Kepler's Three LAWS. Universal Gravitation Chapter 12. Heliocentric Model. Geocentric Model. Other Models. Johannes Kepler

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1 Universl Grvittion Chpter 1 Johnnes Kepler Johnnes Kepler ws Germn mthemticin, stronomer nd strologer, nd key figure in the 17th century Scientific revolution. He is best known for his lws of plnetry motion, During his creer, Kepler ws mthemtics techer t seminry school in Grz, Austri, n ssistnt to stronomer Tycho Brhe, He lso did fundmentl work in the field of optics, invented n improved version of the refrcting telescope (the Keplerin Telescope), nd helped to legitimize the telescopic discoveries of his contemporry Glileo Glilei. Geocentric Model Heliocentric Model A model of the solr system which holds tht the erth is t the centre of the universe nd ll other bodies re in orbit round it. Theory of the universe tht sttes the sun is the centre, nd tht the erth revolves round it. There where lso wide vriety of other different models tht tried to explin the motion of the plnets. Mny of these where very complicted nd hrd to understnd. Other Models Kepler's Three LAWS Kepler's lws of plnetry motion re three mthemticl lws tht describe the motion of plnets in the Solr System 1

2 The pth of the plnets bout the sun re ellipticl in shpe, with the centre of the sun being locted t one focus. Kepler s First Lw Kepler s Second Lw An Imginry line drwn from the centre of the sun to the centre of the plnet will sweep out equl res in equl intervls of time. Kepler s Second Lw Kepler s Third Lw The rtio of the squres of the periods of ny two plnts revolving bout the sun is equl to the rtio of the cubes of their verge distnces from the sun. Thus, if T nd T b re their periods nd r nd r b re their verge distnces from the sun, then we get following eqution. 3 T r = T r b b Astronomicl Units (AU) An AU is unit of distnce tht is defined s the verge distnce between the Sun nd Erth. : (Using Keplers third lw to find n orbitl period) Glileo discovered four moons of Jupiter. Io, which he mesured to be 4. units from the center of Jupiter, hs period of 1.8 dys. He mesured the rdius of Gnymede s orbit to be 10.7 units. Use Kepler s third lw to find the period of Gnymede. 3 T =? T r 3 = 10.7units T Tb r = ( 1.8dys) b 4.units T = 1.8dys b r = 10.7units r = 4.units b r T = Tb rb 3 T = 7.3dys

3 (Using Keplers third lw to find n orbitl rdius) The fourth moon of Jupiter, Cllisto, hs period of 16.7 dys. Find its distnce from Jupiter using the sme units s Glileo used. Copernicus found the period of Sturn to be 9.5 erth yers nd it s orbitl rdius to be 9. AU. Use these mesurements nd units to predict the orbitl rdius of Mrs, whose period is 687 dys. R =18.5 units r b =1.47 AU ewton's Lw of Universl Grvittion Prctice Problems (1-4) Reviewing Concepts (1 & 3) Applying Concepts (1) Problems (1-5) The force of grvity is proportionl to the product of the two msses tht re intercting nd inversely proportionl to the squre of the distnce between their centres 1 Where: F is the Grvittionl Force G is the Grvittionl Constnt ( N m /kg ) m 1 is the mss of first object m is the mss of second object r is the distnce between the objects Determine the force of grvittionl ttrction between the erth (m = 5.98 x 10 4 kg) nd 70-kg physics student if the student is stnding t se level, distnce of 6.37 x 10 6 m from erth's centre. A 65.0 kg stronut is wlking on the surfce of the moon, which hs men rdius of 1.74x10 3 km nd mss of 7.35x10 kg. Wht is the weight of the stronut? F= 688 N 105 N 3

4 Now let s use Newton s lw of universl grvittion to clculte the force of grvity here on Erth. 1 F = g = 9.8m m 6 ( ) As you cn see Newton s lw of universl grvittion is relly nother version of his second lw of motion F=m Pg 580 # s 1-8 Grvittionl Fields So fr we hve studied grvittionl interction in two relted mnners. First, we studied it in terms of energy AKA. grvittionl potentil Energy Then in terms of force. AKA Weight Yet there is nother wy to look t grvittionl interctions. We cn study it in terms of wht is clled grvittionl field. In the simplest form, we define grvittionl field s region in which grvittionl force cn be experienced. For exmple here on erth t se level we cn experience the force of grvity. More specificlly we re sid to be within grvittionl field with field intensity of 9.8 m/s Wht we hve trditionlly referred to s, the vlue of g (g = 9.8 m/s ), is specific cse exmple of the strength of the grvittionl field intensity here on erth t se level. Grvittionl field intensity will chnge in strength s the seprtion between the two mss chnges The following is digrm of the grvittionl field intensity of both the erth nd moon system. We hve lredy seen this in the cse where the vlue of g is lrger t the bottom of trench, nd smller on top of mountin Cn be seen tht both the mgnitude nd direction of the vlue g chnges with loction. 4

