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,At  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 sin90    3Nm

 Get the direction by RHR  Into the page (-z)  Use AMP   L  nett  (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