Elastic & gravitational potential energy

Set Abominae

Hi there.

I have a bead of mass m, which slides down a frictionless parabolic wire in the form y=x^2, and is attached by elastic to the point (0,h), and I want to write down total energy f(x) (= elastic energy + gravitational potential energy) (no mention of kinetic energy...) of the bead at (x,x^2).

I know that g.p.e is mgx^2, and that elastic energy is (k/2)(h^2 + x^2 - 2hx^2 + x^4), but I'm not sure whether I'm adding the g.p.e. to the elastic energy, or subtracting it, despite the fact that I want f(x) = elastic energy + g.p.e.

Any clarification would be great :)

Thanks.

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tiny-tim

Homework Helper
Welcome to PF!

Hi Set Abominae! Welcome to PF!

(try using the X2 tag just above the Reply box )
… I'm not sure whether I'm adding the g.p.e. to the elastic energy, or subtracting it, despite the fact that I want f(x) = elastic energy + g.p.e.

ie will increasing the height make the bead go slower? will increasing the length of the elastic make the bead go slower?
I know that … elastic energy is (k/2)(h^2 + x^2 - 2hx^2 + x^4)
how did you get that?

and what is the unstretched length of the elastic?

Set Abominae

The question states that:
When the elastic has length d its elastic energy is $$\frac{k}{2}$$ $$d^{2}$$, where d> 0.....

So when the bead is at $$(x,x^{2})$$, we have that:
$$d = \sqrt{x^{2}+(h-x^{2})^{2}}$$

(It'a a question from a math assignment, so its probably not totally accurate...)

So that will be add, then?:shy:

tiny-tim

Homework Helper
The question states that:
When the elastic has length d its elastic energy is $$\frac{k}{2}$$ $$d^{2}$$, where d> 0.....
hmm … that's unusual … but if the question says so, i suppose it's ok

(perhaps the elastic starts further back, and goes round a peg at (0,h), and the unstretched length is up to the peg)
So that will be add, then?:shy:
i'm not sure what you're adding to what …

anyway, give a reason, so we can see you're not guessing!

Set Abominae

Actually, thinking about it, I would imagine that I would subtract the g.p.e from the elastic energy to get f(x) (though I'm cautious about doing so, as the question says f(x)=elastic energy + g.p.e.).

Another reason for doing this is that if I sum them to get f(x), when I start drawing my bifurcation diagram later in the question, I get:

A single fixed point $$x=0$$ is stable for $$h<\frac{mg}{k}+\frac{1}{2}$$,

and 3 fixed points

$$x=0, \pm\sqrt{h-\frac{mg}{k}-\frac{1}{2}}$$ are all unstable for $$h>\frac{mg}{k}+\frac{1}{2}$$, and I've never seen a system where all 3 fixed points are unstable before...

Set Abominae

But I also get the same result of 3 unstable fixed points when I subtract the g.p.e. from the elastic energy....

tiny-tim

Homework Helper
Actually, thinking about it, I would imagine that I would subtract the g.p.e from the elastic energy to get f(x) (though I'm cautious about doing so, as the question says f(x)=elastic energy + g.p.e.).
Are you taking into account the fact that gpe is minus mgh?
Another reason for doing this is that if I sum them to get f(x), when I start drawing my bifurcation diagram later in the question …
urggh :yuck: … dunno anythng about bifurcation diagrams

Bob S

Here is a related problem that I. Newton solved in one day back in about 1697. Suppose we have a frictionless bead sliding on a wire from point A at (0,h) to point B at (x,0) by the fastest route. What is the shape of the wire from point A to point B? A straight line is the shortest distance, but a parabola is longer and faster. But a parabola is not the fastest. So what is?

Set Abominae

Are you taking into account the fact that gpe is minus mgh?
I forgot about that! Upon further thought, I would imagine that I would subtract the negative gpe, so be adding it to the elastic energy to give f(x). I say this by considering energy conservation - assuming we don't have any stupidly big oscillations, the higher the bead gets, the larger mgh gets, and the smaller the elastic energy.

Likewise, when the bead is at x=0, it has it least gpe (0 if we take the x-axis as our 'zero height', but the elastic is at maximum extension (again, assuming well-behaved oscillations).

