, GM K n K = + θ This is the point in the analytical poof whee istance squae must be in the enominato of the gavitational foce In oe fo the analysis to conclue with a close cuve we must have istance,, the aius vecto, exactly to the fist powe When we multiply though with,, we will have to the fist powe within the equation It will then be possible fo the aius vecto to tace a smooth cuve as we assume in Figues 1 to 4The secon eivative tem will etemine the shape of the cuve 3 n GM = K + θ K This completes Step 7
17 Analytical Poof of Newton s Foce Laws Page 17 8 Obtain as a Function of θ an Confim Keple's Fist Law Simplify the above equation by eplacing 1 with n n GM + n = 0 θ K This is the iffeential equation to be solve in oe to get as a function of θ We know, fom Section Step, that the eivative of a cosine function is a negative sine function an the eivative of the sine function is a cosine function It appeas that the cosine function of θ will fit into the iffeential equation an solve it The poceue to solve the equation is to let, GM n = A cos θ +, whee A is anothe constant K What is the fist eivative of n with espect to θ? n = θ θ θ θ Acos + GM K = A sin What is the secon eivative of n with espect to θ? n = A cos θ θ To test this solution, put n an the secon eivative of n with espect to θ back into the oiginal equation an check the esult n GM + n = 0 θ K GM GM A cos θ + A cos θ + = 0 K K The left sie of the test equation is also zeo The esult shown in Step 7 is: n K GM = K + θ Substitute Acosθ fo the secon eivative of n with espect to θ K GM = K ( A cos θ ) +
18 Page 18 Analytical Poof of Newton s Foce Laws Solve fo an obtain; = K GM K 1 + GM A cos θ The aius vecto etemining the eath s path aoun the sun is a function of the mass of the sun, the cosine of the geneate angle an constants G, K, an A This is Newton's eive equation fo the planet's path aoun the sun an it has the fom of the pola equation of a conic (See Section 3) This analytically eive path tuns out to be an ellipse an agees with Keple's fist law Notice again that the mass of the eath plays no pat in its equation of motion An object of a fa iffeent size an mass woul occupy the same path if it ha the same aial acceleation This same phenomenon occus in Cavenish s hoizontal penulum expeiment to Weigh the eath The mass of the small bob an the lage attacting sphee both appea to be use in fining G But on close inspection we fin that the mass of the bob is use in calculating its moment of inetia an again in the multiplication of the mass of the bob an the lage attacting sphee fo gavity foce So that the mass of the small bob cancels out Howeve, physics texts often show the mass of the small bob to be necessay fo the calculation of G in the Cavinish expeiment The above equation of motion of the eath s aius vecto was eive using only Newton's foce laws an it was an extemely impotant esult It helpe make Newton's laws the basis of mechanical physics This completes Step 8 an the analytical poof of Newton's foce laws The poofs of Keple s secon an thi laws ae in Sections 4 an 5
19 Analytical Poof of Newton s Foce Laws Page 19 3 Pola Equation of Conics Recall that one pola equation of a conic is; = 1 + ε ε cosθ Whee is the aius vecto, an is the istance fom focus to iectix The aius vecto geneates the angle θ an taces out the conic The planet obit stats in Figue 1 an completes the ellipse in Figue 5 Diectix b Focus =a θ Sun Focus aε a(1ε ) a Figue 5 a ε If the eccenticity, ε, is less than one an geate than zeo, the plotte equation is an ellipse with the sun at a focus (When ε equals one the equation is a paabola When ε is geate than one the equation is an hypebola) If the path of the planet is an ellipse, the planet will etun to some stating point once evey obit The eath etuns to a anomly selecte stating point, as o all the planets So, using only his equations, Newton pove that the path of the eath is an ellipse as Keple ha obseve Theefoe Newton's equations ae pove to be coect Since all the planets an thei moons follow elliptical paths, they all emonstate Newton s laws
