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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. - PowerPoint PPT Presentation
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Rigid-Body Rotation
rotating and revolving
§ 9.1–9.2
Radians
A dimensionless angle measure
= dimensionless!lengthlength
arc lengthdistance from axis
Radian Measurements
• Complete cycle = 2r
• Complete cycle = 2 radians
• 1 radian = 57.3°
r
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. /4.
B. /3.
C. /2.
D. .
Poll Question
What is the equivalent of 45° in radians?
A. /4.
B. /3.
C. /2.
D. .
Angular Position
• Radius r
• Arc length s
s
r 1
2
• Angle = sr
Angular Speed
Rate of change of angular position
• Angular speed
• vT = tangential speed
ddt
sr= 1
rdsdt
=ddt
=vT
r=
Whiteboard Work
A particle moves in a circular path of radius r.
a) What is its angular displacement 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 ?
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 .
• Rotation Axis || .
Angular Acceleration
Rate of change of angular velocity
• a|| = tangential acceleration• Valid for a fixed axis of rotation
(acceleration about the axis)
d2
dt2sr= 1
rd2sdt2=
ddt
=a||
r=
Angular Kinematic Formulas
Constant , ||
= 0 + t
= 0 + 0t + 1/2 t2
2 = 02 + 2( – 0)
Note the similarity to the linear kinematic formulas!
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
A. half the ladybug's
B. the same as the ladybug's
C. twice the ladybug's
D. wicked fast
E. impossible to determine
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
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 Motion
rotation + translationmoments of inertia
§ 9.3–9.4
Rolling without slipping
Center-of-mass speed
v = r
Rolling without slipping
Center-of-mass acceleration
a|| = r
Rolling without slipping
Rim centripetal acceleration
a = v2/r = 2r
Poll Question
Which has the greatest kinetic energy?
A. A bar rotating at angular speed about its long axis.
B. A bar rotating at speed about its middle, perpendicular to its long axis.
C. A bar rotating at speed about its end.
D. All of these have the same kinetic energy.
E. Cannot be determined.
Rotating Kinetic Energy
K = 1/2 I2
I = moment of inertia(rotational analogue of mass)
units?
Moment of Inertia
Of a particle of mass m, distance r from axis
1/2 I2 = 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).