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).