# Spring constant based on change in potential energy

1. The problem statement, all variables and given/known data
A compressed spring has 138 J of energy stored in it. When it is decompressed by 125 cm, it now stores 82.6 J of energy. What is the spring constant

2. Relevant equations
General:
ΔU = 1/2kxf2 – 1/2kxi2
U(x) = 1/2kx2
U1 = 1/2kx12
U2 = 1/2k(x1 + 1.25 m)2

3. The attempt at a solution
U1 = 1/2kx12

k = 2U1/x12

U2 = 1/2 (2U1/x12)(x1 + 1.25 m)2

U2 = U1/x12 (x12 + 2.5m x1 + 1.5625 m2

U2 = U1 + (2.5 m x1/x1) + (1.5625 m2 U1/x12)

x12 ΔU = (2.5m U1/x1 + 1.5625m2 U1/x12) x12

ΔU x12 = 2.5m U1 x1 + 1.5625m2 U1

Substitute values and get them all to one side:
55.4J x12 – 345Jm x1 – 215.625Jm2 = 0

Quadratic formula gives:
x1 = 6.8 cm and -.57 cm

138J = 1/2k (.068 m )2

276J/.00462 m2 = 59740 N/m = k? Obviously that can’t be right.
Any help on this would be great!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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