Section 39 Gravitational Potential Energy & General Relativity

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1 Section 39 Gavitational Potential Enegy & Geneal elativity What is the univese made out of and how do the pats inteact? We ve leaned that objects do what they do because of foces, enegy, linea and angula momentum In the last section, we continued to build an undestanding of the theoy of gavitation by following the development of the theoy though time ecall, Tycho povided good data fo Keple to invent his thee ules which Newton s used as the basis of his Law of Univesal Gavitation This law was missing a numeical value fo the gavitation constant, which the cleve expeiment of Cavendish evealed In this section, we ll complete ou jouney by discussing Einstein s theoy of gavitation, Geneal elativity So fa ou undestanding of gavitational inteactions has not included enegy We ll emedy this by eexamining gavitational potential enegy in light of the Law of Gavitation Expeiment: Tycho Bahe ( ) Spent most of his life compiling accuate ecods of planets and thei obits Theoy: Johannes Keple ( ) A student of Tycho and used his data to poduce thee ules of planetay motion Theoy: Si Isaac Newton ( ) Explained Keple s ules by poposing the Law of Univesal Gavitation Expeiment: Si Heny Cavendish ( ) Measued the stength of the gavitational foce between objects Theoy: Albet Einstein ( ) Explained the foce of gavity in tems of the bending of space by matte by intoducing the Theoy of Geneal elativity Section Outline 1 Einstein s Theoy of Gavity 2 Gavitational Potential Enegy 39-1

2 1 Einstein s Theoy of Gavity When the foce of gavity is the only one acting, the mass of the moving object doesn t seem to matte If a bowling ball and a baseball ae given the same initial velocity, they execute the same motion Einstein found this vey bothesome because, he agued, it means that even massless objects like light should also be bent by gavity He felt that this could only mean that the connection between gavity and mass is, in some sense, atificial vo vo vo bowling ball baseball light We now have expeimental evidence that Einstein s speculation about light being bent by gavity is actually coect Gavitational Lensing is an example of light bent by gavity Conside a distance galaxy that has a lage mass between it and Eath Light leaving the galaxy will be bent by the gavitation of the mass so the light will appea to be coming to Eath fom a diffeent diection as shown at the ight Light doesn t leave the galaxy along just one line It leaves in all diections, so fom Eath we would see the galaxy as a ing, as shown at the ight This is gavitational lensing image image galaxy mass Eath At the ight is a photogaph fom the Hubble Space Telescope Deep Field Camea In it you will see many nomal looking galaxies, but thee ae also many cicula acs of light These ae expeimental evidence of gavitational lensing The easoning behind Einstein s Theoy of Gavity goes something like this: 1 Light is bent by gavity (expeimentally veified) 2 Light always tavels in staight lines because it always coves the shotest distance between two points in the least time (nothing tavels faste than light) 3 The conclusion is that if the light is going staight, then space must be cuved! 4 The amount of cuvatue is popotional to the mass 39-2

3 This is the basis fo the Theoy of Geneal elativity The idea of bent space is potayed in the iconic image at the ight So, the point of the Theoy of Geneal elativity is that gavity isn t eally a foce Instead, it is the esult of the bending of space caused by massive objects The tajectoies of planets, satellites, and such is due to the bending of space and not the esult of a foce Fo ou puposes, we can still teat it like a foce and we ll do so fo the est of the couse This completes ou look at the histoy of the development of the undestanding of gavitation The inteplay between theoy and expeiment known as the Scientific Method ceated ou cuent knowledge of gavitation The biggest steps ae summaized below: 1 Tycho Bahe devoted his life to ceating the most accuate expeimental measuements of the positions of the planets 2 Johannes Keple, his student, used these measuements to design a theoy summaized by Keple s Thee ules of Planetay Motion 3 Isaac Newton built upon Keple s theoy by explaining gavitation in tems of a foce between massive objects, the Law of Univesal Gavitation 4 Heny Cavendish caied out a cleve expeiment to find the missing numeical value of Newton s theoy called the gavitation constant 5 Albet Einstein ealized that the fact that the tajectoy of objects due to gavitation neve depends upon the mass of the object needed to be explained His Theoy of Geneal elativity povides this explanation Thee ae othe featues of gavitation that still need to be explained by new theoies and we ll talk about them in the next section 2 Gavitational Potential Enegy We need to add enegy to ou undestanding of the gavitational diven motion of objects The gavitational potential enegy can be found using the definition of potential enegy, f ΔU W c = F g d s Since the gavitational foce always points adially, d s and be eplaced by d Since we ae moving out, the dot poduct gives a minus sign Using the Law of Univesal Gavitation, the integation can be done, M! F g m! ds! f f f = + F g d = GMm d = GMm 1 2 f U gf U gi = G Mm f + G Mm By setting the initial potential enegy to zeo when the initial position is infinitely fa away, we get the expession fo the potential enegy due to gavity, The Gavitational Potential Enegy U g 39-3

