727
Chapter 10
Conservation of Angular Momentum
Conceptual Problems
*1
•
(
a
) True. The cross product of the vectors
A
r
and
B
r
is defined to be
.
ˆ
sin
n
B
A
φ
AB
=
×
r
r
If
A
r
and
B
r
are parallel, sin
φ
=
0
.
(
b
) True. By definition,
ω
r
is along the axis.
(
c
) True. The direction of a torque exerted by a force is determined by the definition of
the cross product.
2
•
Determine the Concept
The cross product of the vectors
A
r
and
B
r
is defined to be
.
ˆ
sin
n
B
A
φ
AB
=
×
r
r
Hence, the cross product is a maximum when sin
φ
=
1. This
condition is satisfied provided
A
r
and
B
r
are
perpendicular
.
correct.
is
)
(
c
3
•
Determine the Concept
L
r
and
p
r
are related according to
.
p
r
L
r
r
r
×
=
From this
definition of the cross product,
L
r
and
p
r
are perpendicular; i.e., the angle between them
is 90
°
.
4
•
Determine the Concept
L
r
and
p
r
are related according to
.
p
r
L
r
r
r
×
=
Because the
motion is along a line that passes through point
P
,
r
= 0 and so is
L
.
correct.
is
)
(
b
*5
••
Determine the Concept
L
r
and
p
r
are related according to
.
p
r
L
r
r
r
×
=
(
a
) Because
L
r
is directly proportional
to
:
p
r
.
doubles
Doubling
L
p
r
r
(
b
) Because
L
r
is directly proportional
to
:
r
r
.
doubles
Doubling
L
r
r
r
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Chapter 10
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6
••
Determine the Concept
The figure shows
a particle moving with constant speed in a
straight line (i.e., with constant velocity
and constant linear momentum). The
magnitude of
L
is given by
rp
sin
φ
=
mv
(
r
sin
φ
).
Referring to the diagram, note that the distance
r
sin
φ
from
P
to the line along which the
particle is moving is constant. Hence,
mv
(
r
sin
φ
) is constant and so
constant.
is
L
r
7
•
False. The net torque acting on a rotating system equals the change in the system’s
angular momentum; i.e.,
dt
dL
=
net
τ
, where
L
=
I
ω
. Hence, if
net
τ
is zero, all we can say
for sure is that the angular momentum (the product of
I
and
ω
) is constant.
If
I
changes,
so must
ω
.
*8
••
Determine the Concept
Yes, you can.
Imagine rotating the top half of your body with
arms flat at sides through a (roughly) 90
°
angle.
Because the net angular momentum of
the system is 0, the bottom half of your body rotates in the opposite direction.
Now
extend your arms out and rotate the top half of your body back.
Because the moment of
inertia of the top half of your body is larger than it was previously, the angle which the
bottom half of your body rotates through will be smaller, leading to a net rotation.
You
can repeat this process as necessary to rotate through any arbitrary angle.
9
•
Determine the Concept
If
L
is constant, we know that the
net
torque acting on
the system is zero. There may be multiple constant or timedependent torques acting on
the system as long as the net torque is zero.
correct.
is
)
(
e
10
••
Determine the Concept
No.
In order to do work, a force must act over some distance. In
each
″
inelastic collision
″
the force of static friction does not act through any distance.
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 Spring '07
 Licini
 Angular Momentum, Kinetic Energy, Momentum

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