http:csep10'phys'utk'eduastr162lectpulsarspulsars'html

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http//csep10.phys.utk.edu/astr162/lect/pulsars/pu
lsars.html
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Lecture 17chapter 11 continued

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Announcements

• HW 9 due TODAY after lecture
• Office hours today from 1000-1115am
• And on Thursday from 10am-noon, 230pm-4pm.
• Can also make an appointment to meet.
• Midterm 2 FRIDAY Nov 9th
• Time 830am-950am, BH 166
• Will cover chapters 6-9
• One 8.5x11 sheet with formulas allowed,
  calculators allowed
• Closed book, lecture notes exam.

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Example Falling Chimneys

• Can the force of gravity ever cause an object to
  be accelerated faster than g?
• Consider a Falling Chimney
• Calculate torque about base of chimney

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• Two small spheres of mass m3 kg are attached to
  the two ends of a massless, 1 m long rod. The rod
  can rotate on an axle through its center. A
  massless, frictionless pulley of radius r10 cm
  is attached to the axle and a mass M0.6 kg is
  suspended from a string that is wrapped around
  the pulley. The mass M starts at rest at a height
  h1 m above a surface. As the mass descends under
  the influence of gravity, the string unwinds from
  the pulley and the rod begins to rotate.
• Calculate

• the angular acceleration of the rod
• the angular velocity of the rod when the mass M
  reaches the surface

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• the angular acceleration of the rod
• the angular velocity of the rod when the mass M
  reaches the surface

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• Three objects of different shapes (disk, hoop,
  sphere) roll down an incline. Which object will
  arrive first at the bottom of the incline?

• the disk
• the sphere
• the hoop
• all arrive at the same time

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• Three hoops of different mass and size roll down
  an incline. Which hoop will reach the bottom
  first?

• the hoop with the largest mass
• the hoop with the smallest mass
• the hoop with the smallest diameter
• the hoop with the smallest rotational inertia
• they will all reach at the same time

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the speed at the bottom of the incline is larger
the smaller the ratio I/MR2 is (for same h). for
a sphere
for a disk
for a hoop
For different shapes the one with the smallest
ratio will win regardless of mass or size! For
identical shapes this ratio is the same thus
identical shapes have the same speed at the
bottom regardless of mass.
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object rolling down incline
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in x-direction
in y-direction
determine force of friction
for a rolling object
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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

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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
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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

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• Consider a particle that moves along a straight
  line parallel with the x-axis and a distance r0
  away from the origin O with constant momentum p.
  What is its angular momentum with respect to O?

y
r0
x
O

• L0
• L r0p anywhere on its trajectory
• L depends on the position of the particle

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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
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Newtons Second Law with Ang. Mom.

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

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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

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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

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• Two small 600 g masses are mounted on an axle
  with massless rods as shown in the figure. The
  axle rotates at an angular velocity of 30 rad/s.

20 cm
40 cm

• what is the component of the total angular
  momentum along the axle?
• what angle does the total angular momentum vector
  make with the axle?

20 cm
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a)

x z 0.4 m and therefore the angle with the
y-axis is 45o.
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• A uniform thin rod of length 0.5 m and mass 4.0
  kg can rotate in a horizontal plan about a
  vertical axis through its center. The rod is at
  rest when a 3.0 g bullet is fired in the
  horizontal plane of the rod such that it hits one
  end of the rod at a 60o angle. If the bullet
  lodges in the rod and the angular velocity of the
  rod is 10 rad/s immediately after the collision,
  what was the bullets speed just before impact?

axis
60o
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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

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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

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• A spinning wheel is suspended by strings at both
  ends of its axle. What happens to the wheel if
  one string is released?

• The wheel drops under the influence of gravity
  until its center of gravity is exactly below the
  remaining string
• The wheel drops under the influence of gravity
  and swings back and forth from the remaining
  string
• The wheel experiences a torque under the
  influence of gravity and begins to rotate about
  the remaining string.
• The wheel continues to spin in the same way,
  because angular momentum is conserved.

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• gravity gives rise to torque on wheel, that
  points in horizontal direction ? change in
  angular momentum is in direction of torque ?
  precession.

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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
• Gravity acting on your mass gives rise to a
  torque out of the
  screen
• The change in ang. mom ?L points out of the
  screen
• As the angular momentum of the wheel changes
  direction the bike turns left
• 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

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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

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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

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conservation of angular momentum
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• A figure skater stands on one spot on the ice
  (assumed frictionless) and spins around with her
  arms extended. When she pulls in her arms, she
  reduces her rotational inertia and her angular
  speed increases so that her angular momentum is
  conserved. Compared to her original kinetic
  energy, her rotational kinetic energy after she
  has pulled her arms in must be

• the same
• larger because she is rotating faster
• smaller because her rotational inertia is smaller

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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

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Skylab Videos

• Videos made from by astronauts while in orbit
• No external forces or torques that apply
  specifically to the astronauts
• Space ship and astronauts are falling together
  as they undergo their circular orbit
• Astronauts and the space-craft have identical
  accelerations
• Gravity thus doesnt cause any relative motion
  between the space ship and astronauts
• Videos demonstrate conservation of momentum,
  angular momentum

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Conservation Laws

• Conservation of mechanical energy, momentum, and
  angular momentum were derived from Newtons laws
• Mechanical Energy conservation required no
  non-conservative forces
• Momentum conservation required no external forces
• Angular momentum conservation require no external
  torques
• We argued, but didnt prove, that energy was
  always conserved if you included thermal and
  internal energy in the system
• Furthermore, Einsteins theory of general
  relativity lacked the usual continuity equation
  that gives energy conservation
• In 1915, Emmy Noether came up with a remarkable
  piece of mathematical physics that put the
  concept of conservation laws in an entirely new
  light

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Emmy Noether

• Born in 1882 in Erlangen, Bavaria
• Daughter of a distinguished mathematician
• Obtained her Ph.D. in math in 1908
• Moved to Gottingen in 1915, but was refused her
  habilitation that would allow her to earn money
  by teaching
• Hilbert to faculty senate I dont see why the
  sex of the candidate is relevant this is after
  all an academic institution, not a bath house.
• Finally granted her habilitation after WW I, but
  dismissed from her position at Gottingen in 1933
  because she was jewish

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Noethers Theorem

• In essence, it says that if there is a symmetry
  in nature, then this symmetry requires there to
  be an associated conservation law
• There is a minor caveat that the motion be
  described by a Lagrangian
• Each of our conservation laws is associated with
  a symmetry in nature
• Energy conservation time translation symmetry
  (can redefine t 0)
• Momentum conservation space translation
  symmetry
• (can redefine x 0)
• Angular momentum conservation rotational
  symmetry
• (can redefine f 0)
• Noethers theorem shows explicitly the
  conservation law that results from a given
  symmetry
• It also allowed the establishment of energy
  conservation in general relativity based on the
  time translation symmetry

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Symmetries Define our Forces

• The current Standard Model of particle physics
  accurately describes three of the four known
  forces
• Electricity/magnetism, Strong interaction, Weak
  Interaction
• Still working on gravity
• The Standard Model is an example of a Gauge
  Theory
• Once the symmetries are specified in a Gauge
  Theory, all the details that describe forces,
  interactions, etc. are fixed
• Thus, the physical laws are a direct consequence
  of the symmetry principles
• Standard Model symmetry
• U(1) electric charge conservation, laws of
  electricity and magnetism
• SU(2) applies to weak hypercharge, gives rise
  to weak interactions
• SU(3) color of quarks/gluons is conserved,
  gives rise to strong int.

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Good Luck on your midterm!!!