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

2. 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

3. 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°

4. SCI 340 L23 Rotation
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?
SCI 340 L23 Rotation
Question
What is the equivalent of 180° in radians?
What is the equivalent of 45° in radians?

6. 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

7. 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

8. 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)

9. 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?

10. 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.

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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.

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

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

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

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