The Vector Product Week 11: Chapter 11 Angular Momentum There are nstances where the product of two vectors s another vector Earler we saw where the product of two vectors was a scalar Ths was called the dot product The vector product of two vectors s also called the cross product The Vector Product and Torque The torque vector les n a drecton perpendcular to the plane formed by the poston vector and the force vector Fr The torque s the vector (or cross) product of the poston vector and the force vector The Vector Product Defned Gven two vectors, A and B The vector (cross) product of A and B s defned as a thrd vector, C AB C s read as A cross B The magntude of vector C s AB sn s the angle between A and B More About the Vector Product The quantty AB sn s equal to the area of the parallelogram formed by A and B The drecton of C s perpendcular to the plane formed by A and B The best way to determne ths drecton s to use the rght-hand rule Propertes of the Vector Product The vector product s not commutatve. The order n whch the vectors are multpled s mportant To account for order, remember A B BA If A s parallel to B ( = 0 o or 180 o ), then A B 0 Therefore A A 0 1
More Propertes of the Vector Product If A s perpendcular to B, then AB AB The vector product obeys the dstrbutve law A ( B + C) = A B + A C Fnal Propertes of the Vector Product The dervatve of the cross product wth respect to some varable such as t s d d d A B AB BA where t s mportant to preserve the multplcatve order of A and B Vector Products of Unt Vectors jjk k 0 j j k jk kj k k j Vector Products of Unt Vectors, cont Sgns are nterchangeable n cross products A -B AB and j j Usng Determnants The cross product can be epressed as j k Ay Az A A A z Ay AB A Ay Az j k By Bz B B Bz By B B B y z Epandng the determnants gves AB AB AB AB AB j AB AB k y z z y z z y y Vector Product Eample Gven A 23 j; B 2j Fnd A B Result AB (2 3 j) ( 2 j) 2 ( ) 22j 3 j( ) 3j2j 04k 3k 0 7k 2
Torque Vector Eample Gven the force and locaton F (2.00 3.00 j) N r (4.00 5.00 j) m Fnd the torque produced r F[(4.005.00 j)n] [(2.003.00 j)m] [(4.00)(2.00) (4.00)(3.00) j (5.00)(2.00) j(5.00)(3.00) j 2.0k Nm Angular Momentum Consder a partcle of mass m located at the vector poston and movng wth lnear momentum p Fnd the net torque r dp rf r dr Add the term psncet 0 d( rp) Angular Momentum, cont The nstantaneous angular momentum L of a partcle relatve to the orgn O s defned as the cross product of the partcle s nstantaneous poston vector r and ts nstantaneous lnear momentum p L rp Torque and Angular Momentum The torque s related to the angular momentum Smlar to the way force s related to lnear momentum dl The torque actng on a partcle s equal to the tme rate of change of the partcle s angular momentum Ths s the rotatonal analog of Newton s Second Law and L must be measured about the same orgn Ths s vald for any orgn fed n an nertal frame More About Angular Momentum The SI unts of angular momentum are (kg. m 2 )/ s Both the magntude and drecton of the angular momentum depend on the choce of orgn The magntude s L = mvr sn s the angle between p and r The drecton of L s perpendcular to the plane formed by r and p Partcle, Eample The vector L= rp s ponted out of the dagram The magntude s L = mvr sn 90 o = mvr sn 90 o s used snce v s perpendcular to r A partcle n unform crcular moton has a constant angular momentum about an as through the center of ts path 3
System of Partcles The total angular momentum of a system of partcles s defned as the vector sum of the angular momenta of the ndvdual partcles Ltot L1L2 Ln L Dfferentatng wth respect to tme dltot dl System of Partcles, cont Any torques assocated wth the nternal forces actng n a system of partcles are zero dl tot Therefore, et The net eternal torque actng on a system about some as passng through an orgn n an nertal frame equals the tme rate of change of the total angular momentum of the system about that orgn Ths s the mathematcal representaton of the angular momentum verson of the nonsolated system model. System of Partcles, fnal The resultant torque actng on a system about an as through the center of mass equals the tme rate of change of angular momentum of the system regardless of the moton of the center of mass Ths apples even f the center of mass s acceleratng, provded and L are evaluated relatve to the center of mass System of Objects, Eample The masses are connected by a lght cord that passes over a pulley; fnd the lnear acceleraton Conceptualze The sphere falls, the pulley rotates and the block sldes Use angular momentum approach Rotatng Rgd Object Each partcle of the object rotates n the y plane about the z as wth an angular speed of The angular momentum of an ndvdual partcle s L = m r 2 L and are drected along the z as Rotatng Rgd Object, cont To fnd the angular momentum of the entre object, add the angular momenta of all the ndvdual partcles L L mr I 2 z Ths also gves the rotatonal form of Newton s Second Law dlz d et I I 4
