Rigid-Body Rotation

1. Rigid-Body Rotation rotating and revolving Ch. 8

2. arc length distance from axis length = dimensionless! length Radians A dimensionless angle measure

3. Radian Measurements • Complete cycle = 2pr r • Complete cycle = 2p radians • 1 radian = 57.3°

4. Periodic Processes • You will often encounter radians and angular speed for repeating processes • Not restricted to rotation or circular motion

5. Question What is the equivalent of 180° in radians? What is the equivalent of 45° in radians?

6. q • Angle q = s r Angular Position • Radius r • Arc length s 2 s r 1

7. Dq D Dt Dt w = = = = 1 Ds r s Dt r vT r Angular Speed Rate of change of angular position • Angular speed w • vT = tangential speed

8. Dw Dt a = = a|| r Angular Acceleration Rate of change of angular velocity • a|| = tangential acceleration • Valid for a fixed axis of rotation(acceleration about the w axis)

9. Whiteboard Work A particle moves in a circular path of radius r. • What is its angular displacement q after 2.0 complete rotations? • What is its path length s after 2.0 complete rotations? • If it takes time t to complete 2.0 rotations, what is its average tangential speed v? • If it takes time t to complete 2.0 rotations, what is its average angular speed w?

10. Extended right thumb points in the direction of w. • Rotation Axis || w. Angular Velocity What is the direction of angular motion? Right-hand rule: • Curl right-hand fingers in the direction of rotation.

11. Question A ladybug sits at the outer edge of a merry-go-round, and a lordbug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second. The lordbug's angular speed is • half the ladybug's • the same as the ladybug's • twice the ladybug's • wicked fast • impossible to determine

12. Question A ladybug sits at the outer edge of a merry-go-round, and a lordbug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second. The lordbug's tangential speed is • half the ladybug's • the same as the ladybug's • twice the ladybug's • wicked fast • impossible to determine

13. Question A ladybug sits at the outer edge of a merry-go-round that is turning and slowing down. At the instant shown, its centripetal acceleration is • in the +x direction • in the –x direction • in the +z direction • in the –z direction • in the +y direction • in the –y direction

14. Question A ladybug sits at the outer edge of a merry-go-round that is turning and slowing down. At the instant shown, the tangential component of the ladybug's (cartesian) acceleration is • in the +x direction • in the –x direction • in the +z direction • in the –z direction • in the +y direction • in the –y direction

15. Question A ladybug sits at the outer edge of a merry-go-round that is turning and slowing down. At the instant shown, the vector expressing its angular velocity is • in the +x direction • in the –x direction • in the +z direction • in the –z direction • in the +y direction • in the –y direction

16. Question A ladybug sits at the outer edge of a merry-go-round that is turning and slowing down. At the instant shown, the vector expressing its angular acceleration is • in the +x direction • in the –x direction • in the +z direction • in the –z direction • in the +y direction • in the –y direction

17. Angular Kinematic Formulas For constant a, a || w w = w0 + at q = q0 + w0t + 1/2at2 w2 = w02 + 2a(q– q0) Note the similarity to the linear kinematic formulas.

18. Rigid-Body Motion rotation + translation

19. Rolling without slipping Center-of-mass speed v = rw

20. Rolling without slipping Center-of-mass acceleration a|| = ra

21. Rolling without slipping Rim centripetal acceleration a = v2/r = w2r