THE GRAVITATIONAL FORCE INTRO

Transcription

1 SECTION 4: THE RAVITATIONAL FORCE A INTRO In the Newtonian paadigm, the undestanding of fundamental pocesses becomes equivalent to a desciption of the foces at play in the Univese. It tuns out that ou cuent desciption of the Univese is based on fou fundamental foces. Futhemoe, ou undestanding has these foces unified in the past: In the fist s of the existence of the Univese afte the Big Bang, thee is a unique foce descibed by the Theoy of Eveything (TOE) that we don t conclusively undestand yet. At s (when the tempeatue of the Univese was aound 10 3 K) that single unified foce beaks down into foces: gavity and anothe foce descibed by a theoy called the and Unified Theoy (actually thee ae a few candidate theoies fo UT). At s (when the tempeatue of the Univese was aound 10 7 K) the UT foce in tun, becomes two distinct foces: The Stong Nuclea foce and the so-called Electoweak foce (descibed by a theoy called Electoweak theoy) At s (when the tempeatue of the Univese was aound K) the last tansition happens: the Electoweak foce beaks down into the Weak Nuclea foce and the Electomagnetic foce leaving us with the fou foces we see today: gavity, stong and weak nuclea and electomagnetic foce. avity is a vey weak foce with an infinite ange: It takes all the Eath to geneate you weight whee as a small piece of plywood which is held togethe by electomagnetic foces is enough to withstand that weight!!! The electomagnetic foce is much stonge than gavity and has infinite ange as well. The stong nuclea foce has extemely shot ange (about m!) but is so poweful that it is capable to easily ovecome electical epulsion of the potons in the nuclei of atoms and hold the nuclei togethe. This is the foce at play in nukes, the Sun coe and othe dastic phenomena with atomic nuclei. The weak nuclea foce also has extemely shot ange (about m!) and is esponsible fo elementay paticles in the nucleus of adioactive atoms actually tuning into diffeent paticles (like a neuton into a poton in β decay)! Most of ou daily lives is un by gavity and the electomagnetic foce since Nuclea foces can only aely geneate phenomena at ou scale since thei ange is so small. At the Eath s suface the only object big enough and close enough to us to make its gavity obvious is the Eath itself. You ae of couse attacted gavitationally to eveything in the Univese including, say the wall you e looking at, but all these attactive foces ae totally negligible. All othe significant inteactions on Eath ae essentially the esult of electomagnetic inteactions: indeed, since the electomagnetic foce is esponsible fo holding togethe molecules it gives ise to all contact foces we expeience. Since ou couse is a mechanics couse, we ae inteested in the motion of objects acted on by foces. So we e going to study the most common foces on Eath and thei influence on the motion of objects. We will occasionally study the influence of gavity on the motion of celestial objects as well. Let s conside gavity and contact foces in tun.

2 B EXPRESSION FOR F The gavitational foce is defined by its action on a quality of matte called gavitational mass: m Fo bodies with gavitational masses m 1 and m, sepaated by a distance, expeiments suggest that the expession fo the gavitational inteaction between two objects takes the following fom: Magnitude of the gavitational foce: (4.1) m m F = 1 Whee is the gavitational foce constant, which is a chaacteistic of gavity but not of the masses involved. We'll study and get its value in the next sub-section, all we need to know now is that it is constant. The diection of the foce is attactive and acts at the cente of the bodies (we ll see why this is in a late Chapte intoducing the Cente of Mass). The diection is in line with the centes: m 1 m Actually we can combine infomation about magnitude and diection of the gavitational foce by using a vecto notation. To wite the foce on m geneated by m 1 we can use the vecto 1 m 1, m 1 1 F 1 m, m S4/e1 What we get fo the gavitational foce exeted by m 1 on m is (be caeful to ecove the 1/ magnitude): m1m m1m A. F1 = B. F 1 = ( 1 ) m m 1 C. F1 = m m 1 D. F1 = 1 1

3 In ageement with Newton s 3 d law the gavitational foce on each is equal in magnitude and opposite in diection as can be seen by looking at the expession you just obtained. This is stated as : m1m F = F = 1 F1 since = 1 1 m1m F1 = So in summay we can wite the gavitational foce on m due to a gavitational mass M mm (4.) F = 3 m F Now of couse if we define a unit vecto (magnitude 1) in the diection of and call it, we can ewite (4.) as M mm (4.3) F = Which is moe elegant because the dependence is clealy exhibited. Since it is the fist constant of this kind we encounte let us spend a little time undestanding its oigin and significance. C. THE RAVITATIONAL FORCE CONSTANT All foces must have a dimensionality constant. Let s figue out why fom the gavitational example. Foces act on some quality of matte that is defined in tems of a standad, hee the gavitational mass standad. Fo instance let s take a big chunk of lead and choose it to be the gavitational mass standad. By expeiment, let s say we measue its inetia to be m=000kg. Now let s make a cabon copy (o athe lead ha!ha!) of that new gavitational mass standad and let the two inteact gavitationally, positioning them at 1mete fom each othe. S4/e What is the value of gavitational foce that we measue on each if we define units whee the gavitational mass of the gavitational standad has value 1 (let s call these units leaddies o Lds!) A. B. 1 C. 000 D. (000) E. NPA

