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Rigid-Body Rotation

Rigid-Body Rotation. rotating and revolving. § 9.1–9.2. arc length. distance from axis. length. = dimensionless !. length. Radians. A dimensionless angle measure. Radian Measurements. Complete cycle = 2 p r. r. Complete cycle = 2 p radians 1 radian = 57.3°. Periodic Processes.

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Rigid-Body Rotation

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  1. Rigid-Body Rotation rotating and revolving § 9.1–9.2

  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. Poll Question What is the equivalent of 180° in radians? • p/4. • p/3. • p/2. • p.

  6. Poll Question What is the equivalent of 45° in radians? • p/4. • p/3. • p/2. • p.

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

  8. 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

  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. dw dt d2 a = = = = 1 dt2 d2s r s dt2 r 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)

  12. Angular Kinematic Formulas 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!

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

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

  15. Poll Question A ladybug sits at the outer edge of a merry-go-round that is turning and slowing down. At the instant shown, the radial 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

  16. Poll 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

  17. Poll 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 her 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

  18. Poll 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 her 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

  19. Rigid-Body Motion rotation + translationmoments of inertia § 9.3–9.4

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

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

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

  23. Poll Question Which has the greatest kinetic energy? • A bar rotating at angular speed w about its long axis. • A bar rotating at speed w about its middle, perpendicular to its long axis. • A bar rotating at speed w about its end. • All of these have the same kinetic energy. • Cannot be determined.

  24. Rotating Kinetic Energy K = 1/2 Iw2 I = moment of inertia(rotational analogue of mass) units?

  25. Moment of Inertia Of a particle of mass m, distance r from axis 1/2 Iw2 = 1/2 mv2 • What is I?

  26. Poll Question Two cylindrical objects with equal mass and radius are rotated about their axes. Which has the greater moment of inertia? A solid cylinder. A hollow cylinder. Their moments are the same.

  27. Moments of Inertia Usually expressed in the form I = cMR2 c depends on the shape (mass distribution) of the object

  28. Moments of Inertia Source: Young and Freedman, Table 9-2 (p. 291).

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