Solving Motion along a Line: Finding Turning Points, Max U & Frequency

[zebrastripes]

Homework Statement

A particle P of unit mass moves along an x-axis under the influence of the force

F(x)=2(x$^{3}$-x)

Firstly, I find V(x)=x$^{2}$-x$^{4}$/2.
Where equilibrium points are F(x)=0 so x=0 with energy V(0)=0, x=1 with energy V(1)=V(-1)=1/2.
And I have also sketched the graph.

So here are the parts I'm stuck on:

1) Initially P is projected from the point x=1/2 with speed U. Using conservation of energy, find the turning points (where x'=0) as a function of U. Find the maximum value of U for which the resultant motion will be bounded.

2)Stating from Newton's second law, prove that a particle displaced by a small amount from x=0 will perform periodic oscillations with a frequency of $\sqrt{2}$

Homework Equations

T=mx'$^{2}$/2

T+V=E

The Attempt at a Solution

So for 1), I start with

U$^{2}$/2+(1/2)$^{2}$-(1/2)$^{4}$/2=E

Giving E=U$^{2}$+7/16

Then, because energy is conserved and x'=0 at turning points:

x$^{2}$-x$^{4}$/2=U$^{2}$+7/16

And now I'm really stuck for how to find the turning points as functions of U?

I'm guessing I'm going about this the completely wrong way??

And I have no idea what I'm supposed to be doing for 2)

Sorry this is long, but any help will be greatly appreciated

 

[Ray Vickson]

Homework Statement

A particle P of unit mass moves along an x-axis under the influence of the force

F(x)=2(x$^{3}$-x)

Firstly, I find V(x)=x$^{2}$-x$^{4}$/2.
Where equilibrium points are F(x)=0 so x=0 with energy V(0)=0, x=1 with energy V(1)=V(-1)=1/2.
And I have also sketched the graph.

So here are the parts I'm stuck on:

1) Initially P is projected from the point x=1/2 with speed U. Using conservation of energy, find the turning points (where x'=0) as a function of U. Find the maximum value of U for which the resultant motion will be bounded.

2)Stating from Newton's second law, prove that a particle displaced by a small amount from x=0 will perform periodic oscillations with a frequency of $\sqrt{2}$

Homework Equations

T=mx'$^{2}$/2

T+V=E

The Attempt at a Solution

So for 1), I start with

U$^{2}$/2+(1/2)$^{2}$-(1/2)$^{4}$/2=E

Giving E=U$^{2}$+7/16

Then, because energy is conserved and x'=0 at turning points:

x$^{2}$-x$^{4}$/2=U$^{2}$+7/16

And now I'm really stuck for how to find the turning points as functions of U?

I'm guessing I'm going about this the completely wrong way??

And I have no idea what I'm supposed to be doing for 2)

Sorry this is long, but any help will be greatly appreciated

If v = velocity (= dx/dt), then by conservation of energy we have
$$\frac{1}{2} v^2 + V(x) = \frac{1}{2} U^2 + V(1/2) = <br /> \frac{1}{2} U^2 + \frac{7}{32},$$ since $V(1/2) = 7/32.$ At turning points, $v = 0,$ so you get an equation in $x.$

For small |x| the potential V(x) is almost linear in x. If you neglect the x⁴ term in V(x) you get a simple harmonic oscillator.

RGV

 

[zebrastripes]

For small |x| the potential V(x) is almost linear in x. If you neglect the x⁴ term in V(x) you get a simple harmonic oscillator.

RGV

Thanks RGV, but under what assumptions can I say for small |x|?

And would you mind giving me a hint about what to do for part 2)?

Thanks again!