17 The Universal Law of Gravitation

Transcription

1 Chapte 7 The Univesal Law of avitation 7 The Univesal Law of avitation Conside an object eleased fo est an entie oon s diaete above the suface of the oon. Suppose you ae asked to calculate the speed with which the object hits the oon. This poble typifies the kind of poble in which students use the univesal law of gavitation to get the foce exeted on the object by the gavitational field of the oon, and then istakenly use one o oe of the constant acceleation equations to get the final velocity. The poble is: the acceleation is not constant. The close the object gets to the oon, the geate the gavitational foce, and hence, the geate the acceleation of the object. The istake lies not in using Newton s second law to deteine the acceleation of the object at a paticula point in space; the istake lies in using that one value of acceleation, good fo one object-to-oon distance, as if it wee valid on the entie path of the object. The way to go on a poble like this, is to use consevation of enegy. Back in chapte, whee we discussed the nea-suface gavitational field of the eath, we talked about the fact that any object that has ass ceates an invisible foce-pe-ass field in the egion of space aound it. We called it a gavitational field. Hee we talk about it in oe detail. Recall that when we say that an object causes a gavitational field to exist, we ean that it ceates an invisible foce-pe-ass vecto at evey point in the egion of space aound itself. The effect of the gavitational field on any paticle (call it the victi) that finds itself in the egion of space whee the gavitational field exists, is to exet a foce, on the victi, equal to the foce-peass vecto at the victi s location, ties the ass of the victi. Now we povide a quantitative discussion of the gavitational field. (Quantitative eans, involving foulas, calculations, and pehaps nubes. Contast this with qualitative which eans desciptive/conceptual.) We stat with the idealized notion of a point paticle of atte. Being atte, it has ass. Since it has ass it has a gavitational field in the egion aound it. The diection of a paticle s gavitational field at point P, a distance away fo the paticle, is towad the paticle and the agnitude of the gavitational field is given by g = (7-) whee: N is the univesal gavitational constant: =. 7 0 is the ass of the paticle, and is the distance that point P is fo the paticle. In that point P can be any epty (o occupied) point in space whatsoeve, this foula gives the agnitude of the gavitational field of the paticle at all points in space. Equation 7- is the equation fo of Newton s Univesal Law of avitation. Newton s Univesal Law of avitation is an appoxiation to Einstein s fa oe coplicated eneal Theoy of Relativity. The appoxiation is fantastic fo obital echanics fo space vehicles and ost planets, but we need to use the eneal Theoy of Relativity fo a coplete explanation of the obit of Mecuy. 04

2 Chapte 7 The Univesal Law of avitation The Total avitational Field Vecto at an Epty Point in Space Suppose that you have two paticles. Each has its own gavitational field at all points in space. Let s conside a single epty point in space. Each of the two paticles has its own gavitational field vecto at that epty point in space. We can say that each paticle akes its own vecto contibution to the total gavitational field at the epty point in space in question. So how do you deteine the total gavitational field at the epty point in space? You guessed it! Just add the individual contibutions. And because the contibution due to each paticle is a vecto, the two contibutions add like vectos. What the avitational Field does to a Paticle that is in the avitational Field Now suppose that you have the agnitude and diection of the gavitational field vecto g at a paticula point in space. The gavitational field exets a foce on any victi paticle that happens to find itself at that location in space. Suppose, fo instance, that a paticle of ass finds itself at a point in space whee the gavitational field (of soe othe paticle o paticles) is g. Then the paticle of ass is subject to a foce F = g (7-) The avitational Effect of one Paticle on Anothe Let s put the two peceding ideas togethe. Paticle of ass has, aong its infinite set of gavitational field vectos, a gavitational field vecto at a location a distance away fo itself, a point in space that happens to be occupied by anothe paticle, paticle, of ass. The gavitational field of paticle exets a foce on paticle. The question is, how big and which way is the foce? Let s stat by identifying the location of paticle as point P. Point P is a distance away fo paticle. Thus, the agnitude of the gavitational field vecto (of paticle ) at point P is: g = (7-3) The stateent that the vaious contibutions to the gavitational field add like vectos is known as the supeposition pinciple fo the gavitational field. Einstein, in his eneal Theoy of Relativity, showed that the supeposition pinciple fo the gavitational field is actually an appoxiation that is excellent fo the gavitational fields you will deal with in this couse but beaks down in the case of vey stong gavitational fields. 05

3 Chapte 7 The Univesal Law of avitation Now the foce exeted on paticle by the gavitational field of paticle is given by equation 7-, g F =. Using F fo F to ephasize the fact that we ae talking about the foce of on, and witing the equation elating the agnitudes of the vectos we have F = g Replacing the g with the expession we just found fo the agnitude of the gavitational field due to paticle nube we have F = which, with soe ino eodeing can be witten as F = (7-4) This equation gives the foce of the gavitational field of paticle on paticle. Neglecting the iddlean (the gavitational field) we can think of this as the foce of paticle on paticle. You can go though the whole aguent again, with the oles of the paticles evesed, to find that the sae expession applies to the foce of paticle on paticle, o you can siply invoke Newton s 3 d Law to aive at the sae conclusion. Objects, Rathe than Point Paticles The vecto su of all the gavitational field vectos due to a spheically syetic distibution of point paticles (fo instance, a spheically syetic solid object), at a point outside the distibution (e.g. outside the object), is the sae as the gavitational field vecto due to a single paticle, at the cente of the distibution, whose ass is equal to the su of the asses of all the paticles. Also, fo puposes of calculating the foce exeted by the gavitational field of a point object on a spheically syetic victi, one can teat the victi as a point object at the cente of the victi. Finally, egading eithe object in a calculation of the gavitational foce exeted on a igid object by the anothe object: if the sepaation of the objects is vey lage copaed to the diensions of the object, one can teat the object as a point paticle located at the cente of ass of the object and having the sae ass as the object. This goes fo the gavitational potential enegy, discussed below, as well. 0