4 Example 391: Find the gavitational potential enegy of a 100kg mass at the suface of Eath Given: M = 597x10 24 kg, m = 100kg, and = 640x10 6 m Find: U g =? Plugging the known values into the equation fo gavitational potential enegy, U g = ( 667x10 11 ) (597x1024 )(100) 640x10 6 U g = 622x10 7 J What does the minus sign mean? The potential enegy gows (gets less negative) as the mass is moved futhe away When the mass is infinitely fa away the potential enegy is zeo The minus sign indicates that the mass, m, is bound to the Eath What happened to ou old fiend U g = mgh? Suppose we lift the mass up a distance, h Let s find the change in gavitational potential enegy The distance between the cente of Eath and the 1kg mass is initially just the adius of Eath, so U gi The final distance is the adius of Eath plus the height h U gf + h The change in potential enegy is, = U gf U gi + h + G Mm = GMm 1 + h h = GMm ( + h) = GMmh ( + h) The h in the denominato can be neglected in compaison to because even a few hunded metes is much smalle than the adius of Eath Now, GMmh = GMm 2 2 h The quantity in paentheses is just the foce on the mass at the suface of Eath We usually wite that as, F g = GMm = mg 2 Finally, the change in the gavitational potential enegy can be witten as, = GMm 2 h = mgh So, fo small distances fom Eath (small compaed to Eath s adius) mgh woks geat This eally amounts to a choice of whee the zeo fo potential enegy is established The expession fo gavitational potential enegy assumes the zeo is at infinity while mgh uses a zeo nea the suface of Eath m M 39-4

5 Example 392: Find the minimum velocity at which a ocket must leave Eath in ode to escape Eath s gavity Given: M = 597x10 24 kg and = 640x10 6 m Find: v o =? K o = 1 2 mv o 2 U o Applying the Law of Consevation of Enegy, ΔK + ΔU = mv mv o Solving fo the initial velocity, ( ) + 0 G Mm K = 1 2 mv 2 U 0 = mv mv o 2 = G Mm v o = v 2 + 2GM The minimum will occu when the final velocity is zeo This initial velocity is called the escape velocity v e = 2GM = 2(667x10 11 )(598x10 24 ) v 637x10 6 e = 112km / s Kinetic enegy must be added to make the total enegy geate than zeo If the total enegy of a system is less than zeo it is a bound system Example 393: Let s look at the Halley s comet example again ecall, it is closest distance to the sun is 0586AU and the fathest distance is 351AU When it is fathest away it is taveling about 0911km/s Find the speed when it is closest to the sun using enegy methods Compae this answe with the answe fom the Law of Consevation of Angula Momentum Given: = 0586AU = m, = 351AU = m, M = kg and v = 911m/s Find: V =? V v At the fathest distance the total enegy is, E a = 1 2 mv2 G mm At the closest distance the enegy is, 39-5

6 E p = 1 2 mv 2 G mm Using the Law of Consevation of Enegy, E a = E p 1 2 mv2 G mm = 1 2 mv 2 G mm Solving fo the speed at closest appoach, Putting in the numbes, V = V = (911) 2 + 2(667x10 11 )(199x10 30 ) 1 In ageement with ou ealie esults v 2 + 2GM ( 1 1 ) ( ) V = 546 km s x x10 12 Section Summay We completed ou histoical tip though the time to see the pogession of ou undestanding of gavity by looking at Einstein s Theoy of Geneal elativity Gavity is not so much a foce as the cause of the bending of space This bending of space is esponsible fo gavitational lensing Summaizing the histoy: 1 Tycho Bahe devoted his life to ceating the most accuate expeimental measuements of the positions of the planets 2 Johannes Keple, his student, used these measuements to design a theoy summaized by Keple s Thee ules of Planetay Motion 3 Isaac Newton built upon Keple s theoy by explaining gavitation in tems of a foce between massive objects, the Law of Univesal Gavitation 4 Heny Cavendish caied out a cleve expeiment to find the missing numeical value of Newton s theoy called the gavitation constant 5 Albet Einstein ealized that the fact that the tajectoy of objects due to gavitation neve depends upon the mass of the object needed to be explained His Theoy of Geneal elativity povides this explanation We then used the Law of Gavitation to find The Gavitational Potential Enegy U g 39-6

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

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