1. 轉動力學實驗(一)
Rotational Motion

2. Rigid Object ( 剛體 ) A rigid object is one that is nondeformable
The relative locations of all particles making up the object remain constant.
All real objects are deformable to some extent, but the rigid object model is very useful in many situations where the deformation is negligible Q

3. Angular Position
The arc length s and r are related:
s = q r

4. Conversions Comparing degrees and radians
Converting from degrees to radians

5. Angular Displacement and Angular Speed
Average Angular Speed
Instantaneous Angular Speed
( rad/s or s-1 )

6. Angular Acceleration Average angular acceleration
Instantaneous angular acceleration
( rad/s2 or s-2 )

7. Directions of w and a The directions are given by the right-hand rule
out of the plane
The directions are given by the right-hand rule
into the plane

8. Relationship Between Angular and Linear Quantities
Displacements
Tangential Speeds

9. Relationship Between Angular and Linear Quantities
Tangential Acceleration
Centripetal Acceleration

10. Torque ( 力矩 )
Torque, t , is the tendency of a force to rotate an object about some axis.
Torque is a vector
t = r F sin f = F d

11. 轉動力學實驗原理

13. Moments of Inertia of Various Rigid Objects

14. 理想實驗 (忽略軸承之摩擦力矩) T m mg 轉盤 滑輪 轉動( Rotational Motion) 滑輪 (轉動慣量小，忽略) 轉盤
平移運動(Translational Motion) 砝碼 滑輪 T m
轉動( Rotational Motion)
滑輪 (轉動慣量小，忽略) 轉盤 mg
理想實驗
(忽略軸承之摩擦力矩)

15. 修正軸承之摩擦力矩
平移運動達到平衡狀態
利用掛勾與黏土 T
m0g
轉動達到平衡狀態
m0=掛勾與粘土的質量

16. 軸承之摩擦力矩 T 轉盤 滑輪 轉動( Rotational Motion) 滑輪 (轉動慣量小，忽略) 轉盤
平移運動(Translational Motion)
砝碼+掛勾+黏土 滑輪 T
轉動( Rotational Motion)
滑輪 (轉動慣量小，忽略) 轉盤
軸承之摩擦力矩
(m砝碼+m0)g

17. 角動量守恆實驗

18. Angular Momentum(角動量)
The instantaneous angular momentum of a particle relative to the origin O is defined as the cross product of the particle’s instantaneous position vector and its instantaneous linear momentum

19. Conservation of Angular Momentum

20. Angular Momentum of a System of Particles
The total angular momentum of a system of particles is defined as the vector sum of the angular momenta of the individual particles
Differentiating with respect to time

21. Angular Momentum of a Rotating Rigid Object
To find the angular momentum of the entire object, add the angular momenta of all the individual particles

23. End of Lecture

24. Perpendicular-Axis Theorem (垂直軸定理)
Iz = Ix + Iy ri
Prove it
Hint :

25. 圓盤 R M
利用垂直軸定理，系統的軸之轉動慣量 = ?

26. Parallel-Axis Theorem (平行軸定理)
For an arbitrary axis, the parallel-axis theorem often simplifies calculations
Ip = ICM + MD 2
Ip is about any axis parallel to the axis through the center of mass of the object
ICM is about the axis through the center of mass
D is the distance from the center of mass axis to the arbitrary axis D
C.M. P

27. y 
C.M. P x y
x 
y  y
C.M.
r  △m
x 
C.M. R rP P x rP
r  R

28. rP
r  R
C.M. P x y
x 
y 
r  △m R rP

29. r1
r1  R r2
r2 
C.M. P x y
x 
y 
△m2
r2 
r1 
△m1 r2 R r1

30. C.M. P x y
x 
y 
IP = ICM + MR2
(Parallel-Axis Theorem)

31. Perpendicular-Axis Theorem (垂直軸定理)
Iz = Ix + Iy ri
Prove it
Hint :

32. 圓盤 R M
利用垂直軸定理，系統的軸之轉動慣量 = ?

33. Moments of Inertia of Various Rigid Objects