Physics 101 Lecture 13 Angular Momentum
Physics 101Lecture 13Angular MomentumAsist. Prof. Dr. Ali ÖVGÜNEMU Physics Departmentwww.aovgun.com
Angular MomentumTorque using vectors Angular Momentum February 16, 2020
Torque as a Cross Product r F The torque is the cross product of a forcevector with the position vector to its pointof application rF sin r F r F The torque vector is perpendicular to theplane formed by the position vector andthe force vector (e.g., imagine drawingthem tail-to-tail)Right Hand Rule: curl fingers from r to F,thumb points along torque.Superposition: net i ri Fi (vector sum)all iall iCan have multiple forces applied at multiplepoints. Direction of net is angular acceleration axis February 16, 2020
Net torque example: multiple forces at a single point 3 forces applied at point r : F1r r cos ˆi 0 ˆj r sin kˆF1 2 ˆi; F2 2 kˆ ; F3 2 ˆj; r 3; 30zF2 F3 ro yFind the net torque about the origin:τ net r Fnet r ( F1 F2 F3 ) (r ˆi r kˆ ) (2ˆi 2ˆj 2kˆ )xτ netxsetzrx rsin( ) 3sin(30 o ) 1.5rz rcos( ) 3cos(30 o ) 2.6 2rx ˆi ˆi 2rx ˆi ˆj 2rx ˆi kˆ 2rz kˆ ˆi 2rz kˆ ˆj 2rz kˆ kˆ 0 2r kˆ 2r ( )ˆj 2r ˆj 2r ( )ˆi 0 τ netxxz 3ˆi 2.2ˆj 5.2kˆzoblique rotation axisthrough originHere all forces were applied at the same point.For forces applied at different points, first calculatethe individual torques, then add them as vectors, i.e., use: net i ri Fiall iall i(vector sum)ijFebruary 16, 2020k
Angular Momentum Same basic techniques that were used in linearmotion can be applied to rotational motion. F becomes m becomes Ia becomes v becomes ωx becomes θLinear momentum defined as p mv What if mass of center of object is not moving,but it is rotating? Angular momentumL Iω February 16, 2020
Angular Momentum I Angular momentum of a rotating rigid objectL Iω L L has the same direction as *L is positive when object rotates in CCWL is negative when object rotates in CWAngular momentum SI unit: kg-m2/sCalculate L of a 10 kg disk when 320 rad/s, R 9 cm 0.09 mL I and I MR2/2 for diskL 1/2MR2 ½(10)(0.09)2(320) 12.96 kgm2/s*When rotation is about a principal axisFebruary 16, 2020
Angular Momentum IIAngular momentum of a particleL I mr 2 mv r mvr sin rp sin Angular momentum of a particle L r p m(r v ) r is the particle’s instantaneous position vectorp is its instantaneous linear momentumOnly tangential momentum component contributeMentally place r and p tail to tail form a plane, L isperpendicular to this planeFebruary 16, 2020
Angular Momentum of a Particle inUniform Circular MotionExample: A particle moves in the xy plane in a circular path ofradius r. Find the magnitude and direction of its angularmomentum relative to an axis through O when its velocity is v. The angular momentum vectorpoints out of the diagramThe magnitude isL rp sin mvr sin(90o) mvrA particle in uniform circular motionhas a constant angular momentumabout an axis through the center ofits pathOFebruary 16, 2020
Angular momentum III Angular momentum of a system of particlesL net L1 L 2 . L n Li ri piall i all iangular momenta add as vectorsbe careful of sign of each angular momentumfor this case:L net L1 L 2 r1 p1 r2 p 2L net r 1 p1 r 2 p 2February 16, 2020
Example: calculating angular momentum for particlesTwo objects are moving as shown in the figure .What is their total angular momentum about pointO?m2L net L1 L 2 r1 p1 r2 p 2Lnet r1mv1 sin 1 r2 mv2 sin 2 r1mv1 r2 mv2 2.8 3.1 3.6 1.5 6.5 2.2 31.25 21.45 9.8 kg m 2 /sm1Direction of L is out of screen.February 16, 2020
Angular Momentum for a Car What would the angular momentum about point “P” be ifthe car leaves the track at “A” and ends up at point “B”with the same velocity ?A) 5.0 102B) 5.0 A106BC) 2.5 104D) 2.5 106E) 5.0 103PL r p p r pr sin( )February 16, 2020
Recall: Linear Momentum andForce Linear motion: apply force to a massThe force causes the linear momentum to changeThe net force acting on a body is the time rate ofchange of its linear momentumdv dpFnet F ma m dt dtp mvI L Fnet t p tFebruary 16, 2020
Angular Momentum and Torque Rotational motion: apply torque to a rigid bodyThe torque causes the angular momentum to changeThe net torque acting on a body is the time rate ofchange of its angular momentumdpFnet F dtτ netdL τ dtand L are to be measured about the same originThe origin must not be accelerating (must be aninertial frame) τ February 16, 2020
Demonstration dp dLFnet F net dtdt Start from dL d (r p ) m d (r v )dt dtdtExpand using derivative chain rule dLd dr dv m (r v ) m v r m v v r a dtdtdt dt dL m v v r a mr a r (ma ) r Fnet netdtFebruary 16, 2020
What about SYSTEMS of RigidBodies? dL iRotational 2 law for a single body : i dt Total angular momentum angular momenta LiLsys L i individualof a system of bodies:all about same origin dLsys i net torque on particle “i” dLi i internal torque pairs aredtdtincluded in sumindBUT internal torques in the sum cancel in Newton 3rd lawpairs. Only External Torques contribute to Lsys dLsysdt i ,ext netnet external torque on the systemiNonisolated System: If a system interacts with its environment in thesense that there is an external torque on the system, the net externaltorque acting on a system is equal to the time rate of change of itsangular momentum.February 16, 2020
