Angular momentum. Instructor: Dr. Hoi Lam TAM ( 譚海嵐 )


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1 Angular momentum Instructor: Dr. Ho Lam TAM ( 譚海嵐 ) Physcs Enhancement Programme or Gted Students The Hong Kong Academy or Gted Educaton and Department o Physcs, HKBU Department o Physcs Hong Kong Baptst Unversty 1
2 Contents Operaton o vectors Angular momentum Angular momentum o rgd body Conservaton o angular momentum Examples Department o Physcs Hong Kong Baptst Unversty
3 Department o Physcs Hong Kong Baptst Unversty 3
4 Department o Physcs Hong Kong Baptst Unversty 4
5 Angular Momentum l r p m( r v) l mrv sn. Alternatvely, l rp rmv or l r p r mv Newton s Second Law dl dt. The vector sum o all the torques actng on a partcle s equal to the tme rate o change o the angular momentum o that partcle. Department o Physcs Hong Kong Baptst Unversty 5
6 Proo l m( r v) Derentatng wth respect to tme, dl dv dr m r v dt dt dt dl dt m r a vv v v 0 because the angle between v and v s zero. dl mr a r ma dt Usng Newton s law, F ma. Hence dl r F r F. dt Snce dl r F, we arrve at. dt Department o Physcs Hong Kong Baptst Unversty 6
7 The Angular Momentum o a System o Partcles Total angular momentum or n partcles: L l l 1 l Newtons law or angular moton: dl d l dl. dt dt dt ncludes torques actng on all the n partcles. Both nternal torques and external torques are consdered. Usng Newton s law o acton and reacton, the nternal orces cancel n pars. Hence dl ext. dt Department o Physcs Hong Kong Baptst Unversty 7
8 Department o Physcs Hong Kong Baptst Unversty 8
9 The Angular Momentum o a Rgd Body For the th partcle, angular momentum: l l r p sn90 o r m v. z The component o angular momentum parallel to the rotaton axs (the z component): l sn r m v ( r. sn )( m v The total angular momentum or the rotatng body ) L z l z m ( r m v r ) r m r. Ths reduces to L I. Department o Physcs Hong Kong Baptst Unversty 9
10 Conservaton o Angular Momentum ext dl dt dl dt I no external torque acts on the system, 0 L constant. L L. Law o conservaton o angular momentum. Department o Physcs Hong Kong Baptst Unversty 10
11 Department o Physcs Hong Kong Baptst Unversty 11
12 Department o Physcs Hong Kong Baptst Unversty 1
13 Examples A student sts on a stool that can rotate reely about a vertcal axs. The student, ntally at rest, s holdng a bcycle wheel whose rm s loaded wth lead and whose rotatonal nerta I about ts central axs s 1. kgm. The wheel s rotatng at an angular speed wh o 3.9 rev/s; as seen rom overhead, the rotaton s counterclockwse. The axs o the wheel s vertcal, and the angular momentum L wh o the wheel ponts vertcally upward. The student now nverts the wheel; as a result, the student and stool rotate about the stool axs. The rotatonal nerta I b o the student + stool + wheel system about the stool axs s 6.8 kgm. Wth what angular speed b and n what drecton does the composte body rotate ater the nverson o the wheel? Department o Physcs Hong Kong Baptst Unversty 13
14 Usng the conservaton o angular momentum, L wh L ( Lwh ) I L L wh b I wh b b I I b wh 1.38 rev s 1 ()(1.)(3.9) 6.8 Department o Physcs Hong Kong Baptst Unversty 14
15 Examples A cockroach wth mass m rdes on a dsk o mass 6m and radus R. The dsk rotates lke a merrygoround around ts central axs at angular speed I = 1.5 rad s 1. The cockroach s ntally at radus r = 0.8R, but then t crawls out to the rm o the dsk. Treat the cockroach as a partcle. What then s the angular speed? Department o Physcs Hong Kong Baptst Unversty 15
16 Usng the conservaton o angular momentum, we have I I Rotatonal nerta: The dsk: I d I c The cockroach: and 1 m MR I c mr 3mR ( 0.8R) 0.64mR I I I I d d Thereore, I I c c 3.64mR 4mR I I ( 3.64mR )(1.5) rad s 4mR Department o Physcs Hong Kong Baptst Unversty 16
