Toadfish Monastery

Quote from: Swatopluk on November 29, 2009, 09:39:09 AM
I also thought about the kinetic energy solution but was unsure whether that would work.

I think it has to. There's nowhere else for the energy to go and it complies with Physics' inviolate First Lancashire Law "You don't get owt for nowt".

If P is density of Earth, and remembering volume of sphere is 4*pi*r^3/3
If you start with your particle at the centre and push it up a bit the work done is Force * Distance

dw=(G*m*4*pi*P*r^3)/(3*r^2)*dr

integrate

  (G*m*4*pi*P*r^2)/6

and evaluate at bounds 0 and R

(G*m*4*pi*P*R^2)/6

But by definition P is mass/volume so substitute P =M/((4*pi*R^3)/3) gives

(G*m*4*pi*R^2*M*3)/(R^3*^*6*4*pi)

or

(G*m*M)/(2*R)

By Conservation of energy this equals 0.5*m*v^2

so

v=sqrt((GM/R))

which comes out to the same value as before, 'kin fast as it's known in Physics circles.

Quote
I think it will take some time to digest your solution.

Feel free to ask about any bits.
I suppose getting from

d2r/dt2+kr=0

to r=Rcos(wt)

is where I omitted steps. We know  that to match the equation our r expression in t must have the property that its second derivative can cancel the value of its zeroth.  We know that both sin and cos have the property that if you differentiate them twice you get a constant times the original value back, but with the sign negated. Therefore we can propose our r expression is a linear superposition of one or more sin or cos terms, maybe with a constant term.

r=A*cos(wt)+B*sin(wt)+C


But we know at time zero r=R, and dr/dt=0 so there's no reason to keep B and C, as we have no boundary conditions requiring them. 

d2r/dt2=-GM/R^2
gives

-w^2*A*cos(wt)=-(G*M*r)/R^3

which at time zero gives

w^2*A=(G*M)/R^2

A is clearly R giving the value of w as SQRT((G*M)/R^3)) as before.

We could equally well express r in terms of sin but would need a phase shift to get the correct boundary value compliance at time 0.

r=Rsin(wt+(pi/2))

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But what about the more general problem with the acceleration an optional function of x with object accelerated along x?

I think it's always going to boil down to spotting appropriate solutions to the differential equations and matching boundary conditions. For particular situations I suppose choosing Hamiltonian or Lagrangian approaches might give a shortcut, like the energy conservation one.

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I hated differential equations in math at school and at the university and have forgotten most about them apart from the most simple problems.

Don't worry. You still remember far more Physics than I do Chemistry (despite Chemistry being a minor sub-category of Physics  )


edit:

Add dr/dt boundary condition value and fix missing density from integrand.

Now I begin to remember the stuff about the angular functions. I hope those memories don't last 

But there was something worse than 'simple' differential equations: Fourier (+Lagrange) transformation and analysis.
I think when our Prof did Lagrange was one of the few occasions where I suffered what we in Germany call a Filmriß.
Although I was technically awake since I made the checklist how often he used his standard phrases/speech mannerisms (his most favorite popped up up to twice a minute), when I left the room afterward I was unable to recall anything he had said (including the very topic!). Must have been similar to hypnosis I presume. No, there were no embarassing photos of naked dancing

That was my Linear Algebra teacher!

Quote from: Swatopluk on November 30, 2009, 09:17:45 AM
Now I begin to remember the stuff about the angular functions. I hope those memories don't last 

Bonus easy question. How have I cheated with N3 in the energy calculation, and does it matter much?

Quote from: Swatopluk on November 30, 2009, 09:17:45 AM
But there was something worse than 'simple' differential equations: Fourier (+Lagrange) transformation and analysis.

My favourite.   

Quote from: Griffin NoName on November 30, 2009, 11:45:32 PM

Quote from: Swatopluk on November 30, 2009, 09:17:45 AM
But there was something worse than 'simple' differential equations: Fourier (+Lagrange) transformation and analysis.

My favourite.   

What do you expect with stuff designed by two Frenchies? Probably the same people who designed the controls on those old Citroens.

Stick with Newton (known as "Da Man" in Physics circles) or Hamiltonian techniques (so much fun they're named after Nelson's mistress, Lady Hamilton).

Be silent about Hamilton!
Every lesson in quantum mechanics began with the professor writing HΨ=EΨ on the blackboard.
That equation looks so harmless and yet turns out to inevitably lead to pages and pages of cryptic symbols that one has to pretend to comprehend. Even Einstein hated it! (and still got his Nobel for work that now is considered to be part of it*)

Math is not a totally closed book to me but there are limits. Occasionally I manage to get some tricky stuff done (for entertainment usually), at other times I am defeated by elementary stuff. For example I have totally forgotten how I solved the problem to adapt the barometric equation for non-constant g. I did that because I wanted to know what air pressure one could expect at the core of the Earth. On problem I occasionally try to solve is how to hit a ballistic object with another** dependent on limited data. The solution would have to yield values for horizontal and vertical angle, initial velocity and time. I originally wanted to use that for a game.

*still known to remarkably few. It's a notorious killer in quizzes.
**i.e. hitting an incoming shell with one of your own before it comes down on your base.

Quote from: beagle on December 01, 2009, 07:10:24 AM
What do you expect with stuff designed by two Frenchies? Probably the same people who designed the controls on those old Citroens.

You should know that without Fourier & Lagrange we would be still sending messages with pigeons, right?

And the old Citroëns are awesome, active suspensions before everybody else.

Quote from: Sibling Zono (anon1mat0) on December 01, 2009, 03:53:06 PM
You should know that without Fourier & Lagrange we would be still sending messages with pigeons, right?

They still did that during the World Wars

Hmmm. Maybe. It's just nobody since has thought it was a good idea to stick all a car's controls on something obviously inspired by a wine bottle. Although the Swedes are pretty good at hiding the sidelight switch, it has to be said.

http://www.citroenet.org.uk/miscellaneous/prn/satellites.html

Don't let your francophobia take over, those consoles are very cool.


Seriously, I find them cool, although you might like something like this better:

http://www.rrsilvershadow.com/EDashb/Dashb2.htm

I don't. 

I always wondered what my car looks like in the front. You'd think Rolls Royce would know which side to put the steering wheel though.

Quote from: beagleBonus easy question. How have I cheated with N3 in the energy calculation, and does it matter much?

...And the answer is.

N3 says that for every action there is an equal and opposite reaction, and as Galileo said (somewhat quietly) "Eppur si muove". The Earth isn't nailed to the sky so it will pick up some velocity in the opposite direction (Conservation of Momentum will say how much).

No it doesn't matter because:

Swato will never get funding for this thing, even from Dubai.

If he tries to build it he'll be busy with tunnelling opportunities for centuries, and the air conditioning bill will bankrupt him.

For anything he can carry the mass will be so small that the energy picked up by the Earth will be something like twenty plus orders of magnitude smaller than that picked up by the object.

The velocity will be in the opposite direction to the particle so the effect on the relative velocity will be reduced still further (and that's what's going to count if it hits the sides).

We went to France for our holiday this year.  I think the British have gone right off the French because there was hardly a Brit there besides us. 

It's back to the Mumbles next year...at least you can get decent Ice Cream there.

Something to do with the Pound to Euro conversion? 

Non!

Well yes, that did have something to do with it...British MP's expenses claims have put our currency in the toilet.