1. Vector Torque

2. Direction of Angular Velocity  Angular velocity can be clockwise or counterclockwise around the axis of rotation.  It has magnitude and direction – it must be a vector. Only two directions are possible for a fixed axisOnly two directions are possible for a fixed axis

3. Right-hand Rule  Along the axis of rotation there are two directions.  The angular velocity can point either way along the axis of rotation.  By convention the direction follows the thumb if the rotation follows the curve of the right hand.

4. 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 next

5. Direction of Torque  Torque is related to a change in angular acceleration. Torque:  = I Torque:  = I   Since  is a vector,  must be a vector pointing in the same direction.  How is this related to the force and position vectors? Torque:  = r F sin Torque:  = r F sin  The product of these vectors is not a scalar, but a vectorThe product of these vectors is not a scalar, but a vector

6.  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. Vector Cross Product  torque points into the page increasing clockwise angular velocity

7. 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

8. Cross Product Components  Cross products can be written in components. Add clockwise Subtract counterclockwise

9. Vector Form of Action  The angular law of action can now be extended to three dimensions with vectors. next