Specific Heat of a nonlinear, temperature dependent spring

[Infernorage]

Homework Statement

A nonlinear spring has a temperature dependent force law,

F = -\frac{K}{T}(L-L_o)^3

At a temperature T = T_o and length L = L_o the specific heat at a constant length is C_L = C_o. What is the specific heat at T = T_o when the spring is stretched to length 2L_o?

Homework Equations

The Attempt at a Solution

I am really not sure where to start. Am I supposed to use the general dU=dQ-dW equation and somehow work the spring into that using the force equation and then solve for a specific heat equation? Any advice would be great. Thanks.

 

[K space]

Since the force is the negative gradien of its potential energy, in a one dimensional case as this we can write,

F=-\frac{dU}{dL} or

U=-\int FdL

Once obtained the potential energy, use the defenition of the heat capacity at constant length,

C_L=(\frac{dU}{dT})_L

With the given information you should be able to express your result in terms of C_o.

 

[Infernorage]

Since the force is the negative gradien of its potential energy, in a one dimensional case as this we can write,

F=-\frac{dU}{dL} or

U=-\int FdL

Once obtained the potential energy, use the defenition of the heat capacity at constant length,

C_L=(\frac{dU}{dT})_L

With the given information you should be able to express your result in terms of C_o.

If you do that doesn't C_L=(\frac{dU}{dT})_L come out to be C_L = -\frac{K}{4T^2}(L-L_o)^4? How can you have a negative specific heat?