Problem 2 (25 pts.):
A block of mass
M
= 5.4 kg, at rest on a horizontal frictionless table, is
attached to a rigid support by a spring of constant
k
= 6000 N/m which is initially uncompressed.
A bullet of mass
m
= 9.5 g and velocity
arrowrightnosp
v
of magnitude 630 m/s strikes and becomes embedded
in the block. This compresses the spring and sets block and spring in simple harmonic motion.
Determine:
a) (
8 pts.
)
the speed of the block immediately after the collision.
b) (
9 pts.
)
the amplitude of the harmonic motion.
c) (
8 pts.
)
the frequency of the harmonic motion.
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Problem 3
(25 pts.)
The figure to the right shows a block S
with mass
M
. The block is free to move along a horizontal
frictionless surface and connected, by a massless cord that
wraps over a frictionless pulley with mass
m
and radius
(
I=mr
2
/2) to a second block H also of mass
m
.
The hanging
block H falls as the sliding block S accelerates to the right.
a)
(
6 pts.)
Draw force diagrams for blocks S and H and the
pulley.
b)
(
10 pts.)
Find the linear acceleration of the rope.
c)
(
9 pts.)
Find the magnitude of tension in the rope pulling on block S.
m
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r
Problem 4 (26 pts)
As shown in the figure, an
overhead
view of a thin uniform rod of length
0.80 m and mass
M
=1.0 kg rotating horizontally at angular speed 20 rad/s about an axis through
its center. A particle of mass M/3 initially attached to one end is ejected from the rod by a small
explosion and travels along a straight path that is perpendicular to the rod at the instant of
ejection. The particle's final speed
v
p
(after the ejection) is 11.0 m/s.
a)
(
6 pts.
) What is the angular momentum (magnitude and
direction) of the particle about the center of rod at the
moment before the ejection?
b)
(
10 pts.)
What is the final angular velocity (magnitude and direction) of the rod?
c)
(
10 pts.
) What is the change in kinetic energy of the system (the rod plus particle)?
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Problem 5 (27 pts.):
A block of mass
m
1
rests on a frictionless inclined plane at a height
above a flat horizontal plane.
This mass is released and slides down the incline, eventually
colliding elastically with a second block at rest at
x
0 with mass
m
2
, with
m
1
= 3
m
2
/4.
Immediately after
the elastic collision, block number 2 moves to the
right with velocity
v
2f
.
It encounters a region that has
friction with a coefficient of kinetic friction of
μ
k
,
traveling a distance
d
before coming to rest.
Answer
the following questions about these two blocks.
a)
(
7 pts.)
Determine an equation for
v
2f
immediately after the collision (using
m
2
,
h
, and
b)
(
7 pts.)
Determine an equation for
d
(in terms of
m
2
,
h
,
μ
k
, and
g
), the distance that block 2
travels on the friction surface, before coming to a stop.
c)
(
6 pts.)
What is the magnitude (equation) and sign of the work done by the frictional force on
block 2?
d)
(
7 pts.
) What is the
velocity
v
1f
of block 1 after the collision (in terms of
m
2
,
h
, and
g
). Use
vector notation.
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h
=
g
)
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