Gavity Notes fo PHYS 111-1131. Joe Wolfe, UNSW 1 Gavity: whee does it fit in? Gavity [geneal elativity] Electic foce* gavitons photons Weak nuclea foce intemediate vecto bosons Stong nuclea foce Colou foce pions gluons electo-weak Some ties fo classical gavity Gand Unified Theoies Theoies Of Eveything * Electomagnetism "unified" by Maxwell, and also by Einstein: Magnetism can be consideed as the elativistic coection to electic inteactions which applies when chages move. Only gavity and electic foce have macoscopic ("infinite") ange. m gaviton? = m photon = 0 Gavity weakest, but dominates on lage scales because it is always attactive Geeks to Galileo: i) things fall to the gound ('natual' places) ii) planets etc move (vaiety of easons) but no connection (in fact, natual vs supenatual) Newton's calculation: accel n of moon = m ω m = (3.8 10 m)( 8 π 7.3 4 3600) =.7 10-3 m.s - accel n of "apple" = 9.8 ms - a apple a moon = 3600; m R e = 385000 km 6370km = 60; m Re = 3600 Newton's billiant idea: What if the apple and the moon acceleate accoding to the same law? What if evey body in the univese attacts evey othe, invese squae law?
Newton's law of gavity: F = G m 1m Negative sign means F_ // _ Why is it invese squae? Wait fo Gauss' law in electicity. F_ 1 = F_ 1 Newton 3 Newton aleady knew Keple's empiical law: Fo planets, 3 Τ obit adius and peiod Now if F a c 1 then constant = a c = ω = 3 ω Planet fom sun T ω ω 3 ω million km Ms ad.s -1 ms - Mecuy 58 7.6 8.5 10-7 3.95 10-5 1.31 10 0 m 3 s - Venus 108 19.4 3.3 10-7 1.13 10-5 1.3 10 0 m 3 s - Eath 150 31.6 1.99 10-7 5.94 10-6 1.33 10 0 m 3 s - etc
How big is G? Cavendish's expeiment (1798) 3 F = G m 1m Fom deflection and sping constant, calculate F, know m 1 and m, can calculate G. G = 6.67 10-11 Nm kg - ( o m 3 kg -1 s - Now also weight of m: W = mg G m.m e R e Cavendish fist calculated mass of the eath: M e = g e G = 9.8 m.s- x (6.37 10 6 m) 6.67 10-11 Nm kg - = 6.0 10 4 kg see http://www.physicscental.com/action/action-01-5-pint.html and http://physics.usask.ca/~kolb/p404/cavendish/ Some numbes What is foce between two oil tankes at 100 m? F = G m 1m What happens when moe thee ae 3 bodies? Supeposition pinciple. F_ all objects togethe = Σ F_ individual continuous body o F_ 1 = Σ i F_ 1i foce on m 1 due to masses m i F_ 1 = body d F_
Shell theoem A unifom shell of mass M causes the same gavitational foce on a body outside is as does a point mass M located at the cente of the shell, and zeo foce on a body inside it. 4 M F=0 dθ dm R F g m F g = GMm R Example. If ρ eath wee unifom (it isn't), how long would it take fo a mass to fall though a hole though the eath to the othe side? M = ρ. 4 3 π3 F = G mρ.4 3 π3 F = K whee K = Gmρ. 4 3 π motion is SHM with ω = K m Τ = π ω = π π = Gρ.4 3 π =... = 84 minutes GM/R3 Simple Hamonic Motion: discussed late falls though (one half cycle) in 4 minutes (actually faste fo eal density pofile)
Gavity nea Eath's suface W = F g = G Mm R e W = mg o = G M em g o is accel n in an inetial (non-otating) fame g o = G M e Usually, R e, but M g o = G e (R e + h) = g s R e R e + h = g s 1 + h/r whee g s is e g o at suface Othe complications: i) Eath is not unifom (especially the cust) useful fo pospecting ii) Eath is not spheical iii) Eath otates (see Foucault pendulum) 5 (Weight) = (the foce exeted by scales) At poles, F_ N_ = 0 At latitude θ, F_ N_ = ma_ whee a = ω = (R e cos θ)ω We define g_ = =... = 0.03 ms - at equato = 0 at poles N_ m = F_ ma_ m So g_ is geatest at the poles, least at the equato, and does not (quite) point towads cente. hoizontal _ g_ Eath is flattened at poles
Puzzle: How fa fom the eath is the point at which the gavitational attactions towads the eath and that towads the sun ae equal and opposite? Compae with distance eath-moon (380,000 km) 6 Fe = Fs GMem GMsm d = (e - d) Me(e d) = Msd e ed + d = M s Me d Ms Me 1 d + ed e = 0 d =... =? Gavitational field. A field is atio of foce on a paticle to some popety of the paticle. Fo gavity, (gavitational) mass is the popety: F_ gav m = g _ = g_(_) is a vecto field F_ cf electic field elec q = E_(_) late in syllabus Gavitational potential enegy. Revision: Potential enegy Fo a consevative foce F_ (i.e. one whee wok done against it, W = W( _)) we can define potential enegy U by U = W against. i.e. U = i f F_. d nea Eath's suface, F_ g = mg_ constant f = (-mgk_). (dxi_ + dyj i _ + dzk_) = mg k_. k_ i f dz = mg (z f - z i ) choose efeence at z i = 0, so U = mgz Gavitational potential enegy of m and M. M F g m ds f f U g F_g. ds = Fg d i i = i f G Mm d = GMm[1 f 1 i ] Convention: take i = as efeence: U() = GMm U = wok to move one mass fom to in the field of the othe. Always negative. Usually one mass >> othe, we talk of U of one in the field of the othe, but it is U of both.
