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6.101:
a) Denote the position of a piece of the spring by
l
;
0
=
l
is the fixed point and
L
l
=
is the moving end of the spring. Then the velocity of the point corresponding to
l
,
denoted
u
, is
L
v
l
u
1
)
(
=
(when the spring is moving,
l
will be a function of time, and so
u
is an implicit function of time). The mass of a piece of length
dl
is
,
dl
dm
L
M
=
and so
,
2
1
2
1
2
3
2
2
dl
l
L
Mv
dmu
dK
=
=
and
∫
∫
=
=
=
L
Mv
dl
l
L
Mv
dK
K
0
2
2
3
2
.
6
2
b)
,
2
2
1
2
2
1
mv
kx
=
so
s.
m
6.1
m)
10
50
.
2
(
kg)
053
.
0
(
m)
N
3200
(
)
(
2
=
×
=
=

x
m
k
v
.
c) With the mass of the spring included, the work that the spring does goes into the
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Unformatted text preview: kinetic energies of both the ball and the spring, so . 2 6 1 2 2 1 2 2 1 Mv mv kx + = Solving for v , s. m 3.9 m) 10 50 . 2 ( 3 kg) (0.243 kg) 053 . ( m) N 3200 ( 3 2 = × + = + =x M m k v d) Algebraically, J 40 . ) 3 1 ( ) 2 1 ( 2 1 2 2 = + = m M kx mv and J. 60 . ) 3 1 ( ) 2 1 ( 6 1 2 2 = + = M m kx Mv...
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