Calculating Energy and Height in a Vertical Spring System

[jiboom]

2 identical strings of length 1 metres and modulus of elasticity are each fastened to a particle of mass .5 kg. their othe ends are fixed to 2 points 4 metres apart in a vertical line. Find the height of the particle above the lower fixed point A in the equilibrium position. the particle is now pulled down to A and released from rest. find the greatest height above A to which the particle rises.

first part i have done, answer is 1.5 so (top string is of stretched length 2.5)

for second part i have said

at bottom

bottom string is slack so epe=0
top string is stretched to 4m from 1 m so
epe=modulus(extension)^2/(natural length)=4.9(3^2)/2
released from rest so KE=0

so initially total energy is 4.9(9/2)

at greatest height above A. H,say

PE=.5gH=4.9H using g=9.8
KE=0
EPE=modulus(extension)^2/(natural length)=4.9(h-1)^2/2

so energy at top is 4.9H+[4.9(H-1)^2]/2
by conservation

4.9H+[4.9(H-1)^2]/2=4.9(9/2)


so H^2=8 H=2.83

this is correct but i am troubled by the lines in bold.

i didnt know the height so assumed it would be high enough for the top string to be slack so have no EPE. this is how i came to that line.

BUT for this value of H, 2.83, the top string will still be extended,by .17m, so surely it contributes some EPE? is ths so? If it does how do i adjust my solution?

 

[jiboom]

is anyone able to help me with this?
i am struggling to see how energy works and really need help.