Chapter 10 – Rotational Kinematics & Energy

1. Chapter 10 – Rotational Kinematics & Energy

2. 10.1 – Angular Position (θ) • In linear (or translational) kinematics we looked at the position of an object (Δx, Δy, Δd…) • We started at a reference point position (xi) and our definition of position relied on how far away from that position we are. • Likewise, our angular position relies on howfar we’ve rotated (Δθ) from a reference line.

3. 10.1 - Angular Position (θ)

4. 10.1 – Angular Position (θ) Degrees and revolutions:

5. 10.1 – Angular Position (θ) Arc length is how far (length) we’ve moved around the circle (arc). Arc length s, measured in radians:

6. 10.1 – Angular Velocity (ω) Linear Velocity Rotational Velocity • Change in linear position of an objet over time is velocity. • How quickly we change position. • Change in angular position of an object over time is angular velocity. • How quickly angle changes.

7. 10.1 – Angular Velocity (ω) Sign Convention:

8. 10.1 – Angular Velocity (ω)

9. A drill bit in a hand drill is turning at 1200 revolutions per minute (1200 rpm). Express this angular speed in radians per second (rad/s) • 2.1 rad/s • 19 rad/s • 125 rad/s • 39 rad/s • 0.67 rad/s

10. w Bonnie Klyde Question 10.1a Bonnie and Klyde I a) same as Bonnie’s b) twice Bonnie’s c) half of Bonnie’s d) one-quarter of Bonnie’s e) four times Bonnie’s Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one complete revolution every2 seconds. Klyde’s angular velocity is:

11. 10.1 – Angular Velocity (ω)

12. 10.1 – Angular Acceleration (α) Linear Acceleration Angular Acceleration Defined as how our angular velocity (ω)changes per unit time. How fast we rotate, does that speed up or slow down? Ex: airplane propellers Really, really, REALLY dumb idea… • Defined as how quickly our velocity is changing per unit time. • When we speed up or slow down.

13. 10.1 – Angular Acceleration (α)

14. 10.1 – Angular Acceleration (α) Sign Convention:

15. 10.2 – Rotational Kinematics Analogies between linear and rotational kinematics:

16. Example 10.2 (pg. 304) If the wheel is given an initial angular speed of 3.40 rad/s and rotates through 1.25 revolutions and comes to rest on the BANKRUPT space, what is the angular acceleration of the wheel (assuming it’s constant)?

18. 10.3 – Tangential Speed What is tangential speed? Imagine riding a merry-go-round, and suddenly letting go before the ride stops. With what velocity will you fly off the merry-go-round?

19. w Bonnie Klyde Question 10.1b Bonnie and Klyde II a) Klyde b) Bonnie c) both the same d) linear velocity is zero for both of them Bonnie sits on the outer rim of amerry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds. Who has the larger linear (tangential) velocity?

20. 10.3 – Centripetal Acceleration of Rotating Object

21. 10.3 – Tangential Acceleration

22. 10.3 – Tangential & Centripetal Acceleration This merry-go-round has BOTH tangentialand centripetal acceleration.

23. 10.1 – 10.3 Summary Arch Length Average Angular Velocity Instantaneous Angular Velocity Period of Rotation Average Angular Acceleration Instantaneous Angular Acceleration

24. 10.1 – 10.3 Summary Linear Kinematics (a = constant) Rotational Kinematics (α = constant)

25. 10.4 - Rolling Motion If a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds:

26. 10.4 – Rolling Motion We may also consider rolling motion to be a combination of pure rotational AND pure translational motion:

27. 10.5 – Rotational Kinetic Energy Linear Kinetic Energy Rotational Kinetic Energy • Depends on an objects angular speed. • Depends on an objects linear speed. • NOT valid for a rotating object because v is different for points of various distances from the axis of rotation.

28. 10.5 – Moment of Inertia • Rotational Kinetic Energy depends on ω2 and r2. AKA the distribution of mass of the rotating object. • Moment of Inertia (I) – • Rotational Kinetic Energy can be rewritten as

29. 10.5 – Moment of Inertia • Moment of Inertia is the distribution of mass throughout the rotating object.

30. 10.5 – Moment of Inertia Calculate the Moment of Inertia of this object.

31. Conceptual Checkpoint 10-2

33. 10.5 – Moment of Inertia of Various Objects Moments of inertia of various regular objects can be calculated (pg. 314):M = total massR = radiusL = Length

34. 10.5 – Kinetic Energy Comparison

35. 10.6 – Conservation of Energy The total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies:

36. Example 10.5 (pg 316) What’s the total Kinetic Energy of this 1.20 kg rolling object?

37. What’s the speed of this object when it reaches the bottom of the ramp?