1. Angular Vectors

2. Direction of Angular Velocity
Angular velocity can be clockwise or counterclockwise around the axis of rotation.
Two directions along the axis of rotation
Angular velocity can point either way
By convention the direction follows the thumb if the rotation follows the curve of the right hand.

3. Angular Acceleration Vector
The angular acceleration vector is the time derivative of the angular velocity vector.
Along the axis if the angular velocity only changes magnitude
In other directions if the axis changes direction

4. Direction of Torque Torque is another kind of vector multiplication.
Vector cross product yields a vector
The magnitude is rFsinq.
The direction points according to the right-hand rule.
torque points into the page q
increasing clockwise angular velocity

5. Cross Product Properties
The vector cross product applies to any two vectors.
The cross product is perpendicular to the plane holding the two vectors.
The cross product is not commutative.
Reversing the order gives an anti-parallel result

6. Momentum Cross Product
Angular momentum is a vector.
Vector cross product of the lever arm and momentum.
Direction follows the right-hand rule
Magnitude matches single-axis form L p r

7. Single Axis Rotation
An axis of rotation that is fixed in direction gives a single axis rotation.
Simplest case has the axis through the center of mass
Angular momentum vector is parallel to the angular velocity L w

8. Limitations
There are limitations to the relationship between angular momentum and angular velocity.
Moving axis of rotation
Asymmetric axis of rotation
Angular momentum and angular velocity can have different directions. p L r

9. Angular Momentum Vector
The vector form of the law of rotational motion is generalized to use angular momentum vectors.
Correct for all axes
Correct for changes in direction as well as angular velocity

10. Gravitational Torque Tops use torque. Gravity supplies the torque.
The lever arm is the axis of rotation.
Gravity is directed down.
The torque is at right angles to the lever arm and horizontal.
The top will precess in a circle. L t w r mg

11. Gyroscope
A gyroscope acts like a top, and precesses if its axis is at an angle.
If the gyroscope axis is vertical the torque from gravity is zero.
If the base moves, the gyroscope stays vertical. L w
t = 0 mg r

12. Boomerang Boomerangs move due to gravitational torque.
Aerodynamic lift is the force
The lever arm is the length of each arm F t r L w