4 Chapte 7 The Univesal Law of avitation How Does this Fit in with g = 9.80 N/? When we talked about the eath s nea-suface gavitational field befoe, we used a single value fo its agnitude, naely 9.80 N/ and said that it always diected downwad, towad the cente of the eath N/ is actually an aveage sea level value. g vaies within about a half a pecent of that value ove the suface of the eath and the handbook value includes a tiny centifugal pseudo foce-pe-ass field contibution (affecting both agnitude and diection) steing fo the otation of the eath. So how does the value 9.80 N/ fo the agnitude of the gavitational field nea the suface of the eath elate to Newton s Univesal Law of avitation? Cetainly the diection is consistent with ou undestanding of what it should be: The eath is oughly spheically syetic so fo puposes of calculating the gavitational field outside of the eath we can teat the eath as a point paticle located at the cente of the eath. The diection of the gavitational field of a point paticle is towad that point paticle, so, anywhee outside the eath, including at any point just outside the eath (nea the suface of the eath), the gavitational field, accoding to the Univesal Law of avitation, ust be diected towad the cente of the eath, a diection we eathlings call downwad. But how about the agnitude? Shouldn t it vay with elevation accoding to the Univesal Law of avitation? Fist off, how does the agnitude calculated using g = copae with 9.80 N/ at, fo instance, sea level. A point at sea level is a distance = fo the cente of the eath. The ass of the eath is = Substituting these values into ou expession fo g (equation 7-4 which eads g = ) we find: g =. 7 0 N N g = ( which does agee with the value of 9.80 N/ to within about 0.3 pecent. (The diffeence is due in pat to the centifugal pseudo foce-pe-ass field associated with the eath s otation.) We can actually see, just fo the way that the value of the adius of the eath,. 37 0, is witten, that inceasing ou elevation, even by a ile (. k), is not going to change the value of g to thee significant figues. We d be inceasing fo to which, to thee significant figues is still This bings up the question, How high N above the suface do you have to go befoe g = 9.80 is no longe within a cetain pecentage of the value obtained by eans of the Univesal Law of avitation? Let s investigate that question fo the case of a pecent diffeence. In othe wods, how high above sea level would 4 ) 07

5 Chapte 7 The Univesal Law of avitation we have to go to ake g = 99% of g at sea level, that is g = (.99)(9.80 /s ) = 9.70 /s. Letting = E + h whee E is the adius of the eath, and using g = 9.70 N/ so that we can find the h that akes 9.70 N/ we have: Solving this fo h yields E g = ( + h) E h = E g E h =. 7 0 N N h = That is to say that at any altitude above 40 k above the suface of the eath, you should use Newton s Univesal Law of avitation diectly in ode fo you esults to be good to within %. In suay, g = 9.80 N/ fo the nea-eath-suface gavitational field agnitude is an appoxiation to the Univesal Law of avitation good to within about % anywhee within about 40 k of the suface of the eath. In that egion, the value is appoxiately a constant because changes in elevation epesent such a tiny faction of the total eath s-cente-to-suface distance as to be negligible. 08

6 Chapte 7 The Univesal Law of avitation The Univesal avitational Potential Enegy So fa in this couse you have becoe failia with two kinds of potential enegy, the nea-eath s-suface gavitational potential enegy U = gy and the sping potential enegy U S = k x. Hee we intoduce anothe expession fo gavitational potential enegy. This one is petinent to situations fo which the Univesal Law of avitation is appopiate. g U = (7-5) This is the gavitational potential enegy of a pai of paticles, one of ass and the othe of ass, which ae sepaated by a distance of. Note that fo a given pai of paticles, the gavitational potential enegy can take on values fo negative infinity up to zeo. Zeo is the highest possible value and it is the value of the gavitational potential enegy at infinite sepaation. That is to say that U 0. The lowest conceivable value is negative li infinity and it would be the value of the gavitational potential enegy of the pai of paticles if one could put the so close togethe that they wee both at the sae point in space. In atheatical notation: U. li 0 Consevation of Mechanical Enegy Pobles Involving Univesal avitational Potential Enegy fo the Special Case in which the Total Aount of Mechanical Enegy does not Change You solve fixed aount of echanical enegy pobles involving univesal gavitational potential enegy just as you solved fixed aount of echanical enegy pobles involving othe fos of potential enegy back in chaptes and 3. Daw a good befoe and afte pictue, wite an equation stating that the enegy in the befoe pictue is equal to the enegy in the afte pictue, and poceed fo thee. To eview these pocedues, check out the exaple poble on the next page: 09

7 Chapte 7 The Univesal Law of avitation Exaple 7-: How geat would the uzzle velocity of a gun on the suface of the oon have to be in ode to shoot a bullet to an altitude of 0 k? Solution: We ll need the following luna data: Mass of the oon: = ; Radius of the oon: = v =? v = 0 h Moon = =.74 0 Moon = + h = =.84 0 E = E v K + U + = K + U = ( is the ass of the oon and is the ass of the bullet.) v v = = v = v = v =. 7 0 N v = 55 s 0