3-Dimensional Rotation: Gyroscopes

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3-Dimensional Rotation Gyroscopes

• 8.01
• W14D1
• Todays Reading Assignment Young and Freedman
  10.7

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Announcements
Problem Set 11 Due Thursday Dec 8 9 pm Sunday
Tutoring in 26-152 from 1-5 pm W014D2 Reading
Assignment Young and Freedman 10.7
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Demo Gimbaled Gyroscope (B140)

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Rules to Live By Angular Momentum and Torque

• About any fixed point P
• Independent of the CM motion, even if
  and
• are not parallel

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Mini Demo Pivoted Falling Stick

• Magnitude of the angular momentum about pivot
  changes.
• Direction of change of angular momentum about
  pivot is the same as direction of angular
  momentum about pivot

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Demo Bicycle Wheel Two Cases

• Case 1 Magnitude of the angular momentum about
  pivot changes.
• Direction of change of angular momentum about
  pivot is the same as direction of angular
  momentum about pivot
• Case 2 Direction of angular momentum about
  pivot changes

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Time Derivative of a Vector with Constant
Magnitude that Changes Direction
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Concept Question Time Derivative of Rotating
Vector
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Example Time Derivative of Position Vector for
Circular Motion

• Circular Motion position vector points
  radially outward, with constant magnitude but
  changes in direction. The velocity vector points
  in a tangential direction to the circle.

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Generalization Time Derivative of a Vector

• Consider a vector
• where
• Vector can change both magnitude and
  direction.
• Suppose it only changes direction then

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Torque and Time Derivative of Angular Momentum

• Torque about P is equal to the time derivative
  of the angular momentum about P
• If the magnitude of the angular momentum is
  constant then the torque can cause the direction
  of the perpendicular component of the angular
  momentum to change

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Introduction To Gyroscopic Motion
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Gyroscopic Approximation

• Flywheel is spinning with an angular velocity
• Precessional angular velocity
• Gyroscopic approximation the angular velocity
  of precession is much less than the
  component of the spin angular velocity,

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Strategy

• Calculate torque about appropriate point P
• Calculate angular momentum about P
• Apply approximation that to
  decide which contribution to the angular momentum
  about P is changing in time. Calculate
• Apply torque law
• to determine direction and magnitude of
  angular precessional velocity

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Table Problem Gyroscope Forces and Torque

• Gravitational force acts at the center of the
  mass and points downward. Pivot force acts
  between the end of the axle and the pylon. What
  is the torque about the pivot point P due to
  gravitational force

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Table Problem Gyroscope Time Derivative of
Angular Momentum

• What is the time derivative of the angular
  momentum about the pivot point for the gyroscope?
  

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Torque and Time Derivative of Angular Momentum

• Torque about P is equal to the time
  derivative of the angular momentum about P
• Therefore
• Precession angular speed is

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More Detailed Analysis of Angular Momentum for
Gyroscopic Motion
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Angular Momentum About Pivot Point
The total angular momentum about the pivot point
P of a horizontal gyroscope in steady state is
the sum of the rotational angular momentum and
the angular momentum of center of mass
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Angular Momentum about Center of Mass
.
The disk is rotating about two orthogonal axes
through center of mass. It is rotating about the
axis of the shaft, with angular speed ?. The
moment of inertia of a uniform disk about this
axis is I1 (1/2) MR2. The disk is also rotating
about the z-axis with angular speed O. The moment
of inertia of a uniform disk about a diameter is
I2 (1/4)MR2. The angular momentum about the
center of mass is the sum of two contributions
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Angular Momentum Due to Motion of Center of Mass
The angular momentum about the pivot point P due
to the center of mass motion is
where is a unit vector in the positive
z-direction and is the angular speed
about the z-axis
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Angular Momentum of Flywheel about Pivot Point
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Gyroscope Time Derivative of Angular Momentum

• If the angular speed (precession angular speed)
  about the z-axis is constant then only the
  direction of the spin angular momentum
• along the axis of the gyroscope is changing in
  time hence

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Torque and Time Derivative of Angular Momentum

• Torque about P is equal to the time
  derivative of the angular momentum about P
• Therefore
• Precession angular speed is

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Concept Question Gyroscope

• For the simple gyroscope problem we just solved,
• if the mass of the disk is doubled how will the
  new
• precession rate O be related to the original rate
  O0?
• O 4 O0
• 2) O 2 O0
• 3) O O0
• 4) O (1/2) O0
• 5) O (1/4) O0
• .

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Table Problem Tilted Gyroscope
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Demo Gyroscope in a Suitcase

• A gyroscope inside a suitcase is spun up via a
  connection to the outside of the suitcase. The
  suitcase is carried across the lecture hall. When
  the lecturer turns while walking, the gyroscope
  causes the suitcase to rise about the handle.

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Table Problem Suspended Gyroscope