Angular Vectors Direction of Angular Velocity Angular velocity

Direction of Angular Velocity ] Angular velocity can be clockwise or counterclockwise around the

Angular Acceleration Vector ] The angular acceleration vector is the time derivative of the

Direction of Torque ] Torque is another kind of vector multiplication. • Vector cross

Cross Product Properties ] The vector cross product applies to any two vectors. ]

Momentum Cross Product ] Angular momentum is a vector. • Vector cross product of

Single Axis Rotation ] An axis of rotation that is fixed in direction gives

Limitations ] p L There are limitations to the relationship between angular momentum and

Angular Momentum Vector ] The vector form of the law of rotational motion is

Gravitational Torque ] Tops use torque. ] Gravity supplies the torque. • The lever

Gyroscope ] A gyroscope acts like a top, and precesses if its axis is

Boomerang ] Boomerangs move due to gravitational torque. • Aerodynamic lift is the force

Slides: 12

Download presentation

Angular Vectors

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.

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

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

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

Momentum Cross Product ] Angular momentum is a vector. • Vector cross product of the lever arm and momentum. • Direction follows the righthand rule • Magnitude matches singleaxis form L p r

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

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

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

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. L t w r mg ] The top will precess in a circle.

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

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