SCI 340 L23 Rotation rotating and revolving Rigid-Body Rotation rotating and revolving

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

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°

SCI 340 L23 Rotation Periodic Processes You will often encounter radians and angular speed for repeating processes Not restricted to rotation or circular motion

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?

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

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

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)

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?

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.

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

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

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

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

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

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

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.

SCI 340 L23 Rotation rotation + translation Rigid-Body Motion rotation + translation

Rolling without slipping SCI 340 L23 Rotation Rolling without slipping Center-of-mass speed v = rw

Rolling without slipping SCI 340 L23 Rotation Rolling without slipping Center-of-mass acceleration a|| = ra

Rolling without slipping SCI 340 L23 Rotation Rolling without slipping Rim centripetal acceleration a = v2/r = w2r