Physcs 53 Systems of Partcles Everythng should be as smple as t s, but not smpler. Albert Ensten Overvew An object of ordnary sze whch we call a macroscopc system contans a huge number of atoms or molecules. It s out of the queston to attempt to use the laws we have dscussed for a sngle partcle to descrbe separately each partcle n such a system. There are nevertheless some relatvely smple aspects of the behavor of a macroscopc system that (as we wll show) follow from the basc laws for a partcle. We wll be able to show that n any mult-partcle system each of the mechancal quanttes relevant to the state of the system conssts of two parts: 1. One part n whch the system s treated as though t were a sngle partcle (wth the total mass of the system) located at a specal pont called the center of mass. Ths part s often called the CM moton. 2. Another part descrbng the nternal moton of the system, as seen by an observer located at (and movng wth) the center of mass. In ths secton we wll gve a general analyss of the varables to be used n descrbng the moton of any system of partcles, large or small. In later sectons we wll apply ths analyss to the case of sold objects, n the approxmaton that all the partcles n the object have a fxed spatal relaton to each other. In ths approxmaton, the object s called a rgd body. Stll later we wll dscuss fluds, ncludng gases, where thngs are more complcated because the nternal moton s domnant, and where we must take a statstcal approach n the analyss. Center of mass We consder a system composed of N pont partcles, each labeled by a value of the ndex whch runs from 1 to N. Each partcle has ts own mass m and (at a partcular tme) s located at ts partcular place r. The center of mass (CM) of the system s defned by the followng poston vector: PHY 53 1 Systems
Center of mass r CM = 1 M m r where M s the total mass of all the partcles. Ths vector locates a pont n space whch may or may not be the poston of any of the partcles. It s the mass-weghted average poston of the partcles, beng nearer to the more massve partcles. As a smple example, consder a system of only two partcles, of masses m and 2m, separated by a dstance. Choose the CM coordnate system so that the less massve partcle s at the orgn and the other s at x =, as shown n the drawng. Then we have m 1 = m, m 2 = 2m, x 1 = 0, x 2 =. (The y and z coordnates are zero of course.) We fnd from the ( ) = 2 3 defnton x CM = 1 m 0 + 2m 3m. The locaton s ndcated on the drawng. As tme goes on the poston vectors of the partcles r generally change wth tme, so n general the CM moves. The velocty of ts moton s the tme dervatve of ts poston: v CM = 1 M m v. But the sum on the rght sde s just the total (lnear) momentum of all the partcles. Solvng for ths, we fnd an mportant result: N =1 Total momentum of a system p tot = Mv CM Ths s just lke the formula for the momentum of a sngle partcle, so we see that: The total momentum of a system s the same as f all the partcles were located together at the CM and movng wth ts velocty. PHY 53 2 Systems
Total force The total force on the th partcle conssts of the (net) external force on t, plus the net force due to the nteractons wth other partcles n the system, whch we call the nternal forces. We wrte ths out as follows: F = F ext + F /j, j where F / j denotes the force exerted on the th partcle by the jth partcle. To get the total force on the whole system, we smply add up all these forces: ext F tot = F + F /j. In the double sum on the rght the terms cancel n pars by Newton's 3rd law (for example, F 1/2 + F 2/1 = 0). The double sum thus gves zero, so j ext F tot = F. The total force on the system s the sum of only the external forces. Snce there are very many nternal forces, the fact that they gve no net contrbuton to the total force s why t s possble for many applcatons to treat an object of macroscopc sze as a sngle partcle. The nternal forces do play mportant roles n determnng some aspects of the system, such as ts energy. For each partcle ndvdually we have Newton's 2nd law: F = dp dt. Addng these for all the partcles, and usng the above result for p tot, we fnd two forms of the 2 nd law as t apples to systems of partcles: Newton s 2nd law for systems F ext tot = dp tot dt = Ma cm Here a CM = dv CM /dt s the acceleraton of the CM. We see that: The total external force produces an acceleraton of the center of mass, as though all the partcles were located there. PHY 53 3 Systems
Ths s not the only possble effect of the external forces. They can also cause rotatonal moton about the CM, as we wll see later. But the nternal forces do not change the moton of the CM. From the frst formula above we fnd one of the most mportant laws of mechancs: Conservaton of momentum If the total external force on a system s zero, the total momentum of the system s conserved. In physcs, the term s conserved means remans constant n tme. Two mportant ponts about ths conservaton law: Only external forces are at ssue here. There can be (and often are) many nternal forces of great complexty actng on the ndvdual partcles and makng ther ndvdual motons very complcated. But only external forces can change the total momentum of the system. Snce force and momentum are both vectors, ths conservaton law holds separately for each component. Thus f the x component of the total external force s zero, the x component of the total momentum s conserved, regardless of whether the other components are also conserved. CM reference frame We have found that n some mportant respects a system of partcles behaves as though all the partcles were located at the CM. But the partcles are not n fact all at the CM, so some aspects of the moton must be more complcated. To analyze the stuaton further, t s useful to ntroduce a coordnate system wth the CM as ts orgn. Ths defnes the CM reference frame whch may or may not be an nertal frame. We wll call the (nertal) frame of our prevous analyss the lab frame. Shown are the two frames, wth the poston of the th partcle relatve to each frame. The vector trangle shows that the two poston vectors are related by r = r r CM. y r r CM r y CM frame x The veloctes (tme dervatves of the postons) are thus related by v = v v CM. z Lab frame x z PHY 53 4 Systems