20 Page 0 Analytical Poof of Newton s Foce Laws To follow the poofs of Keple s laws, we nee moe infomation on the mathematical chaacteistics of an ellipse In Figue 5: = aius vecto θ = angle geneate by aius vecto, 0 o o When θ goes beyon 360 o the cuve epeats just as the planet epeats its obit a = semimajo axis This is also the "mean" planetsun istance b = semimino axis ε = eccenticity ε of eath obit =017 ε of moon obit = 06 Aea of ellipse = πab Pola equations of ellipse; = 1 + a 1 ε ε an = cos θ 1+ cos θ ε ε The secon equation shows that when the eccenticity, ε, is zeo, the conic is a cicle of aius a Since the planet's obseve istance to the sun vaies, the planet path cannot be a cicle an ε cannot be zeo
21 Analytical Poof of Newton s Foce Laws Page 1 4 Poof of Keple's Secon Law Keple's secon law states that the aius vecto, fom planet to sun, sweeps equal aeas in equal times as the planet obits the sun This law can be shown as follows: The fist step is to etemine the aea of a small tiangula segment, A, of the elliptical shape aea shown in Figue 6 0 θ h Figue 6 A 1 h Fo the small angle, θ, the sine of the angle in aians is h equal to the angle; sin θ = = θ Then; A = 1 θ Inicate the eivative with espect to time on both sies of the equation A = 1 θ Recall in Section 6, θ K = Theefoe; A A = K K =, an A = K Then, ( ) times a specifie time peio The equation shows that aea swept is a constant times an elapse time This is Keple's secon law The sweep of aea by the aius vecto in any elapse time peio is inepenent of whee the planet is in its obit This means that the planet's velocity is faste when it is close to the sun
22 Page Analytical Poof of Newton s Foce Laws 5 Poof of Keple's Thi Law Keple's thi law states that, the squae of the planet's time fo one obit, ivie by the cube of the mean istance of planet to sun, is equal to the same constant fo all planets Fom the poof of Keple's secon law (Section 4) we note that; A = K The planet sweeps though the whole aea of the ellipse, π a b, in the time "T" that it takes to obit the sun The total aea swept by vecto in time T is, π ab K ab = T Then, T = π K Keple's law equies T So, squaing each sie, T K a b 4 = Fom Section 3 iscussion of ellipse chaacteistics an Figue 5: π a = b + a ε b = a a = a ε 1 ε Then T = π a K ε
23 Analytical Poof of Newton s Foce Laws Page 3 Next equate the numeatos of the analytic equations of ε the ellipse; 1 ε = a Then, T a π ε = 4 K 3 The equation of an ellipse in pola cooinates is, = ε ε 1+ cos θ An Newton's planetay obit equation is, = K GM 1+ K GM Acos θ So fo planetay obiting motion, K ε = GM Then, π T a = 4 3 GM Theefoe, fo evey planet T is equal to the same 3 a constant (Note that each planet has a iffeent constant in Keple's secon law) Newton equations again pove an astonomically obseve Keple law, gave the mathematical pinciples involve an etemine exactly the value of the constant
24 Page 4 Analytical Poof of Newton s Foce Laws 6 Newton's Analytical Estimate of G The following poceue is one of Newton s estimate of G base on his own foce laws 1 Inetial foce, F I = m a At the Eath s suface; inetial foce F I = mass of any object times the acceleation ue to eath's gavity The acceleation ue to eath's gavity, g, was foun by Galileo to be 98 metes /sec Inetial foce, F I, at eath's suface = mass of any object X g 3 Gavitational foce, F G M Eath x m Object G = Raius Eath 4 The unit of inetial foce is kgmete pe secon To make the unit of gavitational foce consistent, G has the imension of kgmete pe secon times mete ove kg The unit of foce, kgmete pe secon, is now name the newton 5 F I = F G = m Object X 98 m/sec =G M Eath x m Raius Object Eath
