fisica1-youn-e-freedman-exercicios-resolvidos

6.101: a) Denote the position of a piece of the spring by l; l = 0 is the fixed point and
l = L is the moving end of the spring. Then the velocity of the point corresponding to l,
1
denoted u, is u( l)
= v (when the spring is moving, l will be a function of time, and so u
L
is an implicit function of time). The mass of a piece of length dl is dm = dl,
and so
and
2 1 Mv
dK = dmu =
3
2 2 L
K =
2
1 2
Mv
2L
2 L
2
∫ dK = ∫ l dl =
3
0
l dl,
1 2 1 2
−2
b) kx = mv ,
2 2
so v = ( k m)
x = (3200 N m) (0.053 kg)(2.50 × 10 m) = 6.1m s. .
c) With the mass of the spring included, the work that the spring does goes into the
kinetic energies of both the ball and the spring, so 1 2 1 2 1 2
kx = mv + Mv . Solving for v,
v =
d) Algebraically,
k
m + M
x =
3
(3200 N m)
(0.053 kg) + (0.243 kg)
2
2
Mv
6
2
.
6
(2.50 × 10
3
2
1 2 (1 2) kx
mv =
= 0.40 J and
2 (1 + M 3m)
1 2
Mv
6
2
(1 2) kx
=
= 0.60 J.
(1 + 3m
M )
−2
L
M
m) = 3.9 m s.