From ths we can calculate the total momentum n the CM frame: m v = m v Mv CM. As we saw earler, the two terms on the rght are equal and therefore cancel. So we have The total momentum, as measured n the CM frame, s zero. In some applcatons ths property s taken to be the defnton of the CM frame. Knetc energy of a system In the lab frame, the total knetc energy of all the partcles s 1 2 2 m v = 1 m 2 ( v + v CM ) 2 = 1 2 = 1 2 Mv CM m v 2 2 + v CM + 2 v v CM 2 + 1 2 m v 2 + v CM m v The sum n the very last term s zero, as we have just seen, so we have 1 2 2 m v = 1 2 Mv CM 2 + 1 m 2 v 2. The two terms on the rght have smple nterpretatons: The frst term s what the knetc energy would be f all the partcles really were at the CM and movng wth ts speed. We often call ths the knetc energy of the CM moton. The second term s the total knetc energy as t would be measured by an observer n the CM reference frame. We call ths the knetc energy relatve to the CM, or sometmes the nternal knetc energy. The total knetc energy (n the lab frame) s the sum of these two terms: Knetc energy of a system K = 1 2 Mv CM 2 + K(rel. to CM) Ths breakup of the knetc energy nto that of the CM plus that relatve to the CM s an example of the general property stated on page 1. We wll see later that ths property also holds for angular momentum. It holds for lnear momentum too, but the second part, the total lnear momentum relatve to the CM, s always zero. PHY 53 5 Systems
Impulse and average force The 2nd Law says that the total (external) force s equal to the rate of change of total (lnear) momentum. It follows that dp tot = F tot dt. Integratng both sdes we fnd for the net change n momentum Δp tot = t 1 F tot dt. t 0 Ths ntegral (over tme) of the force s called the mpulse. We have shown a theorem: Impulse-momentum theorem The mpulse of the total force s equal to the change of the total lnear momentum. One use of ths relaton s to defne the average force that acts durng a specfed tme nterval. Let the force act for tme Δt, producng a net change Δp n the total momentum. The average force s gven by Average force F av = Δp tot Δt Ths s useful n cases where the force s an unknown functon of tme and we would lke to descrbe ts average effect over some specfc tme nterval wthout havng to nvestgate the detaled behavor. Two body collsons There are numerous stuatons where two objects nteract through a force that has effect only when the objects are close to each other. Ths s called a short-range force. The normal force between objects n contact s an example. In such cases close or n contact mean that the dstances between atoms on the adjacent surfaces of the two objects are comparable to the dameter of a sngle atom (about 10 10 m). If two objects whch are orgnally well separated move toward each other wthout nteractng untl a short-range force comes nto effect, we have a collson. There are two mportant cases, dfferng by the nature of the short-range forces: PHY 53 6 Systems
The short-range forces between atoms at the surfaces of the objects are strongly repulsve, so the two objects move away from each other after the very short tme durng whch the forces are effectve. Those forces are strongly attractve, causng the objects to stck together. After a very short tme the CM veloctes of the two objects become the same, and they move off together (or come to rest, f the total momentum s zero). In ether case, the collson tme durng whch the states of the partcles of the system change from ntal states to fnal states s very short, typcally mllseconds. Let the two objects taken together consttute the system under consderaton. Then the short-range nteracton between them (the collson force, about whch we are lkely to know very lttle) s an nternal force and therefore does not affect the total momentum. If there are no external forces, then total momentum s conserved. Ths s a great smplfcaton. The collson forces can be qute complcated, and ndeed the nternal forces wthn each body may come nto play n mportant ways f the behavor of each object separately s to be analyzed. But f the objects are treated as parts of a sngle system, then total momentum s conserved f external forces are absent or neglgble. Even f there are external forces (such as gravty), the collson tme may be so short that the momentum change produced by external forces durng that tme can be neglected. We can then reasonably assume that the system's total momentum s approxmately conserved durng the bref tme of the collson although t wll be changed by the external forces over longer perods of tme. Ths stuaton often occurs n practce. In classcal physcs one also assumes that the total mass of the system does not change n the collson, although some mass may be transferred from one object to the other. Remarkably, conservaton of total mass s only a classcal approxmaton, vald when all speeds are small compared to the speed of lght. Specal relatvty shows that mass (or rather rest energy mc 2 ) s not generally conserved, and ndeed can be nterchanged wth other forms of energy. What about conservaton of energy n a collson? We dstngush two cases: Frst, suppose the collson force s conservatve. Durng the bref collson tme the objects have moved only a neglgble dstance, so the potental energy assocated wth the collson force has not changed apprecably, and external forces have done only neglgble work. Thus for conservatve collson forces the total knetc energy s conserved durng the collson. Such a collson s called elastc. If the collson force s not conservatve, then n general knetc energy s not conserved. In that case the collson s called nelastc. Cases n whch the objects stck together after the collson are sometmes called totally nelastc. For macroscopc objects there are always some non-conservatve aspects of the collson, but they may be neglgble n practce. The rules for comparng the stuatons just before and after a collson are these: PHY 53 7 Systems
Two-body collsons Total momentum s conserved. Total mass s conserved. If the collson s elastc, total knetc energy s conserved. PHY 53 8 Systems