1. Rotational Motion About a Fixed Axis
Chapter 10
Rotational Motion About a Fixed Axis

2. A body with a definite shape that doesn’t change
Rigid body
A body with a definite shape that doesn’t change
1. Vibrating or deforming can be ignored
2. Distance between particles does not change
Motion of a rigid body can be expressed as
Translational motion of its CM
+ Rotational motion about its CM
Pure rotational motion: all move in circles
centers of these circles all lie on a line
Axis of rotation
Fixed axis

3. Angular coordinate / Angle
Angular quantities
Angular coordinate / Angle
Angular velocity y x o  R s
Angular acceleration
Radial acceleration
Frequency
Period

4. Hard drive
Example1: The platter of hard disk rotates 5400rpm. a) Angular velocity; b) Speed of reading head located 3cm from the axis;
c) Acceleration at that point; d) How many bits the writing head writes per second at that point if 1 bit needs 0.5µm length.
Solution: a)
angular velocity
b) speed
c) acceleration d)

5. Useful Similarities
Notice the similarities in different motion
Uniformly accelerated rotational motion
Analogous thinking is very helpful

6. Vector nature of angular quantities
Angular velocity and acceleration → vectors 
Points along the axis, follows the right-hand rule
How does change?
Angular acceleration
pulley
spinning top

7. What causes acceleration of rotational motion?
Rotational dynamics
What causes acceleration of rotational motion?
Force: magnitude, direction
and point of action
Push a door
Archimedes’ Lever
“Give me a fulcrum, and I shall move the world. ” —— Archimedes
Lever arm: the perpendicular distance from the axis to the line along which the force acts 7

8. Torque about fixed axis
The effect of force → angular acceleration
Torque = force × lever arm R F
R⊥: lever arm or moment arm
Net torque causes acceleration of rotational motion
positive rotational direction
Balance of rigid body
Play on a seesaw 8

9. Torque and balance
Example2: A 15kg mass locates 20cm from the axis of massless lever. Determine: a) torque on the lever; b) Force required respectively to balance the lever.
Solution: a)
15cm
20cm b)
5cm
25cm
150N
What about F4 ?

10. A particle mi in rigid body
Rotational theorem
A particle mi in rigid body fi Fi mi Ri
Rotational inertia
Net external torque
or moment of inertia
Rotational theorem about fixed axis 10

11. Properties of Rotational theorem
1) Only external torques are effective
Sum of the internal torques is 0 from N-3
2) Rotational equivalent of Newton’s second law.
3) Analogy of rotational and translational motion
change in motion
cause of the change
inertia of motion 11

12. Determining rotational inertia
several particles
continuous object
How mass is distributed with respect to the axis
Example3: Rotational inertia of 3 particles fixed on a massless rod about a, b, c axis.
Solution: c b a m 2m 3m l 12

13. Solution: Choose a segment dx
Uniform thin rod
Example4: Rotational inertia of uniform thin rod with mass m and length l. a) Through center;
Solution: Choose a segment dx C dx dm x o
b) Through end o
Typical result, should be memorized 13

14. Example5: Uniform thin hoop (mass m, radius R)
Uniform circles
Example5: Uniform thin hoop (mass m, radius R)
Solution: Choose a segment dm R dm
Example6: Uniform disk/cylinder (m, R)
Solution: Choose a hoop dm r dr
Homework: Uniform sphere (m, R) (P246) 14

15. Parallel-axis theorem
If I is the rotational inertia about any axis, and IC is the rotational inertia about an axis through the CM, and parallel to the first but a distance l, then I Ic l C M o
Proof:
IC is always less than other I of parallel axes 15

16. Perpendicular-axis theorem
The sum of the rotational inertia of a plane about any two perpendicular axes in the plane
is equal to the rotational inertia about an axes through the point of intersection ⊥ the plane. x y z
1) Only for plane figures or
2-dimensional bodies
2) x ⊥ y ⊥ z and intersect at one point
3) Try to prove it by yourself 16

17. Application of two theorems
Example7: Rotational inertia of uniform thin disk about the line of diameter
Solution: You can choose a dm dm
or apply the ⊥-axis theorem
Example8: Thin hoop about a tangent line
Solution: Perpendicular-axis theorem
Parallel-axis theorem 17

