1. Chapter 11
General Rotation

2. Vector cross product b a 
where
Right-hand rule

3. Expressed by cross product
The torque vector r O P
Expressed by cross product
Torque vector about point O
Example1: A particle is at position ,
Calculate the torque about origin if
Solution: We use the determinant form

4. Angular momentum of a particle
Angular momentum about point O
magnitude
Angular momentum theorem
Rotational equivalent of Newton’s second law 4

5. Angular momentum in circle
Example2: Determine the angular momentum of a particle in uniform circular motion (m, v, r).
Solution: It depends on the choice of point O!
First calculate it about the center of circle
Agreed with the rigid body case O l r O’
Direction?
What about another O’ ?
Component along axis OO’ 5

6. Conservation of angular momentum
The angular momentum of a particle remains constant if there is no net torque acting on it.
Typical case: Acted by a central force
Kepler’s 2nd law: A line from the sun to a planet sweeps out equal area in equal time.
Area: 6

7. Solution: No torque, L about o is conserved
Move in a spiral line
Example3: A mass m connected by a rope moves on frictionless table circularly with uniform  and r. Then one pulls the rope slowly through the center, determine the work done when r changes to r/2.
Solution: No torque, L about o is conserved
Work F  r o m 7

8. Homework
A massless spring (l0=0.2m, k=100N/m) connects mass m=1kg to point o on a horizontal frictionless table. Mass m moves with v0=5m/s ⊥ the spring, and the length of spring becomes l=0.5m after rotating 90°, determine the final velocity v’ and θ.
90°
90° 8

9. Angular quantities for a system
Consider a system of particles (rigid body or not)
The total angular momentum:
The net torque:
Angular momentum theorem for a system:
(about the same origin O)
Valid in a inertial frame or frame of the CM
(about the CM of system) 9

10. For a rigid body rotates about a fixed axis
Rigid body & fixed axis
For a rigid body rotates about a fixed axis
Consider the component along axis
Total angular momentum
Angular momentum theorem for a rigid body:
(rotational theorem)
1) Vector & Component
2) 1-dimensional case 10

11. Conservation of L for system
The total angular momentum of a system remains constant if the net external torque is zero.
Example4: A uniform thin rod (m, l) rotates about fixed axis o with  on frictionless horizontal table, and collides elastically with a resting mass m at its end. Determine the final angular velocity. o
m, l m .
Solution: Elastic collision: 11

12. Rotating about varying axis
Rigid body rotates about a fixed axis
For a body rotating about a symmetry axis
(symmetry axis, through CM)
Angular momentum theorem:
where
may have different directions!
How angular momentum changes?
1) No initial rotation
2) With initial rotation 12

13. Motion of a rapidly spinning top, or a gyroscope
*The spinning top
Motion of a rapidly spinning top, or a gyroscope
External torques are acted by a pair of forces:
We always have L
Only change the direction of W N
It is called precession
Precession of Earth
Bullet, bicycle, … 13

14. Noninertial reference frame
Newton’s first law does not hold in such frames S’ S
To use Newton’s laws, we have to use a trick
Inertial force: a type of fictitious force 14

15. Dynamics in noninertial frame
With considering inertial forces, N-2 is still valid
1) It is not a real force: no object exerts it
2) For rotating frame, also called centrifugal force
(opposite to the centripetal force)
Weight loss 15

16. Solution: Centrifugal force → virtual gravity
Ring form spaceship
Example5: A ring-form spaceship with radius r is rotating to obtain a virtual gravity of g, determine the period.
Solution: Centrifugal force → virtual gravity
r=50m O 16

17. If a body is moving relative to a rotating frame
*The Coriolis effect
If a body is moving relative to a rotating frame
There is another inertial force: Coriolis force
where is the relative velocity
Coriolis effect
River, wind, whirlpool
Falling objects
Foucault pendulum 17

18. *Foucault pendulum 
Earth 1 1 2 2 3 18