# Analytical Proof of Newton's Force Laws

Save this PDF as:

Size: px
Start display at page:

Download "Analytical Proof of Newton's Force Laws"

## Transcription

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(1-ε ) 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 = semi-majo axis This is also the "mean" planet-sun istance b = semi-mino 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 kg-mete pe secon To make the unit of gavitational foce consistent, G has the imension of kg-mete pe secon times mete ove kg The unit of foce, kg-mete 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 moon-eath 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 (semi-majo 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 eath-moon 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 Eath-moon Since the atio of these two equations is anothe equality, a Raial moon 9 8 = GM Distance GM Raius Eath Eath-moon Eath Eath, The aial acceleation of the moon can now be etemine a Raial moon Raius Eath = 9 8 = 007 m / sec Distance Eath-moon 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 semi-majo axis of an ellipse is its mean istance, esignate by the lette, a ) As an equation; V Tangential Mean Distance Eath-moon = 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 Eath-moon 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 planet-moon systems (with the iffeent constant because the elliptical path is iffeent) as well as to the sun-planet 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 a-aε 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(1-ε) 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 semi-majo 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(1-ε) 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 eath-moon 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 (semi-majo 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

### Revision Guide for Chapter 11

Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams

### 2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

### Chapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6

Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe

### Ch. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth

Ch. 8 Univesal Gavitation Pat 1: Keple s Laws Objectives: Section 8.1 Motion in the Heavens and on Eath Objectives Relate Keple s laws of planetay motion to Newton s law of univesal gavitation. Calculate

### PHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013

PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0

### Gravitation. AP Physics C

Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What

### Physics 111 Fall 2007 Electrostatic Forces and the Electric Field - Solutions

Physics 111 Fall 007 Electostatic Foces an the Electic Fiel - Solutions 1. Two point chages, 5 µc an -8 µc ae 1. m apat. Whee shoul a thi chage, equal to 5 µc, be place to make the electic fiel at the

### mv2. Equating the two gives 4! 2. The angular velocity is the angle swept per GM (2! )2 4! 2 " 2 = GM . Combining the results we get !

Chapte. he net foce on the satellite is F = G Mm and this plays the ole of the centipetal foce on the satellite i.e. mv mv. Equating the two gives = G Mm i.e. v = G M. Fo cicula motion we have that v =!

### Episode 401: Newton s law of universal gravitation

Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce

### GRAVITATIONAL FIELD: CHAPTER 11. The groundwork for Newton s great contribution to understanding gravity was laid by three majors players:

CHAPT 11 TH GAVITATIONAL FILD (GAVITY) GAVITATIONAL FILD: The goundwok fo Newton s geat contibution to undestanding gavity was laid by thee majos playes: Newton s Law of Gavitation o gavitational and inetial

### Physics 235 Chapter 5. Chapter 5 Gravitation

Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

### FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing

### Chapter 13 Gravitation

Chapte 13 Gavitation Newton, who extended the concept of inetia to all bodies, ealized that the moon is acceleating and is theefoe subject to a centipetal foce. He guessed that the foce that keeps the

### Chapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43

Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.

### The Role of Gravity in Orbital Motion

! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State

### Chapter 13. Vector-Valued Functions and Motion in Space 13.6. Velocity and Acceleration in Polar Coordinates

13.6 Velocity and Acceleation in Pola Coodinates 1 Chapte 13. Vecto-Valued Functions and Motion in Space 13.6. Velocity and Acceleation in Pola Coodinates Definition. When a paticle P(, θ) moves along

### A) 2 B) 2 C) 2 2 D) 4 E) 8

Page 1 of 8 CTGavity-1. m M Two spheical masses m and M ae a distance apat. The distance between thei centes is halved (deceased by a facto of 2). What happens to the magnitude of the foce of gavity between

### 2. Orbital dynamics and tides

2. Obital dynamics and tides 2.1 The two-body poblem This efes to the mutual gavitational inteaction of two bodies. An exact mathematical solution is possible and staightfowad. In the case that one body

### Gravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2

F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,

### Resources. Circular Motion: From Motor Racing to Satellites. Uniform Circular Motion. Sir Isaac Newton 3/24/10. Dr Jeff McCallum School of Physics

3/4/0 Resouces Cicula Motion: Fom Moto Racing to Satellites D Jeff McCallum School of Physics http://www.gap-system.og/~histoy/mathematicians/ Newton.html http://www.fg-a.com http://www.clke.com/clipat

### F G r. Don't confuse G with g: "Big G" and "little g" are totally different things.

G-1 Gavity Newton's Univesal Law of Gavitation (fist stated by Newton): any two masses m 1 and m exet an attactive gavitational foce on each othe accoding to m m G 1 This applies to all masses, not just

