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 force constant \(k\). The work done to
stretch or compress the spring by a distance \(L\) is \\(\frac{1}{2}\\) \(kX^2\),
where \(X = L - L_0\). 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 \(M\) and \(v\). (\(Hint\): Divide the spring into pieces of
length \(dl\); find the speed of each piece in terms of \(l\), \(v\), and \(L\); find
the mass of each piece in terms of \(dl\), \(M\), and \(L\); and integrate from \(0\)
to \(L\). The result is \(not\) \\(\frac{1}{2}\\) \(Mv^2\), since not all of the
spring moves with the same speed.) In a spring gun, a spring of mass 0.243 kg
and force constant 3200 N/m is compressed 2.50 cm from its unstretched length.
When the trigger is pulled, the spring pushes horizontally on a 0.053-kg ball.
The work done by friction is negligible. Calculate the ball's speed when the
spring reaches its uncompressed length (b) ignoring the mass of the spring and
(c) including, 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?