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Rigid-Body Rotation rotating and revolving § 9.1–9.2 Fixed-Axis Rotation • Much easier than changing-axis • Hard enough on its own Radians A dimensionless angle measure arc length distance from axis length = dimensionless! length Radian Measurements • Complete cycle = 2pr r • Complete cycle = 2p radians • 1 radian = 57.3° Periodic Processes • You will often encounter radians and angular speed for repeating processes • Not restricted to rotation or circular motion Poll Question What is the equivalent of 180° in radians? A. p/4. B. p/3. C. p/2. D. p. Poll Question What is the equivalent of 45° in radians? A. p/4. B. p/3. C. p/2. D. p. Angular Position 2 s r 1 q • Arc length s • Radius r s • Angle q = r Angular Speed Rate of change of position • Angular speed w dq d s 1 ds = vT w= = = r dt dt r dt r • vT = tangential speed Group Work A particle moves in a circular path of radius r. a) What is its angular displacement q after 2.0 complete rotations? b) What is its path length s after 2.0 complete rotations? c) If it takes time t to complete 2.0 rotations, what is its average tangential speed v? d) If it takes time t to complete 2.0 rotations, what is its average angular speed w? Angular Velocity What is the direction of angular motion? Right-hand rule: • Curl right-hand fingers in the direction of rotation. • Extended right thumb points in the direction of w. • Rotation Axis || w. Angular Acceleration Rate of change of angular velocity dw d2 s 1 d2s a|| a= = 2 r = r 2= r dt dt dt • a|| = tangential acceleration • Valid for a fixed axis of rotation (acceleration about the w axis) Angular Kinematic Formulas Constant a, a || w w = w0 + at q = q0 + w0t + 1/2 at2 w2 = w02 + 2a(q – q0) Note the similarity to the linear kinematic formulas! Poll 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 gentlemen bug's angular speed is A. half the ladybug's B. the same as the ladybug's C. twice the ladybug's D. wicked fast E. impossible to determine Poll 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 gentlemen bug's linear speed is A. half the ladybug's B. the same as the ladybug's C. twice the ladybug's D. wicked fast E. impossible to determine 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 A. in the +x direction B. in the –x direction C. in the +z direction D. in the –z direction E. in the +y direction F. in the –y direction 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 A. in the +x direction B. in the –x direction C. in the +z direction D. in the –z direction E. in the +y direction F. in the –y direction 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 A. in the +x direction B. in the –x direction C. in the +z direction D. in the –z direction E. in the +y direction F. in the –y direction 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 A. in the +x direction B. in the –x direction C. in the +z direction D. in the –z direction E. in the +y direction F. in the –y direction Rigid-Body Rotation moments of inertia § 9.3–9.4 Rolling without slipping Center-of-mass (axis) speed v = rw Rolling without slipping Center-of-mass (axis) acceleration a|| = ra Rolling without slipping Rim centripetal acceleration a = v2/r = w2r Poll Question Which has the greatest kinetic energy? A. A bar rotating at speed w about its long axis. B. A bar rotating at speed w about its middle, perpendicular to its long axis. C. A bar rotating at speed w about its end. D. All of these have the same kinetic energy. E. Cannot be determined. Rotating Kinetic Energy K = 1/2 Iw2 I = moment of inertia (rotational analogue of mass) units? Moment of Inertia Of a particle of mass m, distance r from axis 1/2 Iw2 = 1/2 mv2 • What is I? Poll Question Two cylindrical objects with equal mass and radius are rotated about their axes. Which has the greater moment of inertia? A. A solid cylinder. B. A hollow cylinder. C. Their moments are the same. Moments of Inertia Usually expressed in the form I = CMR2 C depends on the shape (mass distribution) of the object Moments of Inertia Source: Young and Freedman, Table 9-2 (p. 291).

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