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# Modern Mechanics PHYS 17200

Purdue

GPA 3.95

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This 24 page Class Notes was uploaded by Mrs. Justyn Cremin on Saturday September 19, 2015. The Class Notes belongs to PHYS 17200 at Purdue University taught by Gabor Csathy in Fall. Since its upload, it has received 25 views. For similar materials see /class/207910/phys-17200-purdue-university in Physics 2 at Purdue University.

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Date Created: 09/19/15

PHYS 172 Modern Mechanics Fall 2009 Lecture 20 Angular Momentum Multiparticle systems Read 105 107 CLICKER QUESTION 1 Reading Question Sections 105 107 This is a closedbook quiz no consulting with neighbors etc A yoyo is in the xy plane You pull up on the string with a force of magnitude 06 N What is the direction of the torque you exert on the yoyo A y y L 2 out of the page B y C z D E Z zero magnitude Exam 3 is next week Tuesday November 17th at 8 pm in Elliott Hall Alternative testing at same time in Room 8 in Physics Lectures both days next week and labs amp recitations as usual Last Time single particle system Lumll k clotkvuse E A a 1 Llutkwise Axl D LAzilXij 131 172 LA LtramA Lrot 39 zm m x a Cenwru uasa LtmmlA rcmA X B0 a a a a a remi r3cm Lrotzn mxpl rrlcmXpZ rn 2 Last Time Angular velocity 6 Right hand Moment of inertia I 2 2 2 I mlru m2ru m3ru quot Lrot Ia Moment of inertia masses not in plane W l g hwmgmmgfm Masses not in plane V a 1 E 1273 akimgi 4h 71 is distance from axis of rotation Example solid disk of uniform density 1 13MR2 In respect to its axle Let s consider what the time derivative of the an ular momentum of a single gt particle is Recall the Momentum Principle F dt net do gtlt 13 da A A X p FA X Product rule for derivatives dt dt dt dFA a a a E X I V X I 0 Vector product of parallel vectors 0 A 77A p 17A x M Linear Momentum Principle dt T A X Em Definition of a new quantity torque di A A A A A thZFA XFneIZZ A OI ALAZGA gtlt Fnet ZZZ39AAZ The Angular Momentum Principle for a Point Particle dEA A A TA or ALA TAAZ d1 The Linear Momentum Principle for a Point Particle 213 Or A1321 At net We will soon see a large class of problems where the Angular Momentum Principle will be of fundamental importance Newton s Second Law Revisited I l a 39 dL 0 Fd p gtM r gtI a dt dt dt dt Today The Angular Momentum Principle Torque Angular Momentum Principle Examples ZeroTorque Systems Merrygoround Spinning IceSkater The iver Torque and the Angular Momentum Principle Forces change momentum What changes angular momentum Ap 2 FM At AL 2 TAM At Linear Momentum Principle Angular Momentum Principle A torque comes from an offcenter force that tends to spin the system The more offcenter the greater this tendency is TA E X Fnet E SJ 2 mm m TA EVA gtltFnez ITAIEIVAI39Fnez sin6 Torque Larger rA largertorque CLICKER QUESTION 2 With respect to the sun Asun what is the magnitude of the torque that the Sun exerts on the planet TA on planet due to sun A rp sin 6 B 0 GM m GM m sun planet pSIH 6 sun2 planet r Sing I C r2 Conservation of Angular Momentum ZeroTorque Systems If i net on system is zero then the system cannot exchange angular momentum with the surroundings and L and Lsurr are conserved separately sys O gt AL At0 At0tsys Amer We ll spend the rest of lecture looking at some examples The Three Fundamental Principles Momentum Angular Momentum Energy of dz L E netext 714 zInetextA AE W Q If external forces momentum changes If external torques angular momentum changes If work is done energy of system changes If no external forces Momentum of system is constant If no external torques Angular momentum of system is constant If no work done Energy of system is constant Location of object doesn t matter Location of object relative to point A is important Location of object doesn t matter Example MerryGoRound wherewe started quotC k initial final How is the final 03 related to v and ri 1 System child merrygo round MGR 2 Ang Mom Principle ALAYW TAymAt 3 Point A center of MGR no nettorque along MGR axis 439 MAJ 0 3 ALAchild AMGR 0 3 LAchildi LAMGRi LAchild f LAMGR f Example MerryGoRound where we started gt 5 Values of L Single particle gt LAchild rAchild X pchild initial final CLICKER QUESTION 3 What is the initial angular A lt0 0 mm momentum of the child B lt O O mer C lt mvri O O D mvri O O zoutofthe page CLICKER QUESTION 4 What is the final speed of the child 0 CorL2 0 D ori y LX z out of the page CLICKER QUESTION 5 What is the final angular A lt 0 0 erLQ momentum of the child about B O O 2 point A lt quotrm LCD C lt O 0 mm A DltO 0 mricogt 0 0 E no magnitude Lx z out ofthe page Example MerryGoRound where we started 5 Single particle Child gt LAchild rAchild X pchild LAchildi Ova mm initial final EAchildf lt0 0 ma 12gt Multiparticle disk 39 39 I Look up in book LAMGR LrotMGR IMGRa or integratel I 2 MR2 LAMGRJ MGR lt000gt 000 disk 2 ZAMGRf MGR 395 lt0903MR20gt Example MerryGoRound where we started 6 Insert values we re now dealing only with zcomponent LAchildzi LAMGRZi LAchildzf LAMGRzf 2 1 2 mer O mari MR a 2 I I ZVI i gt a 2 1 2 mri E 7 Check result as we expect the final angular speed is greater for larger values of v and ri If ri 0 the merrygo round doesn t spin at all Angular momentum application System with no torque a 1 gt1 dLrot a l f 7 0 I 2 2 dl netcm llquotJ1 quotkn1212 L 10 Ifwf 21160 60 ia f 1 If httpwww hepphyssotonao ukooursesphy52006 Angular momentum application A freefalling cat cannot alter its total angular momentum Nonetheless by swinging its tail and twisting its body to alter its moment of inertia the cat can manage to alter its orientation htgMWW uutube ummath7vUaAGh AXdWO See also book example High dive iEEEi

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