So by energy conservation, I'm inclined to write that
$$f(x) = (elastic energy) + mgx^{2}$$

Thoughts?

mirabella

I have the same problem for my maths assignment :) check ur calculations carefully. I have a single unstable fixed point at x=0 for h<mg/k+0.05, second derivative is >0 and 3 stable fixed points for h>mg/k+0.05, second derivative<0. Why do u have the opposite?

Set Abominae

I have the same problem for my maths assignment :) check ur calculations carefully. I have a single unstable fixed point at x=0 for h<mg/k+0.05, second derivative is >0 and 3 stable fixed points for h>mg/k+0.05, second derivative<0. Why do u have the opposite?
$$f(x)=\frac{k}{2}(x^{2}+(h-x^{2})^{2}+mgx^{2} \Rightarrow f'(x)=2kx^{3}+kx-2hkx+2mgx$$
Solve $$f'(x)=0$$ to give $$x=0$$ for $$h\leq\frac{mg}{k}+\frac{1}{2}$$,
and $$x=0, x=\pm\sqrt{h-\frac{mg}{k}-\frac{1}{2}}$$ for $$h>\frac{mg}{k}+\frac{1}{2}$$

Now, $$f''(x)=6kx^{2}+k+2mg-2hk h<\frac{mg}{k}+\frac{1}{2} \Rightarrow f''(0)=2mg+k-2hk<0 \Rightarrow$$ stable.
(Note, $$h=\frac{mg}{k}+\frac{1}{2} \Rightarrow f''(0)=0 \Rightarrow$$ system is structurally unstable.)

$$h>\frac{mg}{k}+\frac{1}{2} \Rightarrow f''(0)>0 \Rightarrow$$ unstable, and
$$f''(\pm\sqrt{h-\frac{mg}{k}-\frac{1}{2}})=4hk-4mg-2k>4k(\frac{mg}{k}+\frac{1}{2})-4mg-2k=0 \Rightarrow$$ unstable.

(Sorry for the mess!)

I don't think there's any mistakes here... How did you go about solving it?

mirabella

[QUOTE f''(0)=2mg+k-2hk
$$h>\frac{mg}{k}+\frac{1}{2} \Rightarrow f''(0)>0 \Rightarrow$$ unstable QUOTE]

multiply h>mg/k+0.05 by 0.05k ( sign stays the same since k>0). U get 2kh>2mg+k i.e
2mg+k-2kh<0 so f''(0)<0 stable ?? Same for the other two.
I know it's a cusp catastrophe but how do u justify that?
Do u have a symmetry question (with a see-saw) later on in your assignment?

mirabella

Set Abominae;2134032[tex said:
f''(\pm\sqrt{h-\frac{mg}{k}-\frac{1}{2}})=4hk-4mg-2k>4k(\frac{mg}{k}+\frac{1}{2})-4mg-2k=0 \Rightarrow[/tex] unstable.
Actually this is ok. I had a wrong sign there, but u still have a mistake for x=0 so check it. We have ether one unstable at x=0 or one stable at x=0 and two unstable at +-sqrt(h-mg-1/2). Bifurcation looks much better now :) Thoughts?

Set Abominae

Actually this is ok. I had a wrong sign there, but u still have a mistake for x=0 so check it. We have ether one unstable at x=0 or one stable at x=0 and two unstable at +-sqrt(h-mg-1/2). Bifurcation looks much better now :) Thoughts?
Yes, I now have:

1 unstable fixed point $$x=0$$ for $$h<\frac{mg}{k}+\frac{1}{2}$$

1 fixed point $$x=0$$ for $$h=\frac{mg}{k}+\frac{1}{2}$$, and system is structurally unstable.

1 stable fixed point $$x=0$$ and 2 unstable fixed points at
$$x=\pm\sqrt{h-\frac{mg}{k}-\frac{1}{2}}$$ for $$h>\frac{mg}{k}+\frac{1}{2}$$

I do indeed have a symmetry question (check your private messages). I wasn't really too sure on whether this was a cusp catastrophe or not. I couldn't use any of the equations for the canonical cusp catastrophe, but was inclined to believe that it's a cusp catastrophe, as we just from a maximum to a mininum and two maximums (basically, the opposite of the canonical cusp catastrophe: just the same situation with all equations made negative). But, that's my only real argument....

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