Example: A Non-isolated System A sphere mass m1 and a blockof mass m2 are connected by alight cord that passes over apulley. The radius of the pulleyis R, and the mass of the thinrim is M. The spokes of thepulley have negligible mass.The block slides on africtionless, horizontal surface.Find an expression for thelinear acceleration of the twoobjects.a a ext m1 gRFebruary 16, 2020
Masses are connected by a light cord. Find thelinear acceleration a. Use angular momentum approach No friction between m2 and table Treat block, pulley and sphere as a non isolated system rotating about pulley axis.aAs sphere falls, pulley rotates, block slides Constraints:Equal v's and a's for block and spherev ωR for pulley α d / dta αR dv / dt aI Ignore internal forces, consider external forces only Net external torque on system: net m1 gR about Angular momentum of system:(not constant)Lsyscenter of wheel m1vR m2 vR Iω m1vR m2 vR MR 2 ωdLsys m1aR m2 aR MR 2 α (m1 R m2 R MR)a τ net m1 gRdtm1 gsame result followed from earlier a method using 3 FBD’s & 2nd lawM m1 m2February 16, 2020
Isolated System Isolated system: net external torque acting ona system is ZERO no external forcesnet external force acting on a system is ZERO τ extLtot constantdLtot 0dtorLi L fFebruary 16, 2020
Angular Momentum ConservationLtot constantLi L forHere i denotes initial state, f is the final state L is conserved separately for x, y, z direction For an isolated system consisting of particles, Ltot L n L1 L 2 L 3 constantFor an isolated system that is deformableI i i I f f constantFebruary 16, 2020
First Example A puck of mass m 0.5 kg isattached to a taut cord passingthrough a small hole in africtionless, horizontal surface. Thepuck is initially orbiting with speedvi 2 m/s in a circle of radius ri 0.2 m. The cord is then slowlypulled from below, decreasing theradius of the circle to r 0.1 m.What is the puck’s speed at thesmaller radius?Find the tension in the cord at thesmaller radius.February 16, 2020
Angular Momentum Conservation m 0.5 kg, vi 2 m/s, ri 0.2 m,rf 0.1 m, vf ?Isolated system?Tension force on m exert zerotorque about hole, why?Li L fL r p r (mv )Li mri vi sin 90 mri vi L f mrf v f sin 90 mrf v fri0.2v f vi 2 4 m/srf0.1v 2f42T mac m 0.5 80 Nrf0 .1February 16, 2020
IsolatedSystem τ net 0 L about z - axis I ωiinitiali L constant IfinalfωfMoment of inertiachangesFebruary 16, 2020
Controlling spin ( ) by changing I (moment of inertia)In the air, net 0L is constantL I i i I f fChange I by curling up or stretching out- spin rate must adjustMoment of inertia changesFebruary 16, 2020
Example: A merry-go-round problemA 40-kg child running at 4.0m/s jumps tangentially onto astationary circular merry-goround platform whose radius is2.0 m and whose moment ofinertia is 20 kg-m2. There isno friction.Find the angular velocity ofthe platform after the childhas jumped on.February 16, 2020
The Merry-Go-Round The moment of inertia of thesystem the moment ofinertia of the platform plus themoment of inertia of theperson. Assume the person can betreated as a particleAs the person moves towardthe center of the rotatingplatform the moment of inertiadecreases.The angular speed mustincrease since the angularmomentum is constant.February 16, 2020
Solution: A merry-go-round problemLtot I i ωi IfωfLi I i i I 0 mc vT r mc vT rL f I f ω f ( I mc r 2 )ω fI 20 kg.m2VT 4.0 m/smc 40 kgr 2.0 m 0 0( I mc r 2 )ω f mc vT rωf mc vT r40 4 2 1.78 rad/s22I mc r10 40 2February 16, 2020
Physics 101 Lecture 13 Angular Momentum Asist. Prof. Dr. Ali ÖVGÜN EMU Physics Department www.aovgun.com. February 16, 2020 Angular Momentum Torque using vectors Angular Momentum. February 16, 2020 The torque is the cross product of a force vector with the position vector to its point
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Introduction of Chemical Reaction Engineering Introduction about Chemical Engineering 0:31:15 0:31:09. Lecture 14 Lecture 15 Lecture 16 Lecture 17 Lecture 18 Lecture 19 Lecture 20 Lecture 21 Lecture 22 Lecture 23 Lecture 24 Lecture 25 Lecture 26 Lecture 27 Lecture 28 Lecture
Angular Kinetics similar comparison between linear and angular kinematics Mass Moment of inertia Force Torque Momentum Angular momentum Newton’s Laws Newton’s Laws (angular analogs) Linear Angular resistance to angular motion (like linear motion) dependent on mass however, the more closely mass is distributed to the
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0. Since the angular measure q is the angular analog of the linear distance measure, it is natural to define the average angular velocity Xw\ as Xw\ q f-q 0 Dt where q 0 is the initial angular position of the object when Dt 0 and q f is the final angular position of the object after a time Dt of angular motion.
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dimensions of an angular momentum, and, moreover, the Bohr quantisation hypothesis specified the unit of (orbital) angular momentum to be h/2 π. Angular momentum and quantum physics are thus clearly linked.“ 3 In this passage angular momentum is presented as a physical entity with a classical and a quantum incarnation.
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