1. Rotational Kinematics

2. Angular Position, Velocity, and Acceleration

3. Angular Position, Velocity, and Acceleration
Degrees and revolutions:

4. Angular Position, Velocity, and Acceleration
Arc length s, measured in radians:

5. 10-1 Angular Position, Velocity, and Acceleration

6. Angular Position, Velocity, and Acceleration

7. Angular Position, Velocity, and Acceleration

8. Angular Position, Velocity, and Acceleration

9. Rotational Kinematics
If the angular acceleration is constant:

10. Rotational Kinematics
Analogies between linear and rotational kinematics:

11. Connections Between Linear and Rotational Quantities

12. Connections Between Linear and Rotational Quantities

13. Connections Between Linear and Rotational Quantities

14. Connections Between Linear and Rotational Quantities
This merry-go-round has both tangential and centripetal acceleration.

15. 10-4 Rolling Motion
If a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds:

16. 10-4 Rolling Motion
We may also consider rolling motion to be a combination of pure rotational and pure translational motion:

17. Torque
From experience, we know that the same force will be much more effective at rotating an object such as a nut or a door if our hand is not too close to the axis.
This is why we have long-handled wrenches, and why doorknobs are not next to hinges.

18. Torque We define a quantity called torque:
The torque increases as the force increases, and also as the distance increases.
Note:  has the same unit (N . M) as work but it is a very different thing!

19. Torque
Only the tangential component of force causes a torque:

20. Torque
This leads to a more general definition of torque:

21. Torque
If the torque causes a counterclockwise angular acceleration, it is positive; if it causes a clockwise angular acceleration, it is negative.

22. Rotational Kinetic Energy and the Moment of Inertia
For this mass,

23. Rotational Kinetic Energy and the Moment of Inertia
We can also write the kinetic energy as
Where I, the moment of inertia, is given by

24. Rotational Kinetic Energy and the Moment of Inertia
Moments of inertia of various regular objects can be calculated:

25. Conservation of Energy
The total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies:
The second equation makes it clear that the kinetic energy of a rolling object is a multiple of the kinetic energy of translation.

26. Conservation of Energy
If these two objects, of the same mass and radius, are released simultaneously, the disk will reach the bottom first – more of its gravitational potential energy becomes translational kinetic energy, and less rotational.

27. Summary
Describing rotational motion requires analogs to position, velocity, and acceleration
Average and instantaneous angular velocity:
Average and instantaneous angular acceleration:

28. Summary
Period:
Counterclockwise rotations are positive, clockwise negative
Linear and angular quantities:

29. Summary Linear and angular equations of motion: Tangential speed:
Centripetal acceleration:
Tangential acceleration:

30. Rolling motion:
Kinetic energy of rotation:
Moment of inertia:
Kinetic energy of an object rolling without slipping:
When solving problems involving conservation of energy, both the rotational and linear kinetic energy must be taken into account.