2012 Semifinal Exam Part A 11 d.Note that the masses will always move symmetrically about the center of mass.Thus,in order to be at minimum separation,their velocities must be perpendicular to the line joining them(and will be oppositely directed).Let the minimum separation be d,and let the speed of each mass at minimum separation be v. d L=2mv2 =mud The initial angular momentum is likewise mvol,and so by conservation of angular momentum mud mvol v=wd By conservation of energy 2. 2mo2、 Gm2 =2.号m2- 1 Gm2 2 2、 Gm=2 d Combining these, o2、Gm Gm 1 d (-嗣)()°+（份-1=。 任(-))=0 so that d=1 or d= -1 The second root is only sensible if and is only smaller than the first if2. (Note that both of these results make sense in light of the previous ones.Thus the minimum separation is l if≤2aud号-otherwise Copyright C2012 American Association of Physics Teachers2012 Semifinal Exam Part A 11 d. Note that the masses will always move symmetrically about the center of mass. Thus, in order to be at minimum separation, their velocities must be perpendicular to the line joining them (and will be oppositely directed). Let the minimum separation be d, and let the speed of each mass at minimum separation be v. L = 2mv d 2 = mvd The initial angular momentum is likewise mv0l, and so by conservation of angular momentum mvd = mv0l v = v0 l d By conservation of energy 2 · 1 2 mv0 2 − Gm2 l = 2 · 1 2 mv2 − Gm2 d v0 2 − Gm l = v 2 − Gm d Combining these, v0 2 − Gm l = v0 2 l 2 d 2 − Gm d  1 − Gm v0 2l  d l 2 + Gm v0 2l  d l  − 1 = 0  d l − 1  1 − Gm v0 2l  d l + 1 = 0 so that d = l or d = l Gm v0 2l − 1 The second root is only sensible if Gm v0 2l > 1, and is only smaller than the first if Gm v0 2l > 2. (Note that both of these results make sense in light of the previous ones.) Thus the minimum separation is l if Gm v0 2l ≤ 2 and l Gm v0 2l −1 otherwise. Copyright c 2012 American Association of Physics Teachers