### Determining solar characteristics using planetary data

Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation

### So we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1)

Lectue 17 Cicula Motion (Chapte 7) Angula Measue Angula Speed and Velocity Angula Acceleation We ve aleady dealt with cicula motion somewhat. Recall we leaned about centipetal acceleation: when you swing

### Exam 3: Equation Summary

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P

### Voltage ( = Electric Potential )

V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage

### PY1052 Problem Set 8 Autumn 2004 Solutions

PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

### Exam I. Spring 2004 Serway & Jewett, Chapters 1-5. Fill in the bubble for the correct answer on the answer sheet. next to the number.

Agin/Meye PART I: QUALITATIVE Exam I Sping 2004 Seway & Jewett, Chaptes 1-5 Assigned Seat Numbe Fill in the bubble fo the coect answe on the answe sheet. next to the numbe. NO PARTIAL CREDIT: SUBMIT ONE

### Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

### Experiment 6: Centripetal Force

Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee

### 4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

### Continuous Compounding and Annualization

Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

### Deflection of Electrons by Electric and Magnetic Fields

Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An

### 8-1 Newton s Law of Universal Gravitation

8-1 Newton s Law of Univesal Gavitation One of the most famous stoies of all time is the stoy of Isaac Newton sitting unde an apple tee and being hit on the head by a falling apple. It was this event,

### ESCAPE VELOCITY EXAMPLES

ESCAPE VELOCITY EXAMPLES 1. Escape velocity is the speed that an object needs to be taveling to beak fee of planet o moon's gavity and ente obit. Fo example, a spacecaft leaving the suface of Eath needs

### Multiple choice questions [60 points]

1 Multiple choice questions [60 points] Answe all o the ollowing questions. Read each question caeully. Fill the coect bubble on you scanton sheet. Each question has exactly one coect answe. All questions

### (a) The centripetal acceleration of a point on the equator of the Earth is given by v2. The velocity of the earth can be found by taking the ratio of

Homewok VI Ch. 7 - Poblems 15, 19, 22, 25, 35, 43, 51. Poblem 15 (a) The centipetal acceleation of a point on the equato of the Eath is given by v2. The velocity of the eath can be found by taking the

### Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E

CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known

### Mechanics 1: Motion in a Central Force Field

Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.

### CHAPTER 9 THE TWO BODY PROBLEM IN TWO DIMENSIONS

9. Intoduction CHAPTER 9 THE TWO BODY PROBLEM IN TWO DIMENSIONS In this chapte we show how Keple s laws can be deived fom Newton s laws of motion and gavitation, and consevation of angula momentum, and

### Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

### Copyright 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

Chapte 5. Foce and Motion In this chapte we study causes of motion: Why does the windsufe blast acoss the wate in the way he does? The combined foces of the wind, wate, and gavity acceleate him accoding

### 12. Rolling, Torque, and Angular Momentum

12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.

### Chapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom

Chapte 7 The Keple Poblem: Planetay Mechanics and the Boh Atom Keple s Laws: Each planet moves in an ellipse with the sun at one focus. The adius vecto fom the sun to a planet sweeps out equal aeas in

### Orbital Motion & Gravity

Astonomy: Planetay Motion 1 Obital Motion D. Bill Pezzaglia A. Galileo & Fee Fall Obital Motion & Gavity B. Obits C. Newton s Laws Updated: 013Ma05 D. Einstein A. Galileo & Fee Fall 3 1. Pojectile Motion

### Lesson 32: Measuring Circular Motion

Lesson 32: Measuing Cicula Motion Velocity hee should be a way to come up with a basic fomula that elates velocity in icle to some of the basic popeties of icle. Let s ty stating off with a fomula that

### Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

### TRIGONOMETRY REVIEW. The Cosines and Sines of the Standard Angles

TRIGONOMETRY REVIEW The Cosines and Sines of the Standad Angles P θ = ( cos θ, sin θ ) . ANGLES AND THEIR MEASURE In ode to define the tigonometic functions so that they can be used not only fo tiangula

### Gravitational Mechanics of the Mars-Phobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning

Gavitational Mechanics of the Mas-Phobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This

### Solutions to Homework Set #5 Phys2414 Fall 2005

Solution Set #5 1 Solutions to Homewok Set #5 Phys414 Fall 005 Note: The numbes in the boxes coespond to those that ae geneated by WebAssign. The numbes on you individual assignment will vay. Any calculated

