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

## Homework Statement

F=-kx+kx32 where k and α are constants and k > 0. Determine U(x) and discuss the motion. What happens when E=kα2/4?

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-3kx22)/2(kx32-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
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## Homework Statement

F=-kx+kx32 where k and α are constants and k > 0. Determine U(x) and discuss the motion. What happens when E=kα2/4?

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-3kx22)/2(kx32-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.