homework 18 – RAHMAN, TARIQUE – Due: Apr 7 2008, 3:00 am
1
Question 1, chap 13, sect 3.
part 1 of 1
10 points
A solid cylinder of mass
M
= 29 kg, radius
R
= 0
.
22 m and uniform density is pivoted on
a frictionless axle coaxial with its symmetry
axis. A particle of mass
m
= 3
.
1 kg and initial
velocity
v
0
= 7
.
8 m
/
s (perpendicular to the
cylinder’s axis) Fies too close to the cylinder’s
edge, collides with the cylinder and sticks to
it.
Before the collision, the cylinder was not ro
tating. What is its angular velocity after the
collision?
Correct answer: 6
.
24483 rad
/
s (tolerance
±
1 %).
Explanation:
Basic Concept:
Conservation of Angu
lar Momentum,
L
particle
z
+
L
cylinder
z
= const
.
The axle allows the cylinder to rotate without
friction around a ±xed axis but it keeps this
axis ±xed. Let the
z
coordinate axis run along
this axis of rotation; then the axle may exert
arbitrary torques in
x
and
y
directions but
τ
z
≡
0. Consequently, the
z
componenent of
the angular momentum must be conserved,
L
z
= const, hence when the particle collides
with the cylinder
L
before
z,
part
+
L
before
z,
cyl
=
L
z,
net
=
L
after
z,
part
+
L
after
z,
cyl
.
Before the collision, the cylinder did not
rotate hence
L
before
z,
cyl
= 0
while the particle had angular momentum
V
L
before
part
=
Vr
×
V
P
0
=
Vr
×
mVv
0
.
Both the radiusvector
Vr
and the velocity
Vv
0
of the particle lie in the
xy
plane (
⊥
to the
z
axis), and according to the picture, at the
moment of collision the radius vector has mag
nitude

Vr

=
R
equal to the cylinder’s radius
and direction perpendicular to the particle’s
velocity.
Hence, its angular momentum is
parallel to the
z
axis and has magnitude

V
L
before
part

=
L
before
z,
part
=
Rmv
0
.
Altogether, before the collision
L
before
z,
net
=
Rmv
0
and therefore, after the collision we should
also have
L
after
z,
net
=
Rmv
0
.
(1)
After the collision, the cylinder and the
particle rotate as a single rigid body of net
moment of inertia
I
net
=
I
cyl
+
I
part
=
1
2
MR
2
+
mR
2
.
(2)
relatve to cylinder’s axis. ²or a rigid rotation
like this, the angular momentum points in
z
direction and its magnitude is
L
z,
net
=
ωI
net
.
(3)
Combining eqs. (1), (2) and (3) together,
we immeditaly obtain
ω
after
=
L
after
z,
net
I
net
=
Rmv
0
1
2
MR
2
+
mR
2
=
v
0
R
×
m
1
2
M
+
m
= 6
.
24483 rad
/
s
.
Question 2, chap 13, sect 3.
part 1 of 2
10 points
A ±gure skater on ice spins on one foot.
She pulls in her arms and her rotational speed