### General Physics (PHY 2130)

Geneal Physics (PHY 130) Lectue 11 Rotational kinematics and unifom cicula motion Angula displacement Angula speed and acceleation http://www.physics.wayne.edu/~apetov/phy130/ Lightning Review Last lectue:

### Section 5-3 Angles and Their Measure

5 5 TRIGONOMETRIC FUNCTIONS Section 5- Angles and Thei Measue Angles Degees and Radian Measue Fom Degees to Radians and Vice Vesa In this section, we intoduce the idea of angle and two measues of angles,

### Lab #7: Energy Conservation

Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 1-4 Intoduction: Pehaps one of the most unusual

### PHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013

PHYSICS 111 HOMEWORK SOLUTION #5 Mach 3, 2013 0.1 You 3.80-kg physics book is placed next to you on the hoizontal seat of you ca. The coefficient of static fiction between the book and the seat is 0.650,

### PRICING MODEL FOR COMPETING ONLINE AND RETAIL CHANNEL WITH ONLINE BUYING RISK

PRICING MODEL FOR COMPETING ONLINE AND RETAIL CHANNEL WITH ONLINE BUYING RISK Vaanya Vaanyuwatana Chutikan Anunyavanit Manoat Pinthong Puthapon Jaupash Aussaavut Dumongsii Siinhon Intenational Institute

### 2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

### 14. Gravitation Universal Law of Gravitation (Newton):

14. Gavitation 1 Univesal Law of Gavitation (ewton): The attactive foce between two paticles: F = G m 1m 2 2 whee G = 6.67 10 11 m 2 / kg 2 is the univesal gavitational constant. F m 2 m 1 F Paticle #1

### Skills Needed for Success in Calculus 1

Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

### LINES AND TANGENTS IN POLAR COORDINATES

LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Pola-coodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and

### Poynting Vector and Energy Flow in a Capacitor Challenge Problem Solutions

Poynting Vecto an Enegy Flow in a Capacito Challenge Poblem Solutions Poblem 1: A paallel-plate capacito consists of two cicula plates, each with aius R, sepaate by a istance. A steay cuent I is flowing

### UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

### Lab 5: Circular Motion

Lab 5: Cicula motion Physics 193 Fall 2006 Lab 5: Cicula Motion I. Intoduction The lab today involves the analysis of objects that ae moving in a cicle. Newton s second law as applied to cicula motion

### Forces & Magnetic Dipoles. r r τ = μ B r

Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent

### CLOSE RANGE PHOTOGRAMMETRY WITH CCD CAMERAS AND MATCHING METHODS - APPLIED TO THE FRACTURE SURFACE OF AN IRON BOLT

CLOSE RANGE PHOTOGRAMMETR WITH CCD CAMERAS AND MATCHING METHODS - APPLIED TO THE FRACTURE SURFACE OF AN IRON BOLT Tim Suthau, John Moé, Albet Wieemann an Jens Fanzen Technical Univesit of Belin, Depatment

### 2008 Quarter-Final Exam Solutions

2008 Quate-final Exam - Solutions 1 2008 Quate-Final Exam Solutions 1 A chaged paticle with chage q and mass m stats with an initial kinetic enegy K at the middle of a unifomly chaged spheical egion of

### Gravity and the figure of the Earth

Gavity and the figue of the Eath Eic Calais Pudue Univesity Depatment of Eath and Atmospheic Sciences West Lafayette, IN 47907-1397 ecalais@pudue.edu http://www.eas.pudue.edu/~calais/ Objectives What is

### Hour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and

Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon

### Physics: Electromagnetism Spring PROBLEM SET 6 Solutions

Physics: Electomagnetism Sping 7 Physics: Electomagnetism Sping 7 PROBEM SET 6 Solutions Electostatic Enegy Basics: Wolfson and Pasachoff h 6 Poblem 7 p 679 Thee ae si diffeent pais of equal chages and

### 1.1 KINEMATIC RELATIONSHIPS

1.1 KINEMATIC RELATIONSHIPS Thoughout the Advanced Highe Physics couse calculus techniques will be used. These techniques ae vey poweful and knowledge of integation and diffeentiation will allow a deepe

### 6.2 Orbits and Kepler s Laws

Eath satellite in unstable obit 6. Obits and Keple s Laws satellite in stable obit Figue 1 Compaing stable and unstable obits of an atificial satellite. If a satellite is fa enough fom Eath s suface that

### Hip Hop solutions of the 2N Body problem

Hip Hop solutions of the N Boy poblem Esthe Baabés baabes@ima.ug.es Depatament Infomàtica i Matemàtica Aplicaa, Univesitat e Giona. Josep Maia Cos cos@eupm.upc.es Depatament e Matemàtica Aplicaa III, Univesitat

### The Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = -W/q 0 1V [Volt] =1 Nm/C

Geneal Physics - PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit

### VISCOSITY OF BIO-DIESEL FUELS

VISCOSITY OF BIO-DIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use

### Gravity. A. Law of Gravity. Gravity. Physics: Mechanics. A. The Law of Gravity. Dr. Bill Pezzaglia. B. Gravitational Field. C.

