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?