Gravitation on the PC

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Cecilia Dean
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1 Pupose: To undestand obital mechanics and the Law of Univesal Gavitation. Equipment: Inteactive Physics Pencil, Pape, Calculato Theoy: It was Galileo, back in the seventeenth centuy, who finally discedited the 000+ yea- old Aistotelian view of motion. Though incedibly caeful expeimentation, olling polished sphees down polished inclined planes, Galileo showed that all objects, no matte thei size o mass, make thei way towad the eath at the same ate. The objects will not, as had peviously been thought, stive hade to each the Eath in popotion to thei mass, and theefoe each the gound soone. While Galileo made geat contibutions to the study of motion, paticulaly in his definition of inetia, he was not able to connect his findings to the motion of the Eath in its obit aound the sun. It would take one of the geatest scientists of all time to make this connection. Newton s fist law of motion is not much moe than a fomal statement of the popety of inetia. Newton s second law explains why all objects acceleate to the Eath at the same ate, even though some may be moe massive than othes. Newton s Fist Law: The acceleation of an object is diectly popotional to the net foce acting on the object, is in the diection of the net foce, and is invesely popotional to the mass of the object. Newton ealized that in the case of an object in fee fall, the net foce acting on the object is the foce of gavity, exeted by the eath on the object. While the foce inceases in popotion to the object s mass; F = mg, the acceleation of the object is invesely popotional to the object s mass; Newton s Second Law: a = F Eq. 1 m Now, fom the time of Aistotle, and pehaps even befoe, it was geneally believed that the ules govening happenings on Eath wee completely diffeent fom the ules coveing the happenings in the celestial ealm. Newton was the fist peson to conside applying the ules of Eath to the Heavens. Assuming, as Newton did, that the foce that causes the apple to fall to the gound is the same foce that causes the Moon to continue in its obit, it should follow that the 1 of 8

2 moon fall aound the Eath at the same ate as the apple falls to the gound. Newton set out to pove this, painstakingly calculating the movement of the moon, conveting tangential displacement to adial displacement, and finally calculating the velocity at which the moon appoaches the Eath. And he found...nothing. His numbes wee completely wong. Discouaged, he shoved his papes into an easily fogotten dawe and took up a stool at a local taven * T. It was hee, twenty yeas late, that Si Edmund Halley (of comet fame), Robet Hooke (of sping fame) and Si Chistophe Wen (not paticulaly famous...he built St. Paul s Cathedal in London) wee discussing celestial mechanics ove a couple of pints, when Hooke suggested that planetay motion could be explained by assuming that a foce that vaied as the invese squae of the distance between the bodies. Wen challenged Hooke to pove it, and when he couldn t, Halley went to Newton (who, accoding to some accounts, claimed to have aleady solved the poblem, but didn t wite it down ). Amed with this new knowledge, Newton pulled out his ealie wok, was able to pove the invese squae elation of the Gavitational Foce in one of the most fa-eaching genealizations in science: the law of Univesal Gavitation. m1m F gavity Eq. It wasn t until a numbe of yeas late that this popotional elation was solidified into an equality with the discovey by Heny Cavendish of the univesal gavitational constant, a.k.a. big G 11 G = N m / kg which bings the univesal law of gavitation to m1m F gavity = G Eq. 3 Slightly afte Galileo but well befoe Newton came Johannes Keple. Building on the measuements of astonome Tycho Bahe (who had a metal nose, and believed in politeness to the point of death), Keple put foth his thee laws of planetay motion: Keple s Fist Law: Each planet moves in an elliptical obit with the sun at one focus of the ellipse. Keple s Second Law: The line fom the sun to any planet sweeps out equal aeas of space in equal time intevals. * Just kidding. He went on to develop the field of geometic optics. Halley, Hooke and Wen wee eally in the taven, though. of 8

