3-Dimensional Rotation: Gyroscopes - PowerPoint PPT Presentation

About This Presentation
Title:

3-Dimensional Rotation: Gyroscopes

Description:

3-Dimensional Rotation: Gyroscopes 8.01 W14D1 Today s Reading Assignment Young and Freedman: 10.7 – PowerPoint PPT presentation

Number of Views:135
Avg rating:3.0/5.0
Slides: 37
Provided by: PeterD219
Learn more at: http://web.mit.edu
Category:

less

Transcript and Presenter's Notes

Title: 3-Dimensional Rotation: Gyroscopes


1
3-Dimensional Rotation Gyroscopes
  • 8.01
  • W14D1
  • Todays Reading Assignment Young and Freedman
    10.7

2
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
3
Demo Gimbaled Gyroscope (B140)

4
Rules to Live By Angular Momentum and Torque
  • About any fixed point P
  • Independent of the CM motion, even if
    and
  • are not parallel

5
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

6
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

7
Time Derivative of a Vector with Constant
Magnitude that Changes Direction
8
(No Transcript)
9
Concept Question Rotating Vector
10
Concept Question Time Derivative of Rotating
Vector
11
Concept Question Time Derivative of Rotating
Vector
12
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.

13
Generalization Time Derivative of a Vector
  • Consider a vector
  • where
  • Vector can change both magnitude and
    direction.
  • Suppose it only changes direction then

14
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

15
Introduction To Gyroscopic Motion
16
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,

17
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

18
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

19
Introduction To Gyroscopic Motion
20
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,

21
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

22
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

23
Table Problem Gyroscope Time Derivative of
Angular Momentum
  • What is the time derivative of the angular
    momentum about the pivot point for the gyroscope?

24
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

25
More Detailed Analysis of Angular Momentum for
Gyroscopic Motion
26
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 angular momentum due to the
rotation about the center of mass and the angular
momentum about the pivot point due to the center
of mass motion
27
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
28
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
29
Angular Momentum of Flywheel about Pivot Point
30
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

31
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
  • .

32
Concept Question Answer Gyroscope
  • Answer 3. Both the torque and the angular
    momentum are proportional to the mass of the
    spinning wheel, so they cancel form both sides of
    the torque equation and thus the precessional
    angular speed is independent of mass.
  • .

33
Table Problem Tilted Gyroscope
34
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.

35
Table Problem Suspended Gyroscope
36
Table Problem Suspended Gyroscope
Write a Comment
User Comments (0)
About PowerShow.com