4 S4/e3 Now assuming that in that expeiment we also measued thei acceleation towad each othe to be a=10-5 m/s, what is the value of (in units of N m /(Lds )) associated with the leaddies system of units? (Hint: use nd law) A. =1/000 B. =10-5 C. = 10 - D. NPA And thus you see that the value of the gavitational constant depends on ou choice of standad and units. One way to look at it is that its ole is to manufactue Newtons out of the quantities involved in the expession of the foce (hee: leaddies and metes). All fundamental foce constants have the same ole. Now because of an incedible esult we actually ae going to identify gav mass and inetia, although thee is ABSOLUTELY NO REASON fo these two concept to be the same at this point!!!! C PROPERTIES OF F NOTE ON THE RAV FORCE: avitational Mass and Inetia ae expeimentally found to be indistinguishable and thus just called mass. Let s see how this is done: m, m M, M S4/e4 Conside an object of inetia m and gav mass m being attacted gavitationally by M, M. Using the nd law whee F is the gav foce on m, you find the acceleation of m to be: A. 9.8m/s mm mm M B. C. D. m M S4/e5 Now eplace m by a second object m. Its acceleation fom the nd law is A. 9.8m/s m' M B. m ' m' C. M M M D. In the special case whee M is the Eath and the objects ae located close to the Eath s suface, then we know, because we have measued it often, that the two acceleations we just computed in e4 and e5 ae the same: both acceleations, as well as the acceleation of anything else we dop at the Eath s suface is 9.81m/s. Doing some othe expeiments elsewhee in the Univese and on many diffeent objects we find that in all cases objects dopped in a given gavitational field fall with the same acceleation. We thus genealize these expeimental esults into a law:

5 All objects falling unde the influence of gavity only, fall with the same acceleation. (Although it is clea fom (4.1) that the specific value of the acceleation depends on thei location and the value of M. ) At the suface of the Eath that value is 9.8m/s. S4/e6 Since the acceleations ae the same, we can equate the answes to execises e5 and e4. Afte doing that, we obtain immediately that: A. m / m=1 B. m / m =1 C. m' m ' m = D. NPA m Now this has a vey pofound consequence since, although we just consideed only diffeent falling objects, the atio is the same fo ALL bodies at any distance fom any M. Thus that atio of gav mass to inetia is the same fo all objects in the univese. This is called the pinciple of equivalence of inetia and gavitational mass. It is a vey pofound statement on which is built the incedibly successful theoy of gavity called eneal Relativity which has been able to pedict most of the gavitational aspects of the evolution of the univese, whee Newton s theoy had failed. Howeve, at ou level, Newton s gavitational model is pefectly fine so we ll use it. It only begins to fail when things go too fast o ae too heavy. Now since the atio of inetia to gav mass is the same fo all objects, we can do the following: Since the gav mass standad can be chosen abitaily, we choose it to be the same as the standad of inetia. And let s define the unit gavitational kilogam such that the standad has a value of 1 gav kg. That put the atio inetia/gav mass at 1 since we have (1kg)/(1gav kg) Since the atio is the same fo any piece of matte we conside, the choice above means that the gavitational value and inetial value of any object will be the same. Finally, since the gavitational value and inetial value of any object ae be the same (and we mean thei numbes ae the same), it becomes possible to identify the gav kilogam with the egula kilogam of inetia. We do so!! And thus fom now on we ll neve make the distinction between gavitational mass and inetia: both ae one and the same. And we decide to measue both using the units of inetia: the kilogam. We ae awae, though, that if some expeiment was to invalidate the pinciple of equivalence, then this identification would become impossible and we would need two sepaate standads and systems of units. As a consequence of this choice, the value of is measued to be: = 6.67x10 Nm / kg 11 And the expession fo the gavitational foce becomes: mm (4.4) F = Whee is the vecto stetching fom m to M and is a unit vecto in that diection.

6 S4/e7 Let s spend a few minutes looking at fee fall on Eath fo a mass m. iven the adius of the Eath R=6378km and its mass M=5.97*10 4 kg compute the magnitude of the gavitational foce on a mass m=10kg located 00km above the suface of the Eath. In Newtons, the value is about: A B C. 9 D. NPA S4/e8 Using the nd law you easily find the acceleation of this fee-falling mass to be in m/s A B. 9. C. 9.8*M/ m D. NPA At the suface of the Eath, is just the adius of the Eath R and thus we can wite the gavitational foce o weight of a mass m: mm (4.5) w = R whee: M g = = 9.81 m/ s R and thus we can wite the weight as: (4.6) w= mg And of couse the weight s diection is downwad towad the cente of the Eath. SECTION 4 SUMMARY: We found that the pinciple of equivalence of inetia and gavitational mass implies that we don't need to define an independent gavitational mass standad and instead just use the kilogam standad of inetia. In a system of units whee this gavitational mass standad has also the value 1kg the gavitational foce constant takes the value: = 6.67x10 Nm / kg 11 And the gavitational foce of a mass m acting on a mass M is expessed as:

7 mm F = Whee is the distance between the centes of the bodies. mm Fo objects at the suface of the Eath: g 9.81 m/ s of mass m at the suface of the Eath can be witten as: = so that the gavitational foce on an object F = mg. It is known as the weight of the object. SECTION 4 PROBLEMS: 1. The Eath has a mass of 5x10 4 kg The moon has a mass of 7.35x10 kg. The distance Eath- Moon 3.84x10 8 m. Is thee a point whee an object would have its gavitational attaction to the Eath exactly cancel its gavitational attaction fom the Moon - If so whee?. Conside a fictitious system of gavitational units called gavi-simple fo which the value of is 1. Assuming that the standad kilogam still seves as the gavitational standad, find the gavitational mass of the standad in gavi-simples. 3. Conside the Eath-Moon system again: a. Find the acceleation of the moon aound the Eath. b. Find the time fo a complete obit of the moon aound the Eath. c. Does it match what you know? 4. Deive the mass of the Sun fom the knowledge of the time it takes the Eath to obit the Sun 5. Explain why you don t feel anything when you jump out of a window..until you hit the gound that is!