3 Keple s Thid Law: The squaes of the peiods of evolution of the planets ae popotional to the cubes of thei aveage distances fom the sun. 3 T R While Keple s laws wee founded in expeimental data, he had no explanation fo why they wee so. Keple was unable to see the coelation between his second law the fact that planets tavel moe slowly when futhe fom the sun and the simple tossing of a ock into the ai as the ock moves upwad (futhe fom the Eath), it slows down. Inteestingly, Galileo and Keple could have helped each othe out. Keple knew of Galileo s concept of inetia, but neve thought to apply it to his wok with the planets. He neve thought that the foce that keeps the planets in motion could be anywhee but along the line of motion. Galileo leaned of Keple s wok but efused to accept it. He held on to his notion of cicula obits fo the planets. 1 Expeiment: 1. Open the Pogam Inteactive Physics.. Set up you expeiment: Click on a cicle fom the tool box, and click again on the white aea (the stage ) to daw a cicle. (While the size is not impotant fo this simulation, it would be useful to keep it small--i.e. less than the width of you thumb.) Repeat fo a second cicle placed a few cm away fom the fist one. These will be the two planets used in this simulation. Double click on the fist cicle. A dialog box will appea, entitled body 1, cicle. In this box you can set physical popeties fo the object, as well as its position and velocity. Set the paticle s location to x = -1.0 m, y = 0, and its mass to 1.0 kg. Its x and y velocity components should be zeo. Hit etun when you ae finished. Double click on the second cicle. Let the mass of Planet # be 10, and its x,y coodinates be 1, 0. Let its velocity components be 0,0. Hit etun when you ae finished. 3. In the menu ba at the top of the sceen you will see the Wold menu. Scoll down to the fist item Gavity. When you elease the mouse, you will see a dialog box. Select Planetay Gavity. Note the value of the Gavitational Constant: N m kg. Now select Run in the uppe left. This uns the simulation, allowing both objects to attact one anothe. 1 The theoy fo this expeiment was witten by Jennife LK Whalen 3 of 8

4 Reset the simulation (in the uppe left) when you ae finished. What is (o isn t) happening and why? (Fo a hint, ead #4.) 4. Unless you ae extaodinaily patient, you may want to incease the foce of gavity by changing the value of the Gavitational Constant. Change its value to 1.0 N-m /kg.. Run the simulation again. 5. In the Menu Ba, select View > System Cente of Mass. An X should appea at the cente of mass. Run the simulation again. Question 1: How does simila motion in the Eath/Moon system contibute to tides on Eath? 6. Now lets get planet one to obit planet two. Reset the simulation and double click on planet one (mass = 1.0 kg). Let its velocity in the y diection be 1.0, and its velocity in the x diection be zeo. Run the simulation again. 7. To keep the planets fom difting fom the field of view, we can view the system fom a fame of efeence moving upwad with the same speed as the cente of mass. Double click on the cente of mass, and select View > new efeence fame. Click OK. An eye will indicate the new efeence fame. Run the simulation again. 8. Anothe way to keep the system fom difting is to give mass two an initial velocity in the opposite diection, so that the velocity of the cente of mass is equal to zeo. Using the elation: v cm = M v M i total i Show that the initial velocity of mass two should be m/s. 9. Reset the simulation, etun to the oiginal (lab) fame of efeence by selecting View>Home. This will etun the system to the old efeence fame. Change the velocity of Planet to v x = 0, v y = -0.1 m/s. Run the simulation again. 4 of 8

5 10. Thee is an even easie way to keep the mass fom difting away into space. Simply dag the ancho fom the toolba on top of planet. This will Ancho it in place. 11. Now vay the initial velocity of planet one by tial, and un the simulation until you get it into as cicula obit as possible about the system cente of mass. Select Wold> Tacing, and then evey fames to see a tack of planet one. Ease Tack will emove the tacks. What speed woked best? 1. Eve since Isaac Newton, howeve, thee s been an analytical way to detemine obits. Show that: the speed of Planet #1 should be v = G masses, and m is the mass of planet ) the numeical value of v should be.36 m 1, (whee 1 is the distance between the Hint: Use the elation: F = ma m1m (whee F = G and 1 v a = ) Double click on Planet #1 and set v y =.36 and v x = 0. Run the simulation. 14. Change the velocity setting on Planet #1 to five othe values of v y (fom 1.0 to 3.0) in intevals of 0.5 m/s. Obseve the new obits. Note how speed and distance ae elated. 15. Now let's put a satellite in obit. Let the mass of Planet # be 500 and let the mass of Planet #1 (you satellite) by equal to 1. The two masses should be 5 metes apat. Ancho Planet # so it won't dift out of the pictue. Duing this section, you may way to double the size of the viewing aea. Use View > View Size and ente 0.0m in the view aea dialog box. 16. Note that if no extenal foces act on the satellite (othe than gavity fom the planet it obits), this total enegy will emain constant. 5 of 8

6 Show that the total enegy can be witten: GMm E = Hint: the expession fo total enegy, E = K + U, whee U Mm mv the elation G = fom Newton s laws. = G Mm, and K = ½ mv 17. Next, give the satellite velocity v y = 10.0 m/s, v x = 0. This will place it in a cicula obit. Click "OK" and veify that it is in a cicula obit. Using the elation descibed ealie, show that the velocity equied fo cicula obit is 10.0 m/s. 18. Now let's fie the eto-ockets and give the satellite some moe kinetic enegy. Let s incease the kinetic enegy of the satellite by 0% (o the velocity 10%). Do so, and un the simulation. Include a sketch of the old and new obits. Using a ule, sceen ule, o gid maks on the sceen, detemine the old and new semi-majo axes; include these measuements in you answe. Since you ae finding a atio, the actual units ae unimpotant. Helpful Hint: You can also hit the Apple + Shift + 3 keys simultaneously to take a snapshot of the sceen like the one below. The pictue will be located on the had dive as Pictue 1 (o, etc.). 6 of 8

7 19. You will now need you semi-majo axis measuements. Show that the enegies of the new and old obit ae elated by: E = E 1 a a 1 Hint: Use the elation deived in Step 16. Now, fom this elation, and you semi-majo axes measuements, show that the % incease in the enegy of the new obit is appoximately 0%. 0. While we ae on the subject of elliptical obits, let s veify Keple s thid law fo a planetay obit: 4π 3 T a GM = Whee M is the much lage mass of two objects. With the velocity of m 1 set at 10 m/s, select Measue > Time and un the simulation fo 5 to 10 obits. If you ae having difficulty veifying this law, ty using a stopwatch to measue time. Divide the total time by the numbe of obits to get T, the obital peiod. Using the value of a you measued ealie, calculate the mass of planet two. Repeat you calculation fo the lage elliptical (v = 11 m/s) obit. Descibe you esults fo both obits. Question : How could this method be used to measue the mass of a planet such as Jupite? Question 3: How must this elation be modified if the satellite is compaable in mass to the planet as in execises 8? Results: Wite at least one paagaph descibing the following: what you expected to lean about the lab (i.e. what was the eason fo conducting the expeiment?) you esults, and what you leaned fom them Think of at least one othe expeiment might you pefom to veify these esults 7 of 8

8 Think of at least one new question o poblem that could be answeed with the physics you have leaned in this laboatoy, o be extapolated fom the ideas in this laboatoy. Clean-Up: Befoe you can leave the classoom, you must clean up you equipment, and have you instucto sign below. How you divide clean-up duties between lab membes is up to you. Clean-up involves: Completely dismantling the expeimental setup Removing tape fom anything you put tape on Dying-off any wet equipment Putting away equipment in pope boxes (if applicable) Retuning equipment to pope cabinets, o to the cat at the font of the oom Thowing away pieces of sting, pape, and othe detitus (i.e. you wate bottles) Shutting down the compute Anything else that needs to be done to etun the oom to its pistine, pe lab fom. I cetify that the equipment used by has been cleaned up. (student s name),. (instucto s name) (date) 8 of 8

History of Astronomy - Part II. Tycho Brahe - An Observer. Johannes Kepler - A Theorist

History of Astronomy - Part II. Tycho Brahe - An Observer. Johannes Kepler - A Theorist Histoy of Astonomy - Pat II Afte the Copenican Revolution, astonomes stived fo moe obsevations to help bette explain the univese aound them Duing this time (600-750) many majo advances in science and astonomy