This preview shows pages 1–3. Sign up to view the full content.
Physics 204A Class Notes
101
Section 10 – Rotation of Rigid Bodies
What do objects do and why do they do it?
We have a very thorough explanation of this in terms of
kinematics, forces, energy and momentum.
This includes three laws of motion and two conservation
laws.
However, we have only considered the motion of objects as a whole (translational motion).
In
general, objects rotate as well as translate.
We need to revisit the questions “what do objects do?” and
“why do they do it?”
The good news is that we will be able to understand rotation by using our
explanation of translational motion as a model.
In fact, the model is so good that we will only need to
introduce one new law (in the next chapter).
Section Outline
1. The Definitions of the Rotational Variables
2. Rotational Kinematics
3. The Laws of Rotational Motion
4. Some Applications of the Laws of Rotational Motion
5. Rotational Kinetic Energy
1.
The Definitions of the Rotational Variables
The rotational variable will be defined by analogy to the translational variables.
Translational Variables
Rotational Variables
Position
: The location of an object with
respect to a coordinate system.
Angle
: The rotational location of an object with respect to
a coordinate system.
Displacement
: A change in position.
Angular Displacement
: A change in angle.
Velocity
: The rate of displacement.
Angular Velocity
: The rate of angular displacement.
Acceleration
: The rate of change of
velocity.
Angular Acceleration
: The rate of change of angular
velocity.
These definitions can be rewritten mathematically,
Translational Variables
Rotational Variables
Relationship
Position
: x
Angle
:
θ
s
=
r
!
Displacement
: dx
Angular Displacement
: d
θ
ds
=
rd
!
Velocity
:
v
!
dx
dt
Angular Velocity
:
! "
d
#
dt
v
!
ds
dt
=
r
d
"
dt
#
v
t
=
r
$
Acceleration
:
a
!
dv
dt
Angular Acceleration
:
! "
d
#
dt
a
!
dv
dt
"
a
t
=
dv
t
dt
=
r
d
#
dt
"
a
t
=
r
$
a
c
=
v
t
2
r
=
(r
!
)
2
r
"
a
c
=
!
2
r
y
x
x
r
!
y
x
s
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentPhysics 204A Class Notes
102
Example 10.1:
A CD 12.0cm in diameter is placed in a drive.
It reaches 200rpm in 1.20s.
Find
(a)the average angular acceleration and (b)the translational acceleration when the rotation rate
is 100rpm.
Given: r = 0.0600m,
ω
i
= 0,
ω
= 100rpm,
ω
f
= 200rpm,
and
∆
t = 1.20s
Find:
α
= ? and
r
a
=
r
a
c
+
r
a
t
The rotation rates in rpm must be converted to angular speeds in
rad/s,
!
f
=
200
rev
min
( )
2
"
rad
rev
( )
min
60s
#
$
%
&
=
20.9rad / s
and
!
=
100
rev
min
( )
2
"
rad
rev
( )
min
60s
#
$
%
&
=
10.5rad / s
.
(a)Using the definition of angular acceleration,
! "
d
#
dt
$
!
=
%#
%
t
=
#
f
& #
i
%
t
=
20.9
&
0
1.20
$
!
=
17.4rad / s
2
.
(b)The angular acceleration is related to the tangential acceleration,
a
t
=
r
!
=
(0.0600)(17.4)
"
a
t
=
1.05m / s
2
.
The centripetal acceleration is related to the angular velocity,
a
c
=
!
2
r
=
(10.5)
2
(0.0600)
"
a
c
=
6.62m / s
2
.
These are the perpendicular components of the total acceleration.
Using the Pythagorean
Theorem,
a
=
c
2
+
a
t
2
=
(6.62)
2
+
(1.05)
2
!
a
=
6.70m / s
2
.
2. Rotational Kinematics
Recall that when and object is translating with a constant acceleration, we developed a set of
equations to describe the motion called the kinematic equations.
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '09
 Kagan
 Energy, Force, Momentum

Click to edit the document details