1. Rotational Motion

2. Axis of rotation
Angular displacement
𝜃= 𝑥 𝑟
θ is measured in radians
1 rev = 360° = 2π rad
Radians is a “ghost” unit

3. Angular velocity
𝜔= 𝑣 𝑟
𝜔 is measured in rad/s
𝜔= 𝑑𝜃 𝑑𝑡

4. Angular acceleration
𝛼= 𝜔 𝑟
𝛼 is measured in rad/s2
𝛼= 𝑑𝜔 𝑑𝑡

5. Rotation with constant angular acceleration
𝑣= 𝑣 0 +𝑎𝑡 𝜔= 𝜔 0 +𝛼𝑡
𝑥= 𝑣 0 𝑡+ 1 2 𝑎 𝑡 2 𝜃= 𝜔 0 𝑡+ 1 2 𝛼 𝑡 2
𝑣 2 = 𝑣 𝑎𝑥 𝜔 2 = 𝜔 𝛼𝜃
𝑥= 𝑣 0 +𝑣 𝑡 𝜃= 𝜔 0 +𝜔 𝑡

6. Period of rotation
Determine the period of a car travelling at 35 m/s around a track with a radius of 75 m.
X s 1 rev = 1 s 35 m × 2π 75 m 1 rev
𝑇= 2𝜋 𝑇= 2𝜋𝑟 𝑣
𝑇=13.5 s

7. Determine the period of a rotating disc with an angular speed of 4
Determine the period of a rotating disc with an angular speed of 4.3 rad/s
X s 1 rev = 1 s 4.3 rad × 2π rad 1 rev
𝑇= 2𝜋 𝑇= 2𝜋 𝜔
𝑇=1.46 s

8. Acceleration A rotating object 𝑎 𝑡 =𝛼𝑟 𝑎 𝑟 = 𝑣 2 𝑟 = 𝜔 2 𝑟
Does have centripetal acceleration
Might have linear (tangential) acceleration
𝑎 𝑡 =𝛼𝑟
𝑎 𝑟 = 𝑣 2 𝑟 = 𝜔 2 𝑟

9. Kinetic energy
If we treat a rotating object as a series of particles
𝐾= 1 2 𝑚 1 𝑣 𝑚 2 𝑣 𝑚 3 𝑣 3 2 +…
𝐾= 𝑚 𝑣 𝑖 2
𝐾= 𝑚 𝜔 𝑟 𝑖 2
𝐾= 𝑚 𝑟 𝑖 2 𝜔 2

10. 𝐼= 𝑚 𝑟 𝑖 2
I is the moment of inertia or rotational inertia
I is the rotational analog for mass
𝐾= 1 2 𝐼 𝜔 2

11. 𝐼= 𝑚 𝑟 𝑖 2
If a body is continuous
𝐼= 𝑟 2 𝑑𝑚

12. Objects for which you need to be able to determine rotational inertia
Thin hoop rotated about its center
Thin rod rotated about an axis through its center and perpendicular to the rod
Solid disk rotated about its center

13. Thin hoop rotated about its center
𝐼= 𝑟 2 𝑑𝑚
𝐼= 𝑅 2 𝑑𝑚
𝐼=𝑚 𝑅 2

14. ALL moments of inertia will have the same form:
𝐼=𝑥𝑀 𝑅 2
Where x ≤ 1

15. Thin rod rotated about an axis through its center and perpendicular to the rod
𝐼= 𝑟 2 𝑑𝑚

16. Consider a thin slice of width dr
The mass of this slice,
dm, is
𝑑𝑚=𝜇𝑑𝑟
Where μ is linear density
𝜇= 𝑀 𝐿
𝑑𝑚= 𝑀 𝐿 𝑑𝑟

17. 𝐼= 𝑀 3𝐿 𝐿 3 4 𝐼= 𝑀 𝐿 2 12 or 𝐼= 1 12 𝑀 𝐿 2 𝐼= 𝑟 2 𝑀 𝐿 𝑑𝑟 𝐼= 𝑀 𝐿 𝑟 2 𝑑𝑟
𝐼= 𝑟 2 𝑀 𝐿 𝑑𝑟
𝐼= 𝑀 𝐿 𝑟 2 𝑑𝑟
𝐼= 𝑀 𝐿 − 𝐿 2 𝐿 2 𝑟 2 𝑑𝑟
𝐼= 𝑀 𝐿 𝑟 − 𝐿 2 𝐿 2
𝐼= 𝑀 3𝐿 𝐿 − − 𝐿 2 3
𝐼= 𝑀 3𝐿 𝐿 3 8 − − 𝐿 3 8
𝐼= 𝑀 3𝐿 𝐿 3 4
𝐼= 𝑀 𝐿 2 12 or
𝐼= 1 12 𝑀 𝐿 2