Physics: Mechanics 1 Gavity D. Bill Pezzaglia A. The Law of Gavity Gavity B. Gavitational Field C. Tides Updated: 01Jul09 A. Law of Gavity 3 1a. Invese Squae Law 4 1. Invese Squae Law. Newton s 4 th law

### Chapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.

Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium-39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming

### 2. An asteroid revolves around the Sun with a mean orbital radius twice that of Earth s. Predict the period of the asteroid in Earth years.

CHAPTR 7 Gavitation Pactice Poblems 7.1 Planetay Motion and Gavitation pages 171 178 page 174 1. If Ganymede, one of Jupite s moons, has a peiod of days, how many units ae thee in its obital adius? Use

### On the Relativistic Forms of Newton's Second Law and Gravitation

On the Relativistic Foms of Newton's Second Law and avitation Mohammad Bahami,*, Mehdi Zaeie 3 and Davood Hashemian Depatment of physics, College of Science, Univesity of Tehan,Tehan, Islamic Republic

### Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field

### Model Question Paper Mathematics Class XII

Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat

### Lesson 7 Gauss s Law and Electric Fields

Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual

### Physics HSC Course Stage 6. Space. Part 1: Earth s gravitational field

Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe

### !( r) =!( r)e i(m" + kz)!!!!. (30.1)

3 EXAMPLES OF THE APPLICATION OF THE ENERGY PRINCIPLE TO CYLINDRICAL EQUILIBRIA We now use the Enegy Pinciple to analyze the stability popeties of the cylinical! -pinch, the Z-pinch, an the Geneal Scew

### Voltage ( = Electric Potential )

V-1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is

### 2 - ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 1

- ELECTROSTATIC POTENTIAL AND CAPACITANCE Page. Line Integal of Electic Field If a unit positive chage is displaced by `given by dw E. dl dl in an electic field of intensity E, wok done is Line integation

### How many times have you seen something like this?

VOL. 77, NO. 4, OTOR 2004 251 Whee the amea Was KTHRN McL. YRS JMS M. HNL Smith ollege Nothampton, M 01063 jhenle@math.smith.eu How many times have you seen something like this? Then Now Souces: outesy

### Chapter 3: Vectors and Coordinate Systems

Coodinate Systems Chapte 3: Vectos and Coodinate Systems Used to descibe the position of a point in space Coodinate system consists of a fied efeence point called the oigin specific aes with scales and

### Trigonometric Functions of Any Angle

Tigonomet Module T2 Tigonometic Functions of An Angle Copight This publication The Nothen Albeta Institute of Technolog 2002. All Rights Reseved. LAST REVISED Decembe, 2008 Tigonometic Functions of An

### Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

### Algebra and Trig. I. A point is a location or position that has no size or dimension.

Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite

### Chapter 23: Gauss s Law

Chapte 3: Gauss s Law Homewok: Read Chapte 3 Questions, 5, 1 Poblems 1, 5, 3 Gauss s Law Gauss s Law is the fist of the fou Maxwell Equations which summaize all of electomagnetic theoy. Gauss s Law gives

### The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges

The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee

### Coordinate Systems L. M. Kalnins, March 2009

Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

### Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing

M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow

### 1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2

Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the

### Sequence Impedances of Transmission Lines R. E. Fehr, May 2004

Sequence Impeances o Tansmission Lines R. E. Feh, May 004 In oe to analye unbalance conitions on tansmission lines, we nee to apply the metho o symmetical components, as escibe by Chales Fotescue in his

### Chapter 26 - Electric Field. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University

Chapte 6 lectic Field A PowePoint Pesentation by Paul. Tippens, Pofesso of Physics Southen Polytechnic State Univesity 7 Objectives: Afte finishing this unit you should be able to: Define the electic field

### Gauss Law. Physics 231 Lecture 2-1

Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

### PY1052 Problem Set 3 Autumn 2004 Solutions

PY1052 Poblem Set 3 Autumn 2004 Solutions C F = 8 N F = 25 N 1 2 A A (1) A foce F 1 = 8 N is exeted hoizontally on block A, which has a mass of 4.5 kg. The coefficient of static fiction between A and the

### Ilona V. Tregub, ScD., Professor

Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation