A system contains a perfectly elastic spring, with an unstretched length of 20 cm and a spring constant of 4 N/cm.

When the length of the spring changes from an initial value of 22.0 cm to a final value, the elastic potential energy it contributes changes by ?0.0800J. Find the final length.

Hi. I'm having difficulty with my text's explanation of this problem. The preceding example suggests that as the spring is stretched, the contributed PE increases, but the answer in the text to this says that this decrease of -0.08 J corresponds to an increase in distance from equilibrium. I don't understand how that's possible.

For U(x) = 0.5kx² :

22cm: U(0.02) = 0.0800 J

22.8cm: U(0.028) = 0.1568 J

23cm: U(0.03) = 0.18 J

19.2cm: U(-0.008) = 0.0128 (this value I got from subtracting the negative result below from 22cm in a desperate attempt to make the math fit.)

Even that last value was used in the previous example. This suggests a clear trend consistent with the numbers, and also that a change of -0.0800 from 22cm should be a return to equilibrium. Intuitively that makes sense, but if that is the case could you show me how that would look mathematically?

I feel like I'm taking crazy pills! ?Thanks for the help!

1 10 8.3 u ( ( Q ) = 0 N = XS X2 = 0. 22 m X, = 0. 22 m EQ = 0. 20 m Key = 0.20 m -0-08 = = h x2 - YOU ( 0. 02 ) = 0.0800 V = - W = - ( - 0 . 0 8 0 0) = 1 K ( x2- 0.02 2) 20 0 -0. 608 0.0004 = X2-0. 0004 0.0 0 08 - X2 10 0 0 08 =2 0. 028