25 Analytical Poof of Newton s Foce Laws Page 5 The mass of the object is on both sies of the equal sign an cancels GM Then, 9 8 = Raius Eath Newton estimate that the aveage ensity of the eath was between 5 an 6 times the ensity of wate Using 55 times the ensity of wate, an an estimate aius an volume of eath, the value of G is etemine to be; 7 X m 3 / kg sec The pesent ay value is 667 X m 3 / kg sec This value of G enable astonomes to estimate the mass of many boies in the sola system an coelate the estimates with the measue istances of moons, planets an sun, an velocities of moons an planets This also inicate that G is a univesal constant 7 Two Methos of Calculating Moon Raial Acceleation Thee ae two methos of calculating the aial acceleation of the moon using Newton s laws: 1 The fist metho equies calculating g times the atio of the eath aius to moon's istance When we know the aial acceleation of the moon at its mean istance fom eath we can calculate G M Eath, an moify the estimates of eath mass o G by using Newton's foce laws The secon metho equies that astonomes povie the tangential velocity of the moon at its mean istance fom the eath The tangential velocity squae ivie by the mean istance mooneath also esults in aial acceleation
26 Page 6 Analytical Poof of Newton s Foce Laws 71 Fist Metho of Calculating Moon Raial Acceleation Recall that the weight of an object on the eath suface is the mass of the object times the acceleation ue to eath's gavity, g By using g an Newton s equations we can calculate the aial acceleation of the moon Astonomical ata: Mean istance (semimajo axis of ellipse; moon cente to eath cente) is 384 X 10 8 metes Moon's tangential velocity is 10 metes pe secon at its mean istance fom eath Eath aius is 638X10 6 metes Acceleation ue to eath's gavity, g, is 98 m/sec Fist metho poceue: Gavitational foce = GM Eath Raius x m Object Eath = m xg Object g is the acceleation expeience by the object on the eath's suface pointing to the eath cente GM g = Eath Raius Eath
27 Analytical Poof of Newton s Foce Laws Page 7 The moon is staying in its obit aoun the eath fo the same eason that the eath stays in its obit aoun the sun The aial acceleation of the moon is exactly equal an opposite the acceleation cause by eathmoon gavity attaction on the moon Equate moon inetial foce to gavitationalfoce m a = G m moon M Eath moon Raial moon Cancel the moon DistanceEath moon mass fom both sies of the equation Hee again we cancel out the small (moon) mass just as the eath mass was cancelle in eveloping Newton s equation of eath motionthe cancelling of the small bob mass in the Cavenish gavity expement follows the same easoning a GM Raial moon = Eath Distance Fom above; 9 8 = GMEath Raius Eath Eathmoon Since the atio of these two equations is anothe equality, a Raial moon 9 8 = GM Distance GM Raius Eath Eathmoon Eath Eath, The aial acceleation of the moon can now be etemine a Raial moon Raius Eath = 9 8 = 007 m / sec Distance Eathmoon It is only at this point in space, when the moon is at its mean istance fom eath, that the moon has exactly this aial acceleation When the moon is close to eath in its elliptical obit, the magnitue of the aial acceleation is geate When the moon is futhe fom eath, the magnitue of aial acceleation is less
28 Page 8 Analytical Poof of Newton s Foce Laws 7 Secon Metho of Calculating Moon Raial Acceleation The secon metho use fo calculating the aial acceleation of the moon, equies iviing the squae of the moon tangential velocity, at its mean istance fom eath, by that istance (The semimajo axis of an ellipse is its mean istance, esignate by the lette, a ) As an equation; V Tangential Mean Distance Eathmoon = 007m / sec This esult confime Newton's atio metho of calculating the aial acceleation of the moon How o we pove analytically that, VTangential = araial moon? Mean Distance Eathmoon An appoximation of aial acceleation can be mae by assuming that the obit of the moon aoun the eath is a cicle But Newton has shown that the path is an ellipse The assumption is then wong fo fou easons: 1 The moon's obit is an ellipse The tangential velocity is not constant 3 The aial acceleation is not constant 4 The aial acceleation of the moon is in line with the focus of an ellipse (the cente of the eath), an not at the cente of a cicula path Using moon's velocity squae ivie by an assume aius gives an assume centipetal acceleation Theefoe we must fin the aial acceleation of the moon, as a function of tangential velocity an the ellipse aius vecto, to show that both methos of calculating the moon's aial acceleation ae coect
