Problem 2 25 pts a block of mass m 54 kg at rest on a

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
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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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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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