Angular Momentum Principle, Torques July 20, 2010 Chapter 11, Section 4-6 Esys W Q psys Fnet t Lsys net t What Causes Angular Momentum If an object is at rest it requires an unbalanced force to make it move Rotation is motion A force is required for angular momentum The location of the applied force is important A Wrench It is better to use a wrench from the end than the middle Applying a force farther away gives more angular momentum Wrenches don’t work if you push inward Angular momentum requires applying a force at an angle Torque Measures “Angular dLA d(rA p) drA dp Force” p rA dt dt dt dt The definition makes dL A v mv r F intuitive sense dt A net Cross product dL A r F Force must be applied at dt A net A an angle Depends on distance from axis Larger the farther away it is applied A rA Fnet Clicker Question #1 Which of these will provide the Most torque A B B C D D All arrows represent equal applied forces C A Clicker Question #2 Which of these will provide the Least torque A B B C D D All arrows represent equal applied forces C A Angular Momentum Principle Compare to the Momentum Principle dp dLA Fnet dt dt A p Fnet t LA net,At Conceptually they are equivalent No change in momentum without some external Force Computationally Angular Momentum is more involved Finding Torque requires doing a cross product [right hand rule] Using the AMP First find the net applied torque r F r F sin90 3Nm Get the direction by RHR Into the page (-z) Use AMP L nett (3zˆ)(3) 9zˆ Calculate final ω 1 1 1 I mr 2 *4*( )2 9zˆ f 2 f 2 2 f rad 18zˆ f s Clicker Question #3 What is the Final angular momentum A) 15 Js into the page B) 15 Js out of the page ωi = 3 r C) 45 Js out of the page F D) 45 Js into the page ωf = ? 5Nm net r t 3 s F I 10 kgm2 More Than One Torque See – Saw Two objects on a massless plank Calculate each torque and then add them up r m g r m g zˆ 1 1 1 1 1 2 r2 m2 g r2m2 g zˆ net 1 2 r1m1 r2m2 g zˆ Equilibrium if the net torque is 0 Torque and Linear Force They are dealt with separately Just like Angular vs. Linear Momentum The Force can also move the COM p Fnet t 18xˆ Ns p m v 4.5xˆ f m s This is in addition to the rad rotation 18zˆ f s Torque and Friction Friction opposes motion So friction creates a torque in opposition to a rotation zˆ r f rfzˆ Often neglected When µ is small When r is small When Δt is small collisions Torque and orbits Does gravitational force apply a Torque on orbits? No Qualitative The Earth’s orbital velocity around the sun is not changing No torque Quantitative GM m F 1 2 rˆ g r 2 GM m r F 1 2 r rˆ g g r 2 r r 0 g 0 Collisions Both Angular Momentum and Linear Momentum are conserved separately Li r pi Rmvi zˆ pi mvi xˆ 1 2 L f I m M R Rmvi zˆ 2 p f m M v f mvi xˆ Allows you to find both rotational and linear velocity mv i zˆ f m .5M R mv v i xˆ f m M Collision Rules Conserve Momentum psys 0 Conserve Total Energy Esys 0 Conserve Angular Momentum Lsys 0 Impact Parameter Where the collision happens is important Determines how fast the object spins Also important for linear momentum 17 Fundamental Principles of Mechanics Momentum dp F External Forces dt net Location is not important Angular Momentum dL External Torques A dt net,A Location relative to some point is important Energy Work or Heat E W Q Location is not important.