18. Solid disk (or cylinder) rotated about its center
𝐼= 𝑟 2 𝑑𝑚

19. Consider a thin ring a distance r from the center with a width of dr
The mass of this slice,
dm, is
𝑑𝑚=𝜎𝑑𝑟
Where σ is areal density
𝜎= 𝑀 𝐴
𝑑𝑚= 𝑀 𝐴 𝑑𝐴

20. 𝑑𝑚= 𝑀 𝐴 𝑑𝐴
𝑑𝑚= 𝑀 𝜋 𝑅 2 𝑑𝐴
Area = ?
“Uncurl” the slice
𝑑𝐴=2𝜋𝑟𝑑𝑟
𝑑𝑚= 𝑀 𝜋 𝑅 2 2𝜋𝑟𝑑𝑟
𝐼= 𝑟 2 𝑑𝑚
𝐼= 𝑟 2 𝑀 𝜋 𝑅 2 2𝜋𝑟𝑑𝑟 dr
2πr

21. 𝐼= 𝑟 2 𝑀 𝜋 𝑅 2 2𝜋𝑟𝑑𝑟
𝐼= 2𝑀 𝑅 𝑟 3 𝑑𝑟
𝐼= 2𝑀 𝑅 𝑅 𝑟 3 𝑑𝑟
𝐼= 2𝑀 𝑅 𝑟 𝑅
𝐼= 2𝑀 𝑅 2 𝑅 4 4
𝐼= 1 2 𝑀 𝑅 2 dr
2πr

22. A rectangular plate about an axis through its center
𝐼= 𝑟 2 𝑑𝑚
𝑑𝑚= 𝑀 𝐴 𝑑𝐴

23. Consider a small piece of the plate of size da × db
𝑑𝑚= 𝑀 𝐴 𝑑𝐴
𝑑𝑚= 𝑀 𝑎𝑏 𝑑𝐴
𝑑𝑚= 𝑀 𝑎𝑏 𝑑𝑎𝑑𝑏
𝐼= 𝑟 2 𝑀 𝑎𝑏 𝑑𝑎𝑑𝑏
𝐼= 𝑀 𝑎𝑏 𝑟 2 𝑑𝑎𝑑𝑏
𝐼= 𝑀 𝑎𝑏 𝑟 𝑏 𝑟 𝑎 2 𝑑𝑎𝑑𝑏

24. 𝐼= 𝑀 𝑎𝑏 𝑟 𝑏 𝑟 𝑎 2 𝑑𝑎𝑑𝑏
𝐼= 𝑀 𝑎𝑏 − 𝑏 2 𝑏 2 𝑟 𝑏 2 − 𝑎 2 𝑎 2 𝑟 𝑎 2 𝑑𝑎𝑑𝑏
𝐼= 𝑀 𝑎𝑏 − 𝑏 2 𝑏 2 𝑟 𝑏 𝑟 𝑎 3 3 𝑏 − 𝑎 2 𝑎 2 𝑑𝑎𝑑𝑏
𝐼= 𝑀 𝑎𝑏 − 𝑏 2 𝑏 2 𝑟 𝑏 𝑎 𝑏− − 𝑎 𝑏 𝑑𝑎𝑑𝑏
𝐼= 𝑀 𝑎𝑏 − 𝑏 2 𝑏 2 𝑟 𝑏 𝑎 3 𝑏 12 𝑑𝑎𝑑𝑏

