## Motion along a line

1. The problem statement, all variables and given/known data

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 $\sqrt{2}$

2. Relevant equations

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

T+V=E

3. The attempt at a solution

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?

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

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 Quote by zebrastripes 1. The problem statement, all variables and given/known data 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 $\sqrt{2}$ 2. Relevant equations T=mx'$^{2}$/2 T+V=E 3. 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
$$\frac{1}{2} v^2 + V(x) = \frac{1}{2} U^2 + V(1/2) = \frac{1}{2} U^2 + \frac{7}{32},$$ 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

 Quote by Ray Vickson 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!

 Tags energy, mechanics, potential well