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

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1 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 the peiods and speeds of obiting objects. Descibe the method Cavendish used to measue G and the esults of knowing G. Johannes Keple Johannes Keple ( ) was an assistant to the Danish astonome Tycho Bahe. Keple was convinced that geomety and mathematics could be used to explain the motion of the planets. Unlike Bahe, he used a heliocentic model. Tycho Bahe I spent 26 yeas making sta chats using this quadant and no telescope. I thought the sun went aound the Eath and the planets went aound the Sun. Keple s Laws Johannes Keple discoveed 3 laws fom the motion of the planets as mapped by Tycho Bahe 1. Planetay obits ae elliptical not cicula 2. Obits sweep out equal aeas in equal times 3. T 2 / 3 = k whee k = constant I used Tycho Bahe s data to come up with my laws. It took him 20 yeas to collect it. 1

2 Keple s 1 st Law 1. The obits of the planets ae ellipses, with the sun at one focus. (Law of Ellipses) Obits ae ellipses Keple s 1 st Law The Sun is at one of the foci The close the planets ae to one anothe, the moe cicula the obit. Keple s 2 nd Law 2. An imaginay line dawn fom the cente of the sun to the cente of the planet will sweep out equal aeas in equal intevals of time. (Law of Equal Aeas) Deals with speed faste when close to the sun. Keple s 2 nd Law 2. Obits sweep out equal aeas in equal times The Sun is at one of the foci Keple s 3 d Law 3. The atio of the squaes of the peiods of any two planets is equal to the atio of the cubes of thei aveage distances fom the sun. (Law of Hamonies) Keple s 3 d Law T 2 / 3 = k whee k = constant T = the peiod fo 1 evolution = the aveage adius of the elliptical obit So fo evey obiting body eveywhee, this atio is tue (T A /T B ) 2 = ( A / B ) 3 2

3 Pat 2: Univesal Gavitation Univesal Gavitation Isaac Newton 24ys old Watching an apple fall to the gound made him wonde if gavity extended beyond Eath Developed a theoy of univesal gavitation Attactive foce between two objects The apple was also attacting the Eath Poposed Law of Univesal Gavitation Law of Univesal Gavitation The foce of attaction between any two masses is constant thoughout the univese F mam G d B 2 G is a univesal gavitational constant between two masses 6.67 x N m 2 /kg 2 Sec. 8.2 Using the Law of Univesal Gavitation Objectives Solve poblems involving obital speed and peiod Relate weightlessness to objects in fee fall Distinguish between inetia mass and gavitational mass Contast Newton s and Einstein s views about gavitation Satellite Motion If a pojectile moves fast enough, it falls at the same ate that the Eath cuves How fast ae satellites moving? F = ma o F = mv 2 / (a c = v 2 /) F = G(m A m B /d 2 ) Solve fo velocity? Set them equal to each othe G(m A m B /d 2 ) = mv 2 / which gives you. 3

4 Peiod of a Satellite Cicling Eath T 2 Gm E o if we know the velocity 2 T v 3 Weightlessness What is gavity in oute space? Whee space shuttle obits g = 8.7m/s 2 How come astonauts ae floating then? g = F/m Histoy Outline 1. Keple used Bahe s s data to make Keple s Laws 2. Newton deived the univesal law using Keples Laws 3. Newton poved his law using the apple and the moon 4. Cavendish measues the univesal constant Histoy Outline 1. Keple used Bahe s s data to make Keple s Laws a) Measued the motion of the stas and planets b) Planets move fom yea to yea but stas stay put c) Keple developed laws to explain the motion of the planets Newton deived the univesal law of gavity He knew: a) T 2 / 3 = k b) v = 2 /T2 c) F = mv 2 / Newton deived the univesal law of gavity He knew: a) T 2 / 3 = k b) v = 2 /T c) F = m( v ) 2 4

5 Newton deived the univesal law of gavity He knew: a) T 2 / 3 = k F = m4 2 b) v = T 2 c) F = m ( 2 /T ) 2 F = m4 2 3 k F = m4 2 T 2 F = 4 2 m k Newton deived the univesal law of gavity He knew: F = 4 2 m But once he got to this pat, he k ealized that evey action had an equal and opposite eaction so, he had to add anothe m F = G m m 2 1 F = Gm 1 m 2 F = the foce of gavity between 2 objects m 1 = mass of object #1 m 2 = mass of object #2 = distance between thei centes of mass G = Univesal Gavitational Constant G was not known but the equation was still poven by the compaison of an apple and the Moon. Newton thought that maybe the Moon moved though the heavens fo the same eason apples fell to the gound If F = Gm 1 m 2 Then if F = ma then m 1 a = Gm 1 m 2 So, a = Gm 2 a = Gm e This equation woks fo any mass attacted to the Eath a = Gm e 60 e a = 9.81m/s 2 fo an apple Since the Moon is 60x futhe away moon = 60 e So, a = Gm e amoon = Gm e a moon (60 e ) e e = a apple

