# 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 = 1/2kx_{f}^{2} – 1/2kx_{i}^{2}

U(x) = 1/2kx^{2}

U_{1} = 1/2kx_{1}^{2}

U_{2} = 1/2k(x_{1} + 1.25 m)^{2}

**3. The attempt at a solution**

U_{1} = 1/2kx_{1}^{2}

k = 2U_{1}/x_{1}^{2}

U_{2} = 1/2 (2U_{1}/x_{1}^{2})(x_{1} + 1.25 m)^{2}

U_{2} = U_{1}/x_{1}^{2} (x_{1}^{2} + 2.5m x_{1} + 1.5625 m^{2}

U_{2} = U_{1} + (2.5 m x_{1}/x_{1}) + (1.5625 m^{2} U_{1}/x_{1}^{2})

x_{1}^{2} ΔU = (2.5m U_{1}/x_{1} + 1.5625m^{2} U_{1}/x_{1}^{2}) x_{1}^{2}

ΔU x_{1}^{2} = 2.5m U_{1} x_{1} + 1.5625m^{2} U_{1}

Substitute values and get them all to one side:

55.4J x_{1}^{2} – 345Jm x_{1} – 215.625Jm^{2} = 0

Quadratic formula gives:

x_{1} = 6.8 cm and -.57 cm

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

276J/.00462 m^{2} = 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**

http://ift.tt/1gYMGTh

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