Angular Vectors - General Physcis - Lecture Slides, Slides of Physics

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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.

ω

 Torque is another kind of vector multiplication.

• Vector cross product yields a vector

 The magnitude is rF sinθ.  The direction points according to the right-hand rule.

Direction of Torque

r F

r τ = ×

θ

torque points into the page F



τ



increasing clockwise angular velocity

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

C A B

= ×

B A A B

× = − ×

C

A

B

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

ω L^ = I

L

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

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.

ω

mg

L

r

τ

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.

ω

mg

L

τ = 0 r