- #1
gelfand
- 40
- 3
Homework Statement
potential energy function of :
$$
U(x) = 4x^2 + 3
$$
And have to
i) Work out the equation of motion
ii) Prove explicitly that the total energy is conserved
Homework Equations
$$F = \frac{dU}{dt}
$$
The Attempt at a Solution
I'm not too sure how to go about this.
I would say that I have the force of
$$
F = 8x
$$
By differentiating the given potential energy function. I need to work out the
equation of motion, what I have an object with mass ##m##.
So this means that I have
$$
F = 8x = ma
$$
Then I have that
$$
a = \frac{8x}{m}
$$
Is this an equation of motion? I mean, it's acceleration, or should I find for
##v(t)## and ##x(t)## as well as this?
In which case I would have
$$
v(t) = \int a(t) dt
$$
Which in this case is found as (having the mass in the equation seems unusual?)
$$
v(t) = v_0 + \frac{1}{2m}8x^2 = v_0 + \frac{4}{m} x^2
$$So then from this I have that
$$
x(t) = x_0 + v_0t + \frac{4}{3m}x^3
$$
And this would be all of the equations of motion for this 1D case?
Then I need to prove that energy is conserved here, and I've no idea how to go
about that.
I've not been given any frictional forces, so it seems like it's just a given
that I'm going to have
$$
W + PE_0 + KE_0 =
PE_f + KE_f + \text{Energy(Lost)}
$$
Here I can remove work ##W## and the energy lost for
$$
PE_0 + KE_0 =
PE_f + KE_f
$$
And I need to do something with these?
Potential energy - I have the potential energy function given as part of the
problem which is
$$
U(x) =
4x^2 + 3
$$
Then I can sub this into the energy expression as
$$
4x_0^2 + 3
+ KE_0 =
4x_f^2 + 3
+ KE_f
$$
Getting rid of the constants seems pretty harmless
$$
4x_0^2
+ KE_0 =
4x_f^2
+ KE_f
$$
Now I'm really not sure what I should do from here, sub in kinetic formulas of
##K = \frac{1}{2}mv^2##?
$$
4x_0^2
+
\frac{1}{2}mv_0^2
=
4x_f^2
+
\frac{1}{2}mv_f^2
$$
I'm not sure if I can arrange this to be 'nicer' in any way either, I'm purely
thinking in algebra at the moment though not physics :S$$
8(x_0^2 - x_f^2) =
m(v_f^2 - v_0^2)
$$
I'm not sure if differentiation should do anything nice here, but I really have
no idea what I'm doing with this.
Thanks