25. 𝐼= 𝑀 𝑎𝑏 − 𝑏 2 𝑏 2 𝑟 𝑏 𝑎 3 𝑏 12 𝑑𝑎𝑑𝑏
𝐼= 𝑀 𝑎𝑏 𝑎 3 𝑏 𝑟 𝑏 3 3 𝑎 − 𝑏 2 𝑏 2
𝐼= 𝑀 𝑎𝑏 𝑎 3 𝑏 𝑏 𝑎− − 𝑏 𝑎
𝐼= 𝑀 𝑎𝑏 𝑎 3 𝑏 𝑏 3 𝑎 12
𝐼= 1 12 𝑀 𝑎 2 + 𝑏 2

26. Annular cylinder about an axis through its center
𝐼= 𝑟 2 𝑑𝑚
𝑑𝑚= 𝑀 𝐴 𝑑𝐴
𝐼= 𝑀 𝐴 𝑟 2 𝑑𝐴
𝐼= 𝑀 𝜋 𝑟 2 2 −𝜋 𝑟 𝑟 2 𝑑𝐴
𝐼= 𝑀 𝜋 𝑟 2 2 −𝜋 𝑟 𝑟 2 2𝜋𝑟 𝑑𝑟
𝐼= 2𝜋𝑀 𝜋 𝑟 2 2 −𝜋 𝑟 𝑟 3 𝑑𝑟

27. 𝐼= 2𝜋𝑀 𝜋 𝑟 2 2 −𝜋 𝑟 𝑟 3 𝑑𝑟
𝐼= 2𝑀 𝑟 2 2 − 𝑟 𝑟 1 𝑟 2 𝑟 3 𝑑𝑟
𝐼= 2𝑀 𝑟 2 2 − 𝑟 𝑟 𝑟 1 𝑟 2
𝐼= 𝑀 2 𝑟 2 2 − 𝑟 𝑟 2 4 − 𝑟 1 4
𝐼= 𝑀 2 𝑟 2 2 − 𝑟 𝑟 2 2 − 𝑟 𝑟 𝑟 1 2
𝐼= 1 2 𝑀 𝑟 𝑟 2 2

28. Thin rod rotated about an axis through its center and perpendicular to the rod
𝐼= 1 12 𝑀 𝐿 2
end ?

29. Consider a thin slice of width dr
The mass of this slice,
dm, is
𝑑𝑚=𝜇𝑑𝑟
Where μ is linear density
𝜇= 𝑀 𝐿
𝑑𝑚= 𝑀 𝐿 𝑑𝑟

30. 𝐼= 𝑟 2 𝑀 𝐿 𝑑𝑟
𝐼= 𝑀 𝐿 𝑟 2 𝑑𝑟
𝐼= 𝑀 𝐿 0 𝐿 𝑟 2 𝑑𝑟
𝐼= 𝑀 𝐿 𝑟 𝐿
𝐼= 𝑀 3𝐿 𝐿 3 − 0 3
𝐼= 𝑀 3𝐿 𝐿 3
𝐼= 𝑀 𝐿 2 3 or
𝐼= 1 3 𝑀 𝐿 2

31. The rotational inertia for a thin rod about and axis through its center and perpendicular to the rod was 𝑀 𝐿 2 . Determine the rotational inertia about a parallel axis through the end of the rod.
𝐼= 𝐼 𝑐𝑜𝑚 +𝑀 𝑑 (Parallel axis theorem)
𝐼= 1 12 𝑀 𝐿 2 +𝑀 𝑑 2
𝐼= 1 12 𝑀 𝐿 2 +𝑀 𝐿 2 2
𝐼= 1 12 𝑀 𝐿 𝑀 𝐿 2
𝐼= 1 12 𝑀 𝐿 𝑀 𝐿 2
𝐼= 4 12 𝑀 𝐿 2 = 1 3 𝑀 𝐿 2

32. Torque is the ability of a force to cause rotation in an object
𝜏=𝑟𝐹 sin 𝜑
r is the distance from the axis of rotation to the point of application of the force
φ is the smaller of the two angles made by r and F

34. Torques are additive
Torques in the clockwise direction are negative, counterclockwise are positive

35. Newton’s second law for rotation
Σ𝜏=𝐼𝛼
Unit for torque
𝑀 𝑅 2 𝑟𝑎𝑑 𝑠 2 =𝑘𝑔∙ 𝑚 𝑠 2
𝑘𝑔∙𝑚 1 𝑠 2 =𝑁
𝑁∙𝑚 (NOT joules!!!!!!!)