6 a moon =.0027m/s 2 accoding to the fomula The eal acceleation of the Moon can be measued: a = v 2 v = 2 /T2 T = 28.5days = 60 e a = v 2 v = 2 /T 2 T = 28.5days = 28.5x24x60x60 = s = 60 e = 60x 6.4x10 6 m = m e = 6.4x10 6 m a = v 2 v = 2 /T2 v = m) / ( s) =980m/s T = 28.5days = 28.5x24x60x60 = s = 60 e = 60x 6.4x10 6 m = m e = 6.4x10 6 m a = v 2 = (980m/s) 2 / m =.0025m/s 2 v = 2 /T2 v = m) / ( s) =980m/s T = 28.5days = 28.5x24x60x60 = s = 60 e = 60x 6.4x10 6 m = m e = 6.4x10 6 m a = v 2 = (980m/s) 2 / m =.0025m/s 2 The eal acceleation of the Moon =.0025m/s 2 The theoetical acceleation =.0027m/s 2 1. Cavendish measues the univesal constant A. G was still unknown fo 100yeas B. Cavendish figued it out using a Tosion Balance This was so close that this became well accepted and Newton went down in histoy as the one who discoveed gavity 6

7 Tosion Balance 1. Imagine twisting the thead aound 100 times Tosion Balance 1. Imagine twisting the thead aound 100 times 2. Then let go 2. Then let go 3. The system would spin in the opposite diection 3. The system would spin in the opposite diection Tosion Balance 1. Now imagine the foce that pulls the ba back Tosion Balance 4. The foce he measued was.0144n F 2. The foce of a twisted wie is called tosion 3. Attach a sping scale to the ba and measue this tosion foce F 5. Now he has to figue out how much tosion a tiny faction of a twist would make. 6. If he twisted the sting 1/60 th of 1 degee, the foce on the scale would be 100times 360 times 60 times smalle. Tosion Balance F 7. The foce of a twist of 1/60 th of a degee =.0144N / (100)(360)(60) = N 8. He used a mio attached to the sting to eflect a beam of light onto a fa away wall. 9. He used this to measue the angle the sting had twisted Each making measued 1/60 th of a degee 7

8 10. He then placed a 1 kg ball at each end of the ba 11. Next, he placed 1kg masses on the table nea the masses on the ba 12. He let go. 13. The foce of gavity twisted the sting 2/60 th s s of a degee. 14. The masses stopped moving. 15. The foce of gavity between the masses = the tosion in the sting 12. He let go. 13. The foce of gavity twisted the sting 2/60 th s s of a degee. 14. The masses stopped moving. 15. The foce of gavity between the masses = the tosion in the sting Top View Tosion Foce F g F g Tosion Foce 16. Thee wee 2 foces fom the two sets of balls 17. Total Tosion Foce = 2F g 18. The total angle it twisted was 2/60 th s s of a degee Top View Tosion Foce F g F g Tosion Foce 16. Tosion Foce = N x 2 = N 20. 2Fg = N 21. F g = N Top View m 1 = 1.0kg =.10m m 2 = 1.0kg F g = 6.67 x 10-9 N 22. m 1 = 1.0kg 23. m 2 = 1.0kg 24. =.10m 25. F g = N 26. F g = Gm 1 m G = F g 2 m 1 m 2 G = 6.67 x Nm 2 kg 2 8

9 F = Gm 1 m 2 F = the foce of gavity between 2 objects m 1 = mass of object #1 m 2 = mass of object #2 = distance between thei centes of mass G = Univesal Gavitational Constant G = 6.67 x Nm 2 kg 2 Hee s a sketch of the Expeiment fom Cavendish s time Sample Poblems 1. No matte how much you say you don t t find someone attactive, the fact is, that all people ae at least gavitationally attactive. If you have a mass of 70kg and the othe peson has a mass of 80kg, what is the foce of gavitational attaction between you both when you ae sitting.50m apat? Sample Poblem No matte how much you say you don t t find someone attactive, the fact is, that all people ae at least gavitationally attactive. If you have a mass m of 70kg and the othe peson has a mass of 80kg, what is the foce of gavitational attaction between you both when you ae sitting.50m apat? F g F g =? m 1 = 70kg m 2 = 80kg =.50m G = 6.67 x Nm 2 kg 2 9

10 Homewok Pg 242 # 1-31 Pg 247 #3a,c, 5 Noth Pole Stas move in a cicle as the Eath otates on its axis evey night. So to explain the motion of the stas, Moon and Sun you need to measue thei positions at a cetain time of day o night Also, the stas move fom season to season though the yea 6am 7am 8am 9am 10am 11am 12am 1pm 2pm 3pm 4pm 5pm Night Day 10

11 Evey yea, at the same time, on the same day, the stas ae in the same position they wee last yea. Night Day 12am 1/22/07 11

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13 Mas Thee ae celestial objects that don t stay in the same place they wande. These ae called wandees, o in Geek, Planets. 12am 1/22/08 Thee ae celestial objects that don t stay in the same place they wande. These ae called wandees, o in Geek, Planets. Thee ae celestial objects that don t stay in the same place they wande. These ae called wandees, o in Geek, Planets. 12am 1/29/08 12am 2/5/08 Thee ae celestial objects that don t stay in the same place they wande. These ae called wandees, o in Geek, Planets. Thee ae celestial objects that don t stay in the same place they wande. These ae called wandees, o in Geek, Planets. 12am 2/12/08 12am 2/19/08 13

14 Thee ae celestial objects that don t stay in the same place they wande. These ae called wandees, o in Geek, Planets. Astonomes tied to explain the moving objects. At fist, they thought these planets wee like the Sun and the Moon. They thought the planets obited aound the Eath in a cicle 12am 2/26/08 12am 3/5/08 14