Page 1 of 6 Conservaton of Angular Momentum = "Spn" We can assgn a drecton to the angular velocty: drecton of = drecton of axs + rght hand rule (wth rght hand, curl fngers n drecton of rotaton, thumb ponts n drecton of.) The drecton of s abstract. There s nothng actually movng n the drecton of the vector. We want to choose the drecton of so that ths gves us nformaton about how the object s spnnng. When a wheel s spnnng on an axs, there s no one drecton of moton (dfferent parts of the wheel are movng n dfferent drectons). But there s always only one drecton of the axs of rotaton. So by gvng both the drecton of the axs (drecton of and how fast the object s spnnng around that axs (magntude of d ), we are descrbng ts spnnng moton exactly. Smlarly, we can assgn a drecton to angular acceleraton : d t t t 2 1 2 1 Drecton of s drecton of, not the drecton of (lke drecton of a s drecton of v, not the drecton of v) We can also assgn a drecton to torque wth the cross product. spnnng, and slowng The cross-product of two vectors A and B s a thrd vector AB C defned lke ths: The magntude s A B A Bsn. The drecton of C AB s the drecton perpendcular to the plane defned by the vectors A and B, plus rght-hand-rule. (Curl fngers from frst vector A to second vector B, thumb ponts n drecton of A B AB A B ^ z ^ k ^ j y ˆ ˆj kˆ ˆ ˆ 0 ˆj kˆ ˆ ˆj ˆj 0 etc. x 3/25/2009 M.Dubson, Unversty of Colorado at Boulder
Page 2 of 6 Vector torque defned as ˆ r F (out of page) orgn r F = vector out of page = vector nto page pont of applcaton of force The vector torque s defned wth respect to an orgn (whch s usually, but not always, the axs of rotaton). So, f you change the orgn, you change the torque (snce changng the orgn changes the poston vector r). Wth these defntons of vector angular acceleraton and vector torque, the fxed-axs equaton I becomes d I I ( lke dv F ma m ) Now, a new concept: angular momentum = "spn". Angular momentum, a vector, s the rotatonal analogue of lnear momentum. So, based on our analogy between translaton and rotaton, we expect I ( lke p mv ). Note that ths equaton mples that the drecton of s the drecton of. Defnton of angular momentum of a partcle wth momentum p = mv at poston r relatve to an orgn s r p ke torque, the angular momentum s defned w.r.t. an orgn, often the axs of rotaton. We now show that the total angular momentum of a object spnnng about a fxed axs s pontng along the +z drecton. We place the orgn at the axs. (out of page) orgn I. Consder a object spnnng about an axs r p partcle 3/25/2009 M.Dubson, Unversty of Colorado at Boulder
Page 3 of 6 y z x v = r v r p r (m v ) tot tot zˆ r m v zˆ m r zˆ I I 2 axs r m If somethng has a bg moment of nerta I and s spnnng fast (bg ), then t has a bg "spn", bg angular momentum. Angular momentum s a very useful concept, because angular momentum s conserved. Important fact: the angular momentum of a object spnnng about a symmetry axs that passes through the center of mass s gven by = I cm ndependent of the locaton of the orgn; that s, even f the orgn s chosen to be outsde the spnnng object, the angular momentum has the same value as f the orgn was chosen to be at the axs. (Proof gven n appendx). Conservaton of Angular Momentum: If a system s solated from external torques, then ts total angular momentum s constant. ext = 0 tot = I = constant ( lke F ext = 0 p tot = constant ) Here s a proof of conservaton of angular momentum: d Frst, we argue that net ( ths s lke Fnet dp ) : d d d r dp d t d t d t d t r p p r. Now, the frst term n the last expresson s zero: dr p v mv m v v 0, snce any vector crossed nto tself s d dp zero. So, we have r r F (snce F net d t d r F net, so we have net d t. dp ). Fnally, 3/25/2009 M.Dubson, Unversty of Colorado at Boulder
Page 4 of 6 So now we have, net f net = 0, then t It turns out that only 4 thngs are conserved: Energy near momentum p Angular momentum Charge q 0 constant. Done. t Conservaton of Angular Momentum s very useful for analyzng the moton of spnnng objects solated from external torques lke a skater or a spnnng star. If ext = 0, then = I = constant. If I decreases, must ncrease, to keep = constant. Example: spnnng skater. I = I f f ( I bg, small ) ( I small, bg ) Example: rotaton of collapsng star. A star shnes by convertng hydrogen (H) nto helum (He) n a nuclear reacton. When the H s used up, the nuclear fre stops, and gravty causes the star to collapse nward. nuclear gravty slow fast!! As the star collapses (pulls ts arms n), the star rotates faster and faster. Star's radus can get much smaller: R 1 mllon mles R f 30 mles 3/25/2009 M.Dubson, Unversty of Colorado at Boulder
Page 5 of 6 2 2 I I f f (Sphere I = 5 M R ) 2 2 2 2 5M R 5M R R f f R 2 2 f f R T 2 ( usng = 2f ) R T T 2 f 2 f f If R >> R f, then T >>>T f. The sun rotates once every 27 days. "Neutron stars" wth dameter of about 30 mles typcally rotate 100 tme per second. et's revew the correspondence between translatonal and rotatonal moton Translaton Rotaton x x v = t v a = t t t F = r F M I = m r 2 F net = M a net = I KE trans = (1/2)M v 2 KE rot = (1/2 ) I 2 p = m v = I F net = p / t net = / t If F ext = 0, p tot = constant If ext = 0, tot = constant 3/25/2009 M.Dubson, Unversty of Colorado at Boulder
Page 6 of 6 Appendx. The angular momentum = I of an object spnnng about a symmetry axs s ndependent of the locaton of the orgn. You can place the orgn anywhere and you wll get the same that you get when you place the orgn at the center-of-mass of the spnnng object. Proof: Consder an object spnnng about a statonary symmetry axs: r v r R r ' y R m poston of CM poston relatve to CM r orgn x r p (R r ' ) p R p r ' p 0 I P 0 tot relatve to orgn at CM More generally, f the center-of-mass R s movng wth velocty V CM 0, then R M V r ' m v' tot cm moton of CM orbtal spn moton relatve to CM 3/25/2009 M.Dubson, Unversty of Colorado at Boulder