18. Massive Pulley
Example9: A box is hanging on a pulley by massless rope, then the system starts to move without slip, determine the angular acceleration and tension. mg 
Solution: T = mg ? m M R 
Free-body diagram T
Rotational theorem:
Newton’s second law:
No slip motion:
Two boxes? 18

19. Solution: a) Gravity → torque
Rotating rod
Example10: A uniform rod of mass M and length l can pivot freely about axis o, released horizontally. Determine: a) α; b) aC; c) force acted by the axis.
Solution: a) Gravity → torque F o Mg C
b) Acceleration of CM
a of the end? c) 19

20. Rotational theorem can be written in terms of L
Angular momentum
Linear momentum p = m v
rotational analog
Angular momentum
Rotational theorem can be written in terms of L
The rate of change in angular momentum of a rigid body is equal to the net torque applied on it.
Comparing with Newton’s second law 20

21. Conservation of angular momentum
Torque & angular momentum
1) It is valid even if I changes
2) Valid for a fixed axis or axis through its CM
Law of conservation of angular momentum:
The total angular momentum of a rotating body remains constant if the net external torque is zero.
1) It holds for inertial frames or frame of CM
2) One of fundamental laws of conservation 21

22. Examples in sports
Figure skating
Diving

23. Helicopter

24. Falling cat
A falling cat can adjust his posture to avoid injury, how to make it?
Angular momentum is conserved
1) Bend his body
2) Rotate his upper part to proper position about blue axis, meanwhile the lower part rotates a less angle
3) Rotate his lower part to proper position about the red axis
4) Get the work done

25. A. increase; B. decrease; C. remain constant
Rotating disk
Example11: A disk is rotating about its center axis, and two identical bullets hit into it symmetrically. The angular velocity of system will ________
A. increase; B. decrease; C. remain constant
Solution: Total angular momentum of the system is conserved  . o
Total rotational inertia increases
So angular velocity will decrease
What if two opposite forces act on the disk? 25

26. Man on rotating platform
Example12: A platform is rotating about its center axis, and a man standing on it (treat as a particle) starts to move. How does  change if he goes: a) to point o; b) along the edge with relative speed v.
Solution: a) Conservation of angular momentum  I o R m
b) choose a positive direction 26

27. Solution: Conservation of total L
Hits on a rod
Example13: A bullet hits into a hanged uniform rod, determine the angular velocity after collision.
Solution: Conservation of total L o
M, l A r mv
Notice: Momentum is not conserved in general!
Require: Force acted by axis remains constant
Only if the bullet hits on position r = 2l / 3 27

28. Rotational kinetic energy
Kinetic energy in translational motion
rotational motion
Total Ek is the sum of Ek of all particles
Work done on a rotating body:
Power 28

29. Energy in rotational motion
Rotational theorem:
Work-energy principle:
Comparing with
Total mechanical energy is conserved in rotational motion if only conservative forces do work.
Potential energy of gravity: 29

30. Solution: Distance oC=l/6
Rotating rod
Example14: A uniform rod (M, l ) can pivot freely about axis o (Ao=l/3), and it is released horizontally. Determine ω at the vertical position.
Solution: Distance oC=l/6 A o . . C
Rotational inertia
Conservation of mechanical energy
Any position?
α = ? 30

31. For general motion of a rigid body
Translational motion of its CM:
+ Rotational motion about its CM:
The total kinetic energy
(Proved in P259)
Examples: 31

32. A typical motion: rolling without slipping
Rolling motion
A typical motion: rolling without slipping
To make sure vC
Relationship between translational and rotational motion
Valid only if no slipping
Motion of pulley, tire, … 32

33. Rolling down an incline
Example15: A uniform cylinder (m, R) rolls down an incline without slipping, determine its speed if the CM moves a vertical height H.
Solution: Conservation of energy
where
Which is faster?
Comparing with sliding: 33

34. Forces acting on the cylinder
Dynamics in rolling F mg N
Forces acting on the cylinder
Translational motion of CM
Rotational motion about CM
We can obtain: aC = 2gsin /3， F = mgsin /3
Static friction causes the rolling motion
It also rearranges the kinetic energy

35. Challenging question
A stick (M, l ) stands vertically on a frictionless table, then it falls down. Describe the motion of its CM, and of each end. 35