Escape "velocity". Escape "velocity" is minimum speed v e equied to escape, i.e. to get to a lage distance ( ). 7 M R v Pojectile in space: no non-consevative foces so consevation of mechancial enegy K i + U i = K f + U f 1 mv e GMm R = 0 + 0 v esc = GM R m Fo Eath: v esc = 6.67 10-11 m 3 kg -1 s - 5.98 10 4 kg 6.37 10 6 m = 11. km.s -1 = 40,000 k.p.h. Put launch sites nea equato: v eq = R e ω e = 0.47 km.s -1 Question In Jules Vene's "Fom the Eath to the Moon", the heos' spaceship is fied fom a cannon*. If the bael wee 100 m long, what would be the aveage acceleation in the bael? v f v i = as a = v e - 0 s = (1.1 104 ms - ) x 100 m = 630,000 ms - = 64,000 g * why? If you bun all the fuel on the gound, you don't have to acceleate and to lift it. Much moe efficient. Planetay motion "The music of the sphees" - Plato Leucippus & Democitus C5 B.C. heliocentic univese Hippachus (C BC) & Ptolemy (C AD) geocentic univese Tycho Bahe (1546-1601) - vey many, vey caeful, naked eye obsevations. Johannes Keple joined him. He fitted the data to these empiical laws: Keple's laws: 1 All planets move in elliptical obits, with the sun at one focus. Except fo Pluto and Oot cloud objects, these ellipses ae cicles. M sun >> m planet, so sun is c.m. A line joining the planet to the sun sweeps out equal aeas in equal time. Slow at apogee (distant), fast at peigee (close) 3 The squae of the peiod the cube of the semi-majo axis Slow fo distant, fast fo close
Newton's explanations: Law of aeas: 8 Aea = 1.δθ i.e. fo same δt, 1 δθ = constant Consevation of angula momentum L_. Sun at c.m. L_ = _ x p_ = _ x mv_ = mv tangential = m.ω = m δθ δt = m δt δθ = constant. momentum = p. see late Consevation of L_ Keple. Law of peiods: (we conside only cicula obits) Keple 3: T 3 Newton F = ma = m ω G Mm = m π T T = 4π GM 3 Keple 3 (woks fo elipses with semi-majo axis a instead of ) Newton & Newton's gavity Keple 3 Newton & Newton's gavity also Keple 1 Example What is the peiod of the smallest eath obit? ( R e ) What is peiod of the moon? ( moon = 3.8 10 8 m) T 1 = = 84 min Keple 3: T 3 T T 1 = 4π GM 3 = 4 π 3/ 1 = 3.8 108 6.37 10 6 3/ = 464 T = 464 T 1 = 7. days 6.67 10-11 5.98 10 4 (6.37 106 ) 3 s Fo othe planets: most have moons, so the mass of the planet can be calculated fom T = 4π GM 3
Obits and enegy No non-consevative foces do wok, so mechanical enegy is constant: E = K + U = 1 mv GMm Let's emove v. Conside cicula obit: v = a c = F m = GMm m 1 mv = 1 GMm E = K + U 1 GMm = = GMm GMm i.e. E = 1 U, o K = 1 U, o K = E. Small U vey negative, K lage. (inne planets fast, oute slow) 9 Example A spacecaft in obit fies ockets while pointing fowad. Is its new obit faste o slowe? F_ // ds Wok done on caft W = F_. ds > 0. E = GMm inceases, i.e. it becomes less negative. (R is lage). K = - E, K smalle, so it tavels moe slowly. Quantitatively: K i = E i K f = E f = (E i + E) K f = K i E 1 mv f = 1 mv f W Looks odd, but need lots of wok to get to a high, slow obit. called "Speeding down"
Manœuving in obit. 10 To catch up, vessel 1 fies engines backwads, and loses enegy. It thus falls to a lowe obit whee it tavels faste, until it catches up. It then fies its engines fowads in ode to slow down (it climbs back to the oiginal, slowe obit). Example: In what obit does a satellite emain above the same point on the equato? Called the Clake Geosynchonous Obit i) Peiod of obit = peiod of eath's otation ii) Must be cicula so that ω constant T = 3.9 hous T = 4π GM = GMT 4π =... = 4,000 km popula obit!