1. Rotational Kinematics Chapter 8

2. Expectations After Chapter 8, students will:  understand and apply the rotational versions of the kinematic equations.  be able to mathematically associate tangential variables with corresponding angular ones  understand and apply the concept of total acceleration in rotational motion  state and use the principle of rolling motion

3. A Brief Review from Chapter 5 Angular displacement: Units: radians (rad)

4. A Brief Review from Chapter 5 Average angular velocity: units: rad/s or: degrees/s, rev/min, etc.

5. Angular Acceleration Average angular acceleration: units: rad/s 2 or: degrees/s 2, rev/min 2, etc.

6. Rotational Kinematic Equations Definition of average angular velocity:

7. Rotational Kinematic Equations Definition of average angular acceleration:

8. Rotational Kinematic Equations A previous result:

9. Rotational Kinematic Equations Solve definition of average acceleration for t: Substitute into a previous result:

10. Comparison: Kinematic Equations Rotational Linear (  = constant) (a = constant)

11. Comparison: Kinematic Equations Same equations, (some) different variables Position, displacement: x  Time: t t Velocity, speed: v  Acceleration: a 

12. Average angular velocity is the angular displacement divided by the time interval in which it occurred. Angular and Tangential Velocity

13. From the definition of linear acceleration: From the definition of angular acceleration: Combining: Angular and Tangential Acceleration

14. From chapter 5: But: Substituting: Angular Velocity, Centripetal Acceleration

15. The tangential and centripetal accelerations are vector components of the total acceleration. Total Acceleration

16. When a circular, cylindrical, or spherical object rolls without slipping over a surface: Rolling Motion: Velocity linear speed of axle wheel radius angular speed of wheel

17. When a circular, cylindrical, or spherical object rolls without slipping over a surface: Rolling Motion: Acceleration linear acceleration of axle wheel radius angular acceleration of wheel

18. Angular displacement, , is not a vector quantity. the reason: addition of angular displacements is not commutative. Where you end up depends on the order in which the angular displacements (rotations) occur. Angular Vectors

19. Angular velocity, , and angular acceleration, , are vectors. Magnitudes: and Directions: Parallel to the axis of rotation, and in the direction given by the right-hand rule: Angular Vectors

20. Right-hand rule direction for  : Angular Vectors

21. Right-hand rule direction for  :  Also parallel to axis of rotation  Same direction as change in  vector  Same direction as  if  is increasing in magnitude  Opposite direction from  if  is decreasing in magnitude Angular Vectors