Kinetic energy of a spring with distributed mass

[adrian116]

the question is following:
To find the effect of the spring's mass, consider a spring with mass M, equilibrium length L0, and spring constant k. When stretched or compressed to a length L, the potential energy is (1/2)Kx^2, where x=L-L0.
a) Consider a spring, as described above, that has one end fixed and the other end moving with speed v. Assume that 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. 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
(1/2)Mv^2, since not all of the spring moves with the same speed.)

my director instructed us that : k=∫dk=1/2ρdl(l/L*v)2=1/6Mv2
i don't understand how to work out this 1/2ρdl(l/L*v)2
also, what is the relationship of k with k.e.?

 

[lightgrav]

If the free end of the spring is moving at speed v_L,
and the fixed end is moving at v_0 = 0 , then
how fast is the middle slice (at x=L/2) moving? _____
What is the middle slice's v^2 ? ______
If the middle slice is dx long, what is its mass? ______
what is the middle slice's Kinetic Energy? ______

Now look at a slice at more general location x between 0 and L .
add the Kinetic Energy of all these general slices by integrating.