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SCI 340 L23 Rotation rotating and revolving
Rigid-Body Rotation rotating and revolving
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Radians A dimensionless angle measure arc length distance from axis
SCI 340 L23 Rotation Radians A dimensionless angle measure arc length distance from axis = dimensionless! length
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Radian Measurements Complete cycle = 2pr Complete cycle = 2p radians
SCI 340 L23 Rotation Radian Measurements Complete cycle = 2pr r Complete cycle = 2p radians 1 radian = 57.3°
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SCI 340 L23 Rotation Periodic Processes You will often encounter radians and angular speed for repeating processes Not restricted to rotation or circular motion
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Question What is the equivalent of 180° in radians?
SCI 340 L23 Rotation Question What is the equivalent of 180° in radians? What is the equivalent of 45° in radians?
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Angular Position Radius r Arc length s s Angle q = r s r 2 1 q
SCI 340 L23 Rotation Angular Position 2 s r 1 q Radius r Arc length s Angle q = s r
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Angular Speed Rate of change of angular position Angular speed w Dq Dt
SCI 340 L23 Rotation Angular Speed Rate of change of angular position Angular speed w Dq Dt w = D Dt s r = 1 r Ds Dt = vT r = vT = tangential speed
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Angular Acceleration Rate of change of angular velocity Dw Dt a = a||
SCI 340 L23 Rotation Angular Acceleration Rate of change of angular velocity Dw Dt a = a|| r = a|| = tangential acceleration Valid for a fixed axis of rotation (acceleration about the w axis)
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Whiteboard Work A particle moves in a circular path of radius r.
SCI 340 L23 Rotation 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?
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Angular Velocity What is the direction of angular motion?
SCI 340 L23 Rotation 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.
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SCI 340 L23 Rotation 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
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SCI 340 L23 Rotation 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
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SCI 340 L23 Rotation Poll 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
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SCI 340 L23 Rotation 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
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SCI 340 L23 Rotation 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 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
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SCI 340 L23 Rotation 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 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
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Angular Kinematic Formulas
SCI 340 L23 Rotation Angular Kinematic Formulas For 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.
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SCI 340 L23 Rotation rotation + translation
Rigid-Body Motion rotation + translation
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Rolling without slipping
SCI 340 L23 Rotation Rolling without slipping Center-of-mass speed v = rw
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Rolling without slipping
SCI 340 L23 Rotation Rolling without slipping Center-of-mass acceleration a|| = ra
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Rolling without slipping
SCI 340 L23 Rotation Rolling without slipping Rim centripetal acceleration a = v2/r = w2r
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