3 Dimensional Rotation Gyroscopes 8 01 W 14
- Slides: 30
3 -Dimensional Rotation: Gyroscopes 8. 01 W 14 D 1 Today’s Reading Assignment Young and Freedman: 10. 7
Announcements Problem Set 11 Due Thursday Dec 8 9 pm Sunday Tutoring in 26 -152 from 1 -5 pm W 014 D 2 Reading Assignment Young and Freedman: 10. 7 2
Demo: Gimbaled Gyroscope (B 140)
Rules to Live By: Angular Momentum and Torque 1) About any fixed point P 2) Independent of the CM motion, even if are not parallel and
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
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
Time Derivative of a Vector with Constant Magnitude that Changes Direction
Concept Question: Rotating Vector
Concept Question: Time Derivative of Rotating Vector
Concept Question: Time Derivative of Rotating Vector
Example: Time Derivative of Position Vector for 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.
Generalization: Time Derivative of a Vector Consider a vector where Vector can change both magnitude and direction. Suppose it only changes direction then
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
Introduction To Gyroscopic Motion
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,
Strategy 1. Calculate torque about appropriate point P 2. Calculate angular momentum about P 3. Apply approximation that to decide which contribution to the angular momentum about P is changing in time. Calculate 4. Apply torque law to determine direction and magnitude of angular precessional velocity
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
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 about center of mass
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 ω. The moment of inertia of a uniform disk about this axis is I 1 = (1/2) MR 2. The disk is also rotating about the z-axis with angular speed Ω. The moment of inertia of a uniform disk about a diameter is I 2 = (1/4)MR 2. The angular momentum about the center of mass is the sum of two contributions
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
Angular Momentum of Flywheel about Pivot Point:
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
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 Direction is the positive z-direction
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 Ω be related to the original rate Ω 0? 1) Ω = 4 Ω 0 2) Ω = 2 Ω 0 3) Ω = Ω 0 4) Ω = (1/2) Ω 0 5) Ω = (1/4) Ω 0
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. .
Table Problem: Tilted Gyroscope
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.
Table Problem: Suspended Gyroscope
Table Problem: Suspended Gyroscope
- Specific rotation of sugar solution
- A circular motion is one dimensional
- Geometric rotation
- Exemple de guidage en rotation
- Sketching hyperbolas
- General pivot point rotation
- Démarrage direct 1 sens de marche
- Rodrigues' rotation formula
- Dynamique de rotation
- Rotation vs revolution
- Super rotation
- "reflection translation rotation"
- 1800 rotation
- Rotation worksheet
- Reflection geometry
- What causes seasons rotation or revolution
- Easterlies
- Formula for 270 degree rotation clockwise
- Volleyball rotation clockwise
- Workshop rotation model
- Improper rotation axis
- Rotation transversale
- 6-2 rotation
- Rotation and revolution
- Subscapularis action
- Spherical harmonics rotation invariant
- Rotation pure
- Job rotation plan template
- Define rotation and revolution
- Pnwu campus map
- Revolution rotation difference