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257
Rotational Motion
CHAPTER OUTLINE
10.1
Angular Position, Speed,
and Acceleration
10.2
Rotational
Kinematics
The Rigid
Object Under Constant
Angular Acceleration
10.3
Relations Between
Rotational and
Translational Quantities
10.4
Rotational Kinetic Energy
10.5
Torque and the Vector
Product
10.6
The Rigid Object in
Equilibrium
10.7
The Rigid Object Under a
Net Torque
10.8
Angular Momentum
10.9
Conservation of Angular
Momentum
10.10
Precessional Motion of
Gyroscopes
10.11
Rolling Motion of Rigid
Objects
10.12
Context
Connection
Turning the
Spacecraft
ANSWERS TO QUESTIONS
Q10.1
1 rev/min, or
π
30
rad/s. Into the wall (clockwise rotation).
α
=
0.
FIG. Q10.1
Q10.2
The speedometer will be inaccurate. The speedometer measures the
number of revolutions per second of the tires. A larger tire will travel
more distance in one full revolution as 2
r
.
Q10.3
The object will start to rotate if the two forces act along different lines. Then the torques of the forces
will not be equal in magnitude and opposite in direction.
Q10.4
A quick flip will set the hard–boiled egg spinning faster and more smoothly. The raw egg loses
mechanical energy to internal fluid friction.
Q10.5
No, only if its angular momentum changes.
Q10.6
IM
R
CM
=
2
,
R
CM
=
2
,
R
=
1
3
2
,
R
CM
=
1
2
2
Q10.7
Since the source reel stops almost instantly when the tape stops playing, the friction on the source
reel axle must be fairly large. Since the source reel appears to us to rotate at almost constant angular
velocity, the angular acceleration must be very small. Therefore, the torque on the source reel due to
the tension in the tape must almost exactly balance the frictional torque. In turn, the frictional torque
is nearly constant because kinetic friction forces don’t depend on velocity, and the radius of the axle
where the friction is applied is constant. Thus we conclude that the torque exerted by the tape on
the source reel is essentially constant in time as the tape plays.
continued on next page
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Rotational Motion
As the source reel radius
R
shrinks, the reel’s angular speed
ω
=
v
R
must increase to keep the
tape speed
v
constant. But the biggest change is to the reel’s moment of inertia. We model the reel as
a roll of tape, ignoring any spool or platter carrying the tape. If we think of the roll of tape as a
uniform disk, then its moment of inertia is
IM
R
=
1
2
2
. But the roll’s mass is proportional to its base
area
π
R
2
. Thus, on the whole the moment of inertia is proportional to
R
4
. The moment of inertia
decreases very rapidly as the reel shrinks!
The tension in the tape coming into the readandwrite heads is normally dominated by
balancing frictional torque on the source reel, according to
TR
≈
τ
friction
. Therefore, as the tape plays
the tension is largest when the reel is smallest. However, in the case of a sudden jerk on the tape, the
rotational dynamics of the source reel becomes important. If the source reel is full, then the moment
of inertia, proportional to
R
4
, will be so large that higher tension in the tape will be required to give
the source reel its angular acceleration. If the reel is nearly empty, then the same tape acceleration
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 Fall '07
 MahaAshourAbdalla
 Acceleration, Energy, Kinetic Energy

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