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 (x i ) and our definition of position relied on how far away from that position we are. Likewise, our angular position relies on how far 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 s, measured in radians: Arc length is how far (length) we’ve moved around the circle (arc).

6. 10.1 – Angular Velocity (ω) Change in linear position of an objet over time is velocity. – How quickly we change position. Linear VelocityRotational Velocity 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) A)2.1 rad/s B)19 rad/s C)125 rad/s D)39 rad/s E)0.67 rad/s

10. Question 10.1aBonnie and Klyde I Bonnie Klyde 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 every 2 seconds. Klyde’s angular velocity is: 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

11. 10.1 – Angular Velocity (ω)

12. 10.1 – Angular Acceleration (α) Linear Acceleration Defined as how quickly our velocity is changing per unit time. – When we speed up or slow down. 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… Really, really, REALLY dumb idea…

13. 10.1 – Angular Acceleration (α)

14. 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. Question 10.1bBonnie and Klyde II Bonnie Klyde a) Klyde b) Bonnie c) both the same d) linear velocity is zero for both of them 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 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 tangential and 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 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. Rotational Kinetic Energy Depends on an objects angular speed.

28. 10.5 – Moment of Inertia Rotational Kinetic Energy depends on ω 2 and r 2. 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 mass R = radius L = Length

34. 10.5 – Kinetic Energy Comparison Kinetic EnergyLinear QuantityAngular Quantity Speed Variablevω Mass VariablemI Final Equation½ mv 2 ½Iω2½Iω2

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?