Announcements

1
Announcements

• Homework 9 is due at end of class
• Homework 10 will be due November 30
• Midterm 2 on Friday, November 18 830 950 AM
  in BH 166
• Will cover chapters 7 9 HRW
• Expect 4 problems of varying difficulty (sort of
  like homework)
• Think problems through if you apply a formula,
  make sure its valid
• Be sure to show your work
• Partial credit given for answers that arent 100
  correct
• No credit for guessing the right answer!
• What you may use during the exam
• 1 sheet of 8.5x11 paper on which you can write
  anything you like
• Basic calculator to obtain numerical answers
• What you may not use
• Books, notes, magnifying glass,
• Any sort of computer, PDA, tablet PC, etc.
• Anything with wireless technology (please turn
  off cell phone, pager, PDA,)

2
Angular Momentum

• For translational motion, momentum conservation
  proved to be a powerful tool
• Many aspects of motion could be understood
  without knowing the detailed forces that were
  involved
• For rotational motion, we will see that
  conservation of angular momentum is an equally
  powerful tool
• Definition of angular momentum for a particle
• Angular momentum depends on coordinate system
  used
• Non-zero angular momentum indicates vector r is
  rotating about origin
• In the plane formed by vectors r and p

3
Angular Momentum

• Can think of angular momentum two ways
• Displacement from origin times the tangential
  momentum
• Momentum times moment arm r?
• Note that even a particle moving in a straight
  line can have angular momentum!
• Unit of angular momentum is (kg?m2/s)

Origin
Origin
4
Example Circular Motion

• For a particle of mass m moving with a velocity v
  in a circle of radius r about the origin
• r and v are always perpendicular
• In this example, angular momentum points out of
  the page
• Pick your coordinate system wisely!
• Angular momentum about some other point would be
  much more complicated

5
Newtons Second Law with Ang. Mom.

• Can re-cast Newtons second law (yet again) in
  terms of angular momentum
• Can perhaps guess what to expect by comparing
  definitions for translational and rotational
  quantities

Translation
Rotation
6
Newtons Second Law with Ang. Mom.

• Proof
• Take time derivative of angular momentum
• Thus,

7
Angular Momentum for a System

• If we have a system of several particles, the
  total angular momentum vector L is given by the
  sum of the angular momentum vectors li for each
  particle
• Apply second law to each particle in system
• Applying a torque to a system changes the systems
  angular momentum

8
Angular Momentum of a Rigid Body

• Consider a rigid body rotating about a fixed axis
• Angular momentum gets contributions from each
  element of the body
• We are interested in the angular momentum along
  the rotation axis, Lz
• The total angular momentum along the rotation
  axis is obtained by integrating over all parts of
  the body

9
Conservation of Angular Momentum

• For an isolated system in which no net external
  torque is applied
• Such a system has a constant angular momentum
• Angular momentum is conserved
• Example Ice skater increases rate of rotation by
  pulling in her arms
• Moment of inertia decreases, angular frequency
  must increase
• Example Hoberman sphere
• Another example of changing the moment of inertia

10
Conservation of Angular Momentum

• Example turning a spinning bicycle wheel while
  sitting on a chair
• Bicycle wheel carries significant angular
  momentum
• Applying a force to the axel generates a torque
  on the turning wheel, causing angular momentum to
  change direction
• Force also creates a torque about the stool axel,
  causing entire system to rotate, conserving
  angular momentum

11
Conservation of Angular Momentum

• Example Neutron star
• When a star of a few solar masses runs out of
  fuel, it will collapse under its gravitational
  force to form a neutron star a few km in size
• Angular momentum is conserved, but moment of
  inertia is vastly smaller
• Radius goes from 109 m to 104 m, I decreases by
  factor of 1010
• Rapidly rotating neutron stars produce pulsed
  radio waves (pulsars)
• Example Crab Nebula seen by Chinese
  astronomers in 1054AD

12
Gyroscope

• A spinning gyro has an angular momentum along its
  axis of rotation
• Align the coordinate system with the axis of
  rotation
• If we apply a torque to the gyro, the angular
  momentum will change
• Gravitational force on mass produces a torque
• Torque points into the screen
• Torque is always perpendicular to ang. mom. L
• Torque causes gyro to precess about the vertical
  axis
• The change in L is perpendicular to L, so the
  magnitude of L is constant but its direction
  changes (or precesses)
• Similar to circular acceleration change in
  velocity perpendicular to velocity

13
Example Spinning Bicycle Wheel

• Spinning bicycle wheel has considerable angular
  momentum
• A torque is required to change this angular
  momentum, which stabilizes the bicycle and
  prevents you from easily falling over (as would
  happen at rest)
• As you go into a turn, you tilt the bicycle so
  its no longer vertical, but you still dont fall!
• Imagine you are riding into the screen, turning
  to the left
• As you make your turn, the angular momentum of
  the wheel changes direction
• The change in ang. mom DL points out of the
  screen
• There must then be a torque pointing out of the
  screen
• Gravity acting on your mass gives such a torque
• You dont fall over because the net torque on the
  bicycle equals the rate of change in angular
  momentum as you turn
• Example spinning bicycle wheel supported from
  one end

14
Summary of Rolling, Torque, and Ang Mom

• Torque is a vector quantity
• Rolling object combines translation rotation
• COM velocity and angular velocity are related
• Kinetic energy includes both motions
• Angular momentum and Newtons 2nd law
• Angular momentum for a rigid body
• Angular momentum is conserved for an isolated
  system