Given force as a function of x, how do I find the total energy?

[GriffinC]

Homework Statement

F=-kx+kx³/α² where k and α are constants and k > 0. Determine U(x) and discuss the motion. What happens when E=kα²/4?

Homework Equations

F=ma=mv²d/dx
U=-∫Fdx

The Attempt at a Solution

The first part is easy.
U(x) = kx²/2-kx⁴/4α²
Now I'm looking for what happens when E=kα²/4
I know E=T+U, so T=kα²/4-kx²/2+kx⁴/4α²
To find T, I need to know velocity since T=1/2mv²
Solving for v, F=mv²d/dx, v=(k-3kx²/α²)/2(kx³/α²-kx)^3/2

Here's where I'm not sure I'm going in the right direction. If I find T, I introduce an m term. The potential energy is independent of mass, so why would kinetic energy depend on mass? It also seems as though the whole thing is independent of time.

 

[PeroK]

Homework Statement

F=-kx+kx³/α² where k and α are constants and k > 0. Determine U(x) and discuss the motion. What happens when E=kα²/4?

Homework Equations

F=ma=mv²d/dx
U=-∫Fdx

The Attempt at a Solution

The first part is easy.
U(x) = kx²/2-kx⁴/4α²
Now I'm looking for what happens when E=kα²/4
I know E=T+U, so T=kα²/4-kx²/2+kx⁴/4α²
To find T, I need to know velocity since T=1/2mv²
Solving for v, F=mv²d/dx, v=(k-3kx²/α²)/2(kx³/α²-kx)^3/2

Here's where I'm not sure I'm going in the right direction. If I find T, I introduce an m term. The potential energy is independent of mass, so why would kinetic energy depend on mass? It also seems as though the whole thing is independent of time.

Did you think of drawing a graph of $U$?

You're right that expressing $T = \frac12 mv^2$ isn't going to help.