5 We cn lso see grvittionl field intensity by looking t Newton s lw of universl grvittion 1 We cn now see tht the grvittionl field intensity (g) cn be found by the mnner 1 If we now substitute in the vlues for Erth t se level we get F = g m 6 ( ) F = g m 6 ( ) Now simplified to get = 9.8m = 9.8m From this it cn be seen tht the universl formul for grdtionl field intensity is g = G m r 1 Or equivlently, if the grvittionl force (weight) is known nd rdius is not. g = F m g A mss of 4.60 kg is plced 6.37x10 6 m from the center of plnet nd experiences grvittionl force of ttrction of 45.1 N. Clculte the grvittionl field intensity t this loction. 9.8 m/s An stronut is sitting on the set of kg lunr rover, on the surfce of the Moon. The set is 50.0 cm bove the centre of mss of the rover. Wht grvittionl field intensity does the rover exert on the stronut? Do # s pg N/kg [down] 5

6 Tying universl grvittion to circulr motion Since the plnets re not flying off into spce (ie in stright line) there must be force cusing them to sty in orbit, which would hve to be some sort of centripetl force. Here the grvittionl force of the sun cn be thought of s tht centripetl force which is cusing the circulr motion. s p F c = mv p r So if the grvittionl force is the centripetl force, we cn equte them to get s p mpv G = r r G m s v r = which gives us formul for clculting orbitl velocity m v= G s r We lso know tht for circulr motion Therefor by substituting this in for the velocity we get Then rerrnge to get ms πr G = r T 3 r Gm = s T 4π v r = π T Where m s is the mss of the plnet or str which the object is orbiting round How fst is the moon moving s it orbits Erth t distnce of 3.84 x 10 5 km from erth s centre? A stellite in low Erth orbit is 5 km bove the surfce. Wht is it s orbitl velocity? m v= G s r 7.78 km/s 1.0x10 3 m/s 6

7 Prctice Problems (5-8) Reviewing Concepts (4-9) Applying Concepts (-11) Problems (6-19) Weightlessness To help nswer this lets exmine the fllowing scenrio. If spce sttion hs n orbit of 6 km, nd n stronut hs mss of 65 kg use Newton's lw of grvittion to find their weight s p Fct or Myth? In ctul fct there is no such thing s weightlessness, NASA coined the phse micro grvity to describe the condition of pprent weightlessness. This is the feeling n object would experience during free fll nd is cused by simply not hving norml force to counterct the force of grvity. Simply put person cn be weightless right here on Erth simply by removing their norml force. AKA. If they re in free fll. 7

8 Question? Wht keeps stellite up in orbit? Wht prevents it from flling out of the sky? Answer: Nothing! It is flling! It just keeps missing the erth. Wht is stellite? An object tht revolves round plnet in circulr or ellipticl pth is termed stellite. The moon is Erth's originl stellite but there re mny mnmde stellites, mostly closer to Erth. The pth tht stellite follows is termed n orbit. An object, such s jvelin, tht is projected horizontlly will fll to erth describing prbolic rc. A bullet fired by rifle is projected t higher velocity thn the jvelin so will trvel further but must still fll to erth describing prbolic rc. The prt tht is different here is the fct tht the Erth is in fct round. For this reson the curvture of the Erth itself becomes significnt, nd llows the bullet to gin extr rnge before lnding. If we could, however, fire rocket with lrge enough velocity, the rocket would cover enough distnce in short mount of time, so tht the curvture of the Erths would fll out from under the rocket. And the rocket would continully miss the surfce of the erth s it flls. The Erth would still cuse grvittionl pull which would hve the effect of continuously chnging the rockets direction. If the direction is continuously chnging while the speed remins constnt then we hve circulr motion where the centripetl force is cused be the grvittionl force. This is wht we refer to s n object in orbit. 8

9 In short, the rocket is lwys flling to the Erth, but it keeps missing ewton s Cnnon Fire pig out of cnnon from the top of high mountin. The pig flls towrds the erth. If too low of initil speed, the pig noseplnts into the erth. However, there is certin speed t which the pig flls towrd the erth t the sme rte s the erth's surfce curves wy. The pig then "misses" the erth nd keeps "flling round it", (i.e. pigs in spce) Geosttionry Orbit A geosttionry orbit is one in which stellite orbits the erth t exctly the sme speed s the erth turns nd t the sme ltitude, specificlly zero, the ltitude of the equtor. A stellite orbiting in geosttionry orbit ppers to be hovering in the sme spot in the sky, nd is directly over the sme ptch of ground t ll times. THE E D Reviewing Concepts (10-13) Problems (0-8) 9