Simple Nature - Light and Matter

interaction with the felt surface is not important) and that theirrotation is unimportant, so that we’ll be able to treat each one as asingle particle. By symmetry, the only place their center of mass canbe is half-way in between, at an x coordinate equal to the averageof the two balls’ positions, x cm = (x 1 + x 2 )/2.Figure j makes it appear that the center of mass, marked withan ×, moves with constant velocity to the right, regardless of thecollision, and we can easily prove this using conservation of momentum:v cm = dx cm / dt= 1 2 (v 1 + v 2 )= 12m (mv 1 + mv 2 )= p totalm totalSince momentum is conserved, the last expression is constant, whichproves that v cm is constant.Rearranging this a little, we have p total = m total v cm . In otherwords, the total momentum of the system is the same as if all itsmass was concentrated at the center of mass point.Sigma notationWhen there is a large, potentially unknown number of particles,we can write sums like the ones occurring above using symbols like“+ . . .,” but that gets awkward. It’s more convenient to use theGreek uppercase sigma, Σ, to indicate addition. For example, thesum 1 2 + 2 2 + 3 2 + 4 2 = 30 could be written asn∑j 2 = 30 ,j=1read “the sum from j = 1 to n of j 2 .” The variable j is a dummyvariable, just like the dx in an integral that tells you you’re integratingwith respect to x, but has no significance outside the integral.The j below the sigma tells you what variable is changing from oneterm of the sum to the next, but j has no significance outside thesum.As an example, let’s generalize the proof of p total = m total v cm tothe case of an arbitrary number n of identical particles moving inone dimension, rather than just two particles. The center of massis atx cm = 1 nn∑x j ,j=1l / No matter what point youhang the pear from, the stringlines up with the pear’s centerof mass. The center of masscan therefore be defined as theintersection of all the lines madeby hanging the pear in this way.Note that the X in the figureshould not be interpreted asimplying that the center of massis on the surface — it is actuallyinside the pear.m / The circus performershang with the ropes passingthrough their centers of mass.Section 3.1 Momentum In One Dimension 141