17 Precesson o a Gyroscope Torque due to the gravtatonal orce o Mgrsn 90 Mgr Angular momentum L I For a rapdly spnnng gyroscope, the magntude o L s not aected by the precesson, dl Ld dl dt d Mgr L L dt dl dt d L dt Usng Newton s second law or rotaton, s the precesson rate where d dt Mgr L Mgr I Department o Physcs Hong Kong Baptst Unversty 17
18 Department o Physcs Hong Kong Baptst Unversty 18
19 Department o Physcs Hong Kong Baptst Unversty 19
20 Example Rollng o a Hexagonal Prsm (1998 IPhO) Consder a long, sold, rgd, regular hexagonal prsm lke a common type o pencl. The mass o the prsm s M and t s unormly dstrbuted. The length o each sde o the crosssectonal hexagon s a. The moment o nerta I o the hexagonal prsm about ts central axs s I = 5Ma /1. a) The prsm s ntally at rest wth ts axs horzontal on an nclned plane whch makes a small angle wth the horzontal. Assume that the suraces o the prsm are slghtly concave so that the prsm only touches the plane at ts edges. The eect o ths concavty on the moment o nerta can be gnored. The prsm s now dsplaced rom rest and starts an uneven rollng down the plane. Assume that rcton prevents any sldng and that the prsm does not lose contact wth the plane. The angular velocty just beore a gven edge hts the plane s whle s the angular velocty mmedately ater the mpact. Show that = s and nd the value o s. b) The knetc energy o the prsm just beore and ater mpact s K and K. Show that K = rk and nd r. c) For the next mpact to occur, K must exceed a mnmum value K,mn whch may be wrtten n the orm K,mn = Mga. Fnd n terms o and r. d) I the condton o part (c) s satsed, the knetc energy K wll approach a xed value K,0 as the prsm rolls down the nclne. Show that K,0 can be wrtten as K,0 = Mga and nd. e) Calculate the mnmum slope angle 0 or whch the uneven rollng, once started, wll contnue ndentely. P P E 30 o Department o Physcs Hong Kong Baptst Unversty 0
21 a) Angular momentum about edge E beore the mpact L I CM o M ( a )( asn 30 ) 11 1 Ma Angular momentum about edge E ater the mpact L I E 5 1 Ma Ma 17 1 Ma Usng the conservaton o angular momentum, L = L 11 1 Ma 17 1 Ma Thus, s P P 30 o E Department o Physcs Hong Kong Baptst Unversty 1
22 b) The knetc energy o the prsm just beore and ater mpact s K and K. Show that K = rk and nd r. b) K Ma Ma 17 4 Ma K Ma Ma 17 4 Ma K K 11 K K P r P 30 o E Department o Physcs Hong Kong Baptst Unversty
23 c) For the next mpact to occur, K must exceed a mnmum value K,mn whch may be wrtten n the orm K,mn = Mga. Fnd n terms o and r. c) Ater the mpact, the center o mass o the prsm rases to ts hghest poston by turnng through an angle 90 o ( + 60 o ) = 30 o. Hence o rk, mn Mga Mgacos(30 ) 1 o 1 cos(30 ) r Department o Physcs Hong Kong Baptst Unversty 3
24 d) I the condton o part (c) s satsed, the knetc energy K wll approach a xed value K,0 as the prsm rolls down the nclne. Show that K,0 can be wrtten as K,0 = Mga and nd. d) At the next mpact, the center o mass lowers by a heght o asn. Change n the knetc energy K Mgasn Knetc energy mmedately beore the next mpact ( K ) K K,0 Mgasn rk When the knetc energy approaches K,0, Thus Mga rk, 0 sn,0 Mgasn 1 r sn 1 r Department o Physcs Hong Kong Baptst Unversty 4
25 e) For the rollng to contnue ndentely, K K, 0,mn sn 1 r 11 cos(30 r o ) 1 r r sn 1 3 cos 1 sn r 1 r 1 sn 3 cos 1 A 1/ 3/ 4cos usn sn u cos 1 where A = r/(1 r) = 11/168 and 3 / 3 sn u A 1/ 3/ 4 (05/84) u o sn( u) A 1/ 3/ 4 (05/84) u o o. Department o Physcs Hong Kong Baptst Unversty 5
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