29 Analytical Poof of Newton s Foce Laws Page 9 71 Consevation of Obital Enegy In oe to obtain an equation of aial acceleation as a function of tangential velocity, we have to consie the consevation of obital, ie mechanical, enegy of the moon as it obits the eath We will combine the obital enegy equation with Newton's planet ellipse equations in oe to obtain an equation fo the moon's tangential velocity Then we can equate this tangential velocity to Newton's equation an obtain the moon's aial acceleation The consevation of enegy concept, as applie to the moon, means that the total mechanical enegy of the moon must be constant fo the moon to maintain its elliptical obit The total obital enegy of the moon is the algebaic sum of its kinetic enegy, K E, an its potential enegy, P E Thee is an exchange of small pecentages of P E an K E as the moon obits the eath
30 Page 30 Analytical Poof of Newton s Foce Laws 71a Development of Fist Obital Enegy Equation The P E of the moon is consiee to be zeo at an infinite istance fom the eath When the infinite istance fom the eath is selecte as the efeence level, the P E of the mass of the moon, m at a istance, fom the eath, M, is: P E = G M m This equation shows that the incease in the moon's P E, when its mass was bought to the obital position, is equal to the negative of the wok one by the eath's gavity fiel The K E of the moon is equal to 1 mv, whee v is the tangential o obital velocity that vaies uing the elliptical obit E (total mechanical o obital enegy) = K E + P E E Obital 1 mv G M m = The equation shows that when the moon appoaches its peigee an gets close to eath, the P E changes to a lage negative value but the K E gows lage as the moon's velocity is inceasing, an the obital enegy emains constant The above obital enegy equation is the fist equation, of the two that ae neee, fo this calculation metho It shows obital enegy as a function of the ellipse aius vecto an tangential velocity G, M, an m ae constant The secon equation will give the obital enegy at a cetain point in the obit, (at peigee) But since obital enegy is constant, the esignate obital enegies can be equate
31 Analytical Poof of Newton s Foce Laws Page 31 71b Development of Secon Obital Enegy Equation We will now obtain the secon equation of E Obital These two equations will enable us to finally show the connection between tangential velocity an aial acceleation As eive below, the iffeential calculus equation fo the vaiable tangential velocity of an object moving some istance along any smooth cuve (such as an ellipse) is: V Tangential = + θ This equation is evelope by consieing a small incease in angle θ an istance, with a small incease in time, inicate in Figue 7 h s 0 θ θ s istance Figue 7 Fo small angles, whee sin θ equals θ, h = θ ( ) = ( ) + ( ) s θ Designate a change of, s, an θ ue to a incease in time: s t = + θ t t
32 Page 3 Analytical Poof of Newton s Foce Laws As the change in time is mae smalle an appoaches zeo, we obtain the calculus expession fo the tangential velocity along the cuve V Tangential = + θ The change in swept angle with espect to time, is the same function of foun in section 6 ; θ = k This equation applies to the planetmoon systems (with the iffeent constant because the elliptical path is iffeent) as well as to the sunplanet system Substituting; V Tangential k = + As seen in Figue 6, thee ae seveal things to note concening the moon s elliptical path at peigee Tangential velocity, V a Moon =a θ Focus Eath cente aε min peigee aaε Figue 6
33 Analytical Poof of Newton s Foce Laws Page 33 The moon's aius vecto goes though a minimum length at peigee The aius vecto length at peigee is, a(1ε) At the peigee the ate of change of the aius vecto length is zeo Fo the ellipse, with θ being geneate by the aius vecto stating at the peigee, the eivative of with espect to time is 0, when θ =0 0 Theefoe, is zeo an can be emove fom the stana equation fo the object tangential velocity on an ellipse at peigee The tangential velocity equation euces to; V Tangential k = Peigee The pola equation fo the moon's elliptical obit is the same fom as the one that Newton etemine fo eath = k GMEath k 1 + Bcosθ GM Eath Fom Section 3, a geneal equation fo an ellipse is: a 1 ε = "a" is the semimajo axis of the ellipse an 1+ ε cosθ is calle the mean istance of an ellipse
