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Angular Momentum Angular Momentum July 19, 2010 Esys W Q Chapter 11, Section 1-3 psys Fnet t Lsys net t Angular Momentum Measures rotational motion – Both spin and orbits Earth orbits the sun and spins on its axis Is a vector L funnel.sfsu.edu/courses/gm309/labs/seasons/facts.html Types of Angular Momentum Angular momentum is split into 2 types Translational Angular Momentum – Angular Momentum relative to some point – Earth’s orbit around the sun Rotational Angular Momentum – An object is spinning around some axis – A spinning top Translational Angular Momentum Defined relative to some point LA rA p Ltrans,A rA p sin Center of mass moves around some point A cross product Direction determined by Right Hand Rule Right Hand Rule Right Hand Rule 2 Clicker Question #1 What direction is the angular momentum vector? – A) Into the board P=mv – B) Out of The board – C) Left r – D) Right Clicker Question #2 What direction is A x B? – A) Into the board – B) Out of The board – C) Left – D) Right θ B A Cross Products Multiply 2 vectors and get another vector L will always be perpendicular to both r and p – For any cross product L r p L ry pz rz py , rz px rx pz , rx py ry px The cross product will give the direction Sample Cross Product Put the vectors tail to tail r ryˆ p mvxˆ L r p 0*(r) 0*(mv) mv*0 0*0 0*0 (r)mv L 0 0 rmv You get the correct direction for L by doing the cross product Translational Angular Momentum An object traveling in a straight line can still have angular momentum – Relative to some point – Important for dealing with collisions Rotational Angular Momentum Object spins around an axis through its center of mass – Bike tire, Earth L I Formula derived from: L r p – Find Ltrans for each particle in a multi-particle system and add them up r p m1r1v1 sin1 m2r2v2 sin2 m3r3v3 sin2 r r sin v r 2 2 2 Lrot [m1r1 m2r2 m3r3 ] I Translation and Moment of Inertia The two equations are equivalent I r p Rotational I is not the same as translational I 2 – Ex: a sphere I Mr 2 rot 5 2 Itrans MR Can have different ω Example: Ball on a String 2 ways of solving Translational Ltrans r mv mvr Rotational L I v r I mr 2 L mrv In both cases the direction is given by the right hand rule upload.wikimedia.org/wikipedia/commons/9/97/Angular_momentum_circle.png Example: Bike Tire All mass is at the edge Split into 20 parts M M L r p R vsin90 R2 trans 20 20 M L 20 R 2 MR2 tot 20 Ltot I L is into the page Starting with Translational you can get rotational Total Angular Momentum Add the two together Ltot Ltrans Lrot Ltrans rA pcm Lrot I Example: The Earth The Earth has an orbit and spin 2 2 L MR2 7.11033 Js rot 5 Earth 24hours 2 L MR2 2.661040 Js trans Sun,Earth 365days Notice that both I and ω are different for each type of L Drawing 3D in 2D Angular momentum will always require 3D – It is a cross product How to draw this easily? Into the page – Arrow feathers going away Out of Page – Arrow head coming towards you Clicker Question #3 Which diagram is the correct representation of L r p – A – B C) p – C B) L – D r r L p A) p p D) L r L r.
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