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 rotationTwo directions along the axis of rotation Angular velocity can point either wayAngular 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 magnitudeAlong the axis if the angular velocity only changes magnitude In other directions if the axis changes directionIn other directions if the axis changes direction

4.  Torque is another kind of vector multiplication. Vector cross product yields a vector  The magnitude is rF sin .  The direction points according to the right-hand rule. Direction of Torque  torque points into the page 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 resultReversing 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.Vector cross product of the lever arm and momentum. Direction follows the right- hand ruleDirection follows the right- hand rule Magnitude matches single- axis formMagnitude matches single- axis form r p L

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 massSimplest case has the axis through the center of mass Angular momentum vector is parallel to the angular velocityAngular momentum vector is parallel to the angular velocity  L

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. L r p

9. Angular Momentum Vector  The vector form of the law of rotational motion is generalized to use angular momentum vectors. Correct for all axesCorrect for all axes Correct for changes in direction as well as angular velocityCorrect 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.The lever arm is the axis of rotation. Gravity is directed down.Gravity is directed down. The torque is at right angles to the lever arm and horizontal.The torque is at right angles to the lever arm and horizontal.  The top will precess in a circle.  mg L r 

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.  mg L  r

12. Boomerang  Boomerangs move due to gravitational torque. Aerodynamic lift is the forceAerodynamic lift is the force The lever arm is the length of each armThe lever arm is the length of each arm  L r  F