837
Chapter 9
Rotation
Conceptual Problems
1
•
Two points are on a disk that is turning about a fixedaxis through its
center, perpendicular to the disk and through its center, at increasing angular
velocity. One point is on the rim and the other point is halfway between the rim
and the center. (
a
) Which point moves the greater distance in a given time?
(
b
) Which point turns through the greater angle? (
c
) Which point has the greater
speed? (
d
) Which point has the greater angular speed? (
e
) Which point has the
greater tangential acceleration? (
f
) Which point has the greater angular
acceleration? (
g
) Which point has the greater centripetal acceleration?
Determine the Concept
(
a
) Because
r
is greater for the point on the rim, it moves
the greater distance. (
b
) Both points turn through the same angle. (
c
) Because
r
is
greater for the point on the rim, it has the greater speed. (
d
) Both points have the
same angular speed. (
e
) Both points have zero tangential acceleration. (
f
) Both
have zero angular acceleration. (
g
) Because
r
is greater for the point on the rim, it
has the greater centripetal acceleration.
2
•
True or false:
(
a
) Angular speed and linear speed have the same dimensions.
(
b
) All parts of a wheel rotating about a fixed axis must have the same angular
speed.
(
c
) All parts of a wheel rotating about a fixed axis must have the same angular
acceleration.
(
d
) All parts of a wheel rotating about a fixed axis must have the same centripetal
acceleration.
(
a
) False. Angular speed has the dimensions
[ ]
T
1
whereas linear speed has
dimensions
[]
T
L
.
(
b
) True. The angular speed of all points on a wheel is
d
θ
/
dt.
(
c
) True. The angular acceleration of all points on the wheel is
d
ω
/
dt.
(
d
) False. The centripetal acceleration at a point on a rotating wheel is directly
proportional to its distance from the center of the wheel
3
•
Starting from rest and rotating at constant angular acceleration, a disk
takes 10 revolutions to reach an angular speed
. How many additional
revolutions at the same angular acceleration are required for it to reach an angular
speed of 2
? (
a
) 10 rev, (
b
) 20 rev, (
c
) 30 rev, (
d
) 40 rev, (
e
) 50 rev?
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838
Picture the Problem
The constantacceleration equation that relates the given
variables is
θ
α
ω
ω
Δ
+
=
2
2
0
2
. We can set up a proportion to determine the number
of revolutions required to double
ω
and then subtract to find the number of
additional revolutions to accelerate the disk to an angular speed of 2
.
Using a constantacceleration
equation, relate the initial and final
angular velocities to the angular
acceleration:
θ
α
ω
ω
Δ
+
=
2
2
0
2
or, because
2
0
ω
= 0,
θ
α
ω
Δ
=
2
2
Let
Δ
θ
10
represent the number of
revolutions required to reach an
angular speed
:
10
2
2
θ
α
ω
Δ
=
(1)
Let
Δ
2
represent the number of
revolutions required to reach an
angular speed
:
( )
ω
θ
α
ω
2
2
2
2
Δ
=
(2)
Divide equation (2) by equation (1)
and solve for
Δ
2
:
( )
10
10
2
2
2
4
2
θ
θ
ω
ω
θ
ω
Δ
=
Δ
=
Δ
The number of
additional
revolutions
is:
()
rev
30
rev
10
3
Δ
3
Δ
Δ
4
10
10
10
=
=
=
−
θ
θ
θ
and
)
(
c
is correct.
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 Spring '12
 Goussiou
 Physics

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