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1
Rotational Motion and
Equilibrium
Chapter 8
Translational and Rotational
Motion
•
The most general motion of a
rigid body
can be analyzed as a
translation
of the center of mass plus a
rotation
about its center of
mass.
Rolling Without Slipping
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Rolling Without Slipping
Linear velocity of the center of mass is:
v
CM
= (s
f
– s
i
)/
∆
t =
∆
s/
∆
t
But
∆
s = r
∆θ
b
v
CM
= r(
∆θ
/
∆
t)
Thus
v
CM
= r
ϖ
Acceleration of the center of mass is:
a
CM
=
∆
v
CM
/
∆
t = r
∆ϖ
/
∆
t
a
CM
= r
α
Example 81
2
v
CM
v
CM
R = 0.12m
V
CM
= 0.10m/s
∆
t = 2.0s
Find
ϖ
:
ϖ
= v
CM
/r = [0.10m/s]/0.12m = 0.083 rad/s
Find
∆
θ:
∆
θ =
ϖ∆
t = [0.083rad/s][2.0s] =
1.7 rad
(
π
/2 = 1.57 rad)
Newton’s 2’
nd
Law of Motion
Applied to Rotational Motion
c
A
force
is required to produce a change in
rotational motion.
c
But the rate of change of motion depends not
only on the magnitude and direction of the
force, but also on the socalled
moment arm
c
The
moment arm
is the perpendicular
distance from the line of force to the axis of
rotation
3
Definition of Moment Arm
and Torque
c
The moment arm, r
⊥
, is the
perpendicular
distance from the
extension of the force vector to the axis
of rotation.
c
r
⊥
= r·sin(θ)
c
Torque is defined to be a force “F”
acting through a moment arm “r
⊥
”
τ ≡
F·r
⊥
= F·r·sin(
θ
)
c
θ
= angle between
r
and
F
c
SI Units are N·m
c
US Units are ft·lb
c
Torque results in rotational motion
F
r
r
Figure 83
Torque and Moment Arm
More Definitions
c
Torque is a vector quantity
c
The direction is
perpendicular to the plane
determined by the moment arm and the force
c
The torque is taken to be
positive
if the turning
tendency of the force is
counterclockwise
,
c
If a torque produces a
clockwise
turning tendency,
that torque is said to be
negative
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More About Torque
c
The net torque is the sum of all the torques
produced by all the forces
c
τ
net
=
τ
1
+
τ
2
+
τ
3
+ …
τ
n
c
Remember to account for the direction of the
tendency for rotation
c
Counterclockwise torques are positive
c
Clockwise torques are negative
Example 82
c
r = 0.04 m
F = 600 N
c
r
⊥
= r
.
cos(30
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This note was uploaded on 01/05/2012 for the course PHYS 1401 taught by Professor Jamesr.boyd during the Spring '09 term at Collins.
 Spring '09
 JamesR.Boyd
 Center Of Mass, Mass

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