34 Page 34 Analytical Poof of Newton s Foce Laws Equating the numeatos of the equivalent ellipse equations: k GM Eath = a 1 ε Fom Figue 6, at peigee = a(1ε) The next step is to calculate the moon's enegy at peigee Enegy at peigee = K E  P E Tangential Since V = k Peigee, Fom above; 1 E m k GMm Peigee = Peigee k = GM a 1 ε Peigee Substituting fo k in the enegy at peigee equation an simplifying; E Peigee GMm = = EObital a The obital enegy of the moon at peigee is the same as the obital enegy at any othe place in its obit This is the secon equation of E Obital that is neee to show the elationship between V Tangential an aial acceleation
35 Analytical Poof of Newton s Foce Laws Page Relation of Moon Tangential Velocity an Raial Acceleation We will fist obtain the moon's tangential velocity as a function of its aius vecto by equating the two enegy equations Fist equation; E Obital 1 = mvtangential GMm V Tangential GMm Secon equation; EObital = a Equate the obital enegy equations an solve fo V GM 1 Tangential = a We will now fin anothe expession fo aial acceleation In Section 4, we foun that the geneal expession fo the aial acceleation of eath obiting the sun (o moon obiting the eath) is; a Raial = GM Wheein, M is the mass of the sun In the eathmoon system, M is the mass of the eath a Raial = GM Eath When = mean istance, a, a Raial GMEath = a
36 Page 36 Analytical Poof of Newton s Foce Laws We just foun that, V GM = a Eath Tangential Diviing both sies by mean istance a esults in; V Tangential a GMeath = a This is the aial acceleation of the moon at the mean istance, a, fom the eath as poven in Section 71 We have just shown why V a can be use fo the aial acceleation of the moon an the equation looks like the equation fo the centipetal acceleation of an object evolving in a cicle But we must always use V Tangential at the mean (semimajo axis) istance of moon to eath (o eath to sun) an, of couse, use the semimajo axis fo the istance An this aial acceleation is tue at only two places (o times) on each obit As shown above, we have evolve two moe equations of planetay (o moon ) motion: V Tangential a One: The aial acceleation of a planet (o moon) is at the mean istance, a, fom the sun (o planet) Two: V GM 1 Tangential = is applicable to obsevations a of all the planets (o moons) an enables accuate cosschecking of obseve istances an tangential velocities
37 Analytical Poof of Newton s Foce Laws Page 37 8 Conclusions Shown by this Analysis of Newton s Laws The conclusion to be awn hee is that each one of Newton's laws is poven by his analysis of planetay motion He confime exactly the empiical ata of Keple, an in aition, he has shown mathematically why the ata is tue If any pat of Newton's wok was incoect, he coul not have aive at the equation of planet elliptical motion about the sun When equating foce of gavity an inetial foce, we foun that the mass of the eath cancele out This means that inetial mass an gavity mass ae exactly the same Also, any size object in the eath's obit will obit the sun exactly as eath if it has the same aial acceleation as eath Newton pove conclusively that the obital path of a planet (o moon) is an ellipse with the sun (o planet) at a focus This is a emakable fact in itself But moe emakable is that Newton pove his own laws at the same time, an intouce calculus into his wok He pove that all the planets, an the moons o satellites obiting the planets (an comets) follow his laws Newton pove F = ma, m m the foce of gavity = G 1, an action ( Distance m to m ) 1 equals eaction, in that the planet attacts the sun with the same foce that the sun attacts the planet Newton pove that the foce of gavity eceases exactly as 1/ Newton estimate G by using his own laws, an showe that G is a univesal constant Newton's laws ae always in use in poblem solving They fom the basis of mechanical physics an engineeing, momentum, wok, satellite positioning, moon lanings an calculations involving binay stas
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