Math Is Fun Forum

mikau

Member

Registered: 2005-08-22

Posts: 1,504

tough physics problem involving dot product equation

here's a physics problem i can't figure out for the life of me,

You are given3 stationary particles in 3d space, their x,y,z coordinates, and masses. You are given a 4th particle with a given mass. You must find the location to place the 4th particle such that the gravitational forces balance and the object remains stationary.

Sounds simple enough, but working it out has proved challenging.

Consider the location of each of the three particles as vectors A,B, C, and call the forth vector D. The formula for gravitational force is

(with r being the distance between the two, m[sub]1[/sub] and m[sub]2[/sub] the mass of the two particles, and G the gravitational constant, all except r are known in this problem)

likewise, the force of attraction between vectors A and D would be given by:

where • denotes the dot product.

Thats fine, but if we're going to be adding these vectors together we need to know their directions. Since the force vector from D to A has the same direction as the vector (A-D), we could normalize this vector:

and then multiplying it by the magnitude of the force, to obtain the force vector. Doing this gives us:

This is just the force between A and D, if we consider all three we get:

which is what i might call a hideous equation, and i don't think this can be solved by any elementary methods.

So, any other ideas?

Last edited by mikau (2008-04-07 18:44:55)

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A logarithm is just a misspelled algorithm.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: tough physics problem involving dot product equation

Would the center of mass of the orginal triangle of 3 masses not work?
Or is it off by some quadratic method?

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igloo myrtilles fourmis

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: tough physics problem involving dot product equation

The only way I can think of from here is to just plug in all your values of A, B, C, m_a, etc. and work it out from there. If it helps, each of the denominators can be written as ||A-D||³, and so on.

I *think* that once you've done that, splitting it into x, y and z co-ordinates will give you three linear equations in three variables (which are therefore solvable).
(Note that you can also ignore m_d and G throughout, until you get to the last step)

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Why did the vector cross the road?
It wanted to be normal.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: tough physics problem involving dot product equation

Yeah, except its a really horrible system of three equations in three variables, that I think can't be solved without divine inspiration.

I was hoping someone could either provide an alternate equation, or divine inspiration!

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A logarithm is just a misspelled algorithm.

Dragonshade

Member

Registered: 2008-01-16

Posts: 147

Re: tough physics problem involving dot product equation

In my opinion, 3 points could determine a plane in a 3D space, since the 3 particle are in equilibrium, you could put the fourth particle as a point in inside the triangle, and with the same distance to all 3 particles, that point should be the center of a circle inscribing the triangle

Last edited by Dragonshade (2008-04-08 03:22:02)

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: tough physics problem involving dot product equation

yes, but you are assuming that all the points have equal mass. They do not.

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A logarithm is just a misspelled algorithm.

Dragonshade

Member

Registered: 2008-01-16

Posts: 147

Re: tough physics problem involving dot product equation

How about considering them lying on the same plane

C and C_2 can be calculated, I think the point is the intersection of two circles and it lies inside the triangle?
still thinking about it ..XD

Last edited by Dragonshade (2008-04-08 03:56:13)

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: tough physics problem involving dot product equation

hmm... F[sub]db[/sub] - F[sub]da[/sub] = F[sub]ab[/sub]

that might be useful. I'll have to think about that for a while.

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A logarithm is just a misspelled algorithm.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: tough physics problem involving dot product equation

For fun, just do 2 stationary points in 1-D, and then to find the middle gravitation point,
you end up using quadratic formula.   m1/(d-x)^2 = m2/x^2, solve for x; d and m's are constants.

Then next, do 3 stationary points in 1-D, a straight line again.  I haven't done this yet...

See if it leads somewhere.

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igloo myrtilles fourmis