Rotatng Rgd Object, fnal The rotatonal form of Newton s Second Law s also vald for a rgd object rotatng about a movng as provded the movng as: (1) passes through the center of mass (2) s a symmetry as If a symmetrcal object rotates about a fed as passng through ts center of mass, the vector form holds: L I where L s the total angular momentum measured wth respect to the as of rotaton Bowlng Ball The momentum of nerta of the ball s 2/5MR 2 The angular momentum of the ball s L z = I The drecton of the angular momentum s n the postve z drecton Conservaton of Angular Momentum The total angular momentum of a system s constant n both magntude and drecton f the resultant eternal torque actng on the system s zero Net torque = 0 -> means that the system s solated Ltot = constant or L = L f For a system of partcles, L = L = constant tot n Conservaton of Angular Momentum, cont If the mass of an solated system undergoes redstrbuton, the moment of nerta changes The conservaton of angular momentum requres a compensatng change n the angular velocty I = I f f = constant Ths holds for rotaton about a fed as and for rotaton about an as through the center of mass of a movng system The net torque must be zero n any case Clcker Queston A person sttng on a rotatng char s holdng two heavy dumbbells n hs two hands. Intally, the poston of dumbbells s n front of hs chest. Suddenly, he etends hs arms wth dumbbells n hands. What wll happen to ths rotatng system (char-man-dumbbells)? Frcton force s gnored. A. It wll rotate faster. B. It wll rotate more slowly. C. It wll stop rotatng. D. Rotaton speed wll not change. E. Need more condtons to decde. Conservaton Law Summary For an solated system - (1) Conservaton of Energy: E = E f (2) Conservaton of Lnear Momentum: p pf (3) Conservaton of Angular Momentum: L L f 5
Conservaton of Angular Momentum: The Merry-Go-Round The moment of nerta of the system s the moment of nerta of the platform plus the moment of nerta of the person Assume the person can be treated as a partcle As the person moves toward the center of the rotatng platform, the angular speed wll ncrease To keep the angular momentum constant Moton of a Top The only eternal forces actng on the top are the normal force and the gravtatonal force The drecton of the angular momentum s along the as of symmetry The rght-hand rule ndcates that the torque s n the y plane rf rmg Moton of a Top, cont The net torque and the angular momentum are related: d L A non-zero torque produces a change n the angular momentum The result of the change n angular momentum s a precesson about the z as The drecton of the angular momentum s changng The precessonal moton s the moton of the symmetry as about the vertcal The precesson s usually slow relatve to the spnnng moton of the top Gyroscope A gyroscope can be used to llustrate precessonal moton The gravtatonal force produces a torque about the pvot, and ths torque s perpendcular to the ale The normal force produces no torque Gyroscope, cont The torque results n a change n angular momentum n a drecton perpendcular to the ale. The ale sweeps out an angle d n a tme nterval. The drecton, not the magntude, of the angular momentum s changng The gyroscope eperences precessonal moton Gyroscope, fnal To smplfy, assume the angular momentum due to the moton of the center of mass about the pvot s zero Therefore, the total angular momentum s due to ts spn Ths s a good appromaton when s large 6
Precessonal Frequency Analyzng the prevous vector trangle, the rate at whch the ale rotates about the vertcal as can be found d Mgh p I p s the precessonal frequency Ths s vald only when p << Gyroscope n a Spacecraft The angular momentum of the spacecraft about ts center of mass s zero A gyroscope s set nto rotaton, gvng t a nonzero angular momentum The spacecraft rotates n the drecton opposte to that of the gyroscope So the total momentum of the system remans zero New Analyss Model 1 Nonsolated System (Angular Momentum) If a system nteracts wth ts envronment n the sense that there s an eternal torque on the system, the net eternal torque actng on the system s equal to the tme rate of change of ts angular momentum: dl tot New Analyss Model 2 Isolated System (Angular Momentum) If a system eperences no eternal torque from the envronment, the total angular momentum of the system s conserved: L Lf Applyng ths law of conservaton of angular momentum to a system whose moment of nerta changes gves I = I f f = constant 7