CALC A Spring with Mass. We usually ignore the
kinetic energy of the moving coils of a spring, but let's try to get
a reasonable approximation to this. Consider a spring of mass
$M,$ equilibrium length $L_{0},$ and spring constant $k .$ The work done
to stretch or compress the spring by a distance $L$ is $\frac{1}{2} k X^{2}$ , where
$X=L-L-L_{0}$ . Consider a spring, as described above, that has one
end fixed and the other end moving with speed $v .$ Assume that
the speed of points along the length of the spring varies linearly
with distance $l$ from the fixed end. Assume also that the mass $M$
of the spring is distributed uniformly along the length of the
spring. (a) Calculate the kinetic energy of the spring in terms of the
$M$ and $v .$ (Hint: Divide the spring into pieces of length $d l ;$ find
the speed of each pivide in terms of $l, v,$ and $L ;$ find the mass of
each piece in terms of $d l, M,$ and $L ;$ and integrate from 0 to $L .$
The result is $n o t \frac{1}{2} M v^{2},$ since not all of the spring moves with the
same speed.) In a spring gun, a spring of mass 0.243 $\mathrm{kg}$ and force
constant 3200 $\mathrm{N} / \mathrm{m}$ is compressed 2.50 $\mathrm{cm}$ from its unstretched
length. When the trigger is pulled, the spring pushes horizon-
tally on a 0.053 -kg ball. The work done by friction is negligible.
Calculate the ball's speed when the spring reaches its uncom-
pressed length (b) ignoring the mass of the spring and (c) includ-
ing, using the results of part (a), the mass of the spring. (d) In
part (c), what is the final kinetic energy of the ball and of the
spring?