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Old Oct 16th 2012, 06:00 AM   #1
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Gravity

I have a problem where two stars are at a certain time r0 apart from each other. They both has the mass of m, and at that time one of them is at rest and the other has the speed of v0 in the opposite direction of the gravity force that the other star exerts upon him. I need to find the maximal distance between them. now it's quite easy using energy and momentum conservation, but what I don't understand is why in the solution they say that the potential gravitational energy of this system at the beginning is -Gm*m/r0. this is the potential energy of each one of the stars, so isn't this suppose to be doubled for the whole system? I'm confused...
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Old Oct 16th 2012, 09:27 AM   #2
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Originally Posted by assaftolko View Post
in the solution they say that the potential gravitational energy of this system at the beginning is -Gm*m/r0.
Right - the system consists of two stars, and both together have this PE.

Originally Posted by assaftolko View Post
this is the potential energy of each one of the stars, so isn't this suppose to be doubled for the whole system?
No - the PE of each star is not GM^2/R separately. The mistake in thinking that both stars start with PE = Gm^2/r0 is that the derivation of that equation is based on the energy required to separate one object from the other a distance r0. Imagine that you hold star 1 steady wh1le moving star 2 away from it - let's call this case 1. Energy is force times distance, so star 2 requires energy to move it distance r0 while star 1 is stationary (and hence star 2 doesn't gain or lose PE). The total change in PE for the system is GM^2/r_0.

Alternatively consider case 2 - suppose the two stars begin life touching, and then you move them apart by having each move a distance ro/2 in opposite directions. If we define x as the distance each has moved, then the gravitational force betwen them at any point is GM^2/(2x)^2, and the energy required to move each to r0/2 is (1/2)GM^2/r0. Hence each gains half the PE of case 1, but the total change in PE for the system is the same.

One thing to remember is that PE is not an absolute number - unlike KE there really is no place where PE=0. What we must be careful to talk about is the change in PE from one state to another, not an absolute value for PE in either state. It's a subtle distinction, but it's one reason why you can't "add" the PE of two different objects unless you know that they both have the same starting reference point.
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Old Oct 16th 2012, 12:39 PM   #3
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Originally Posted by ChipB View Post
Right - the system consists of two stars, and both together have this PE.



No - the PE of each star is not GM^2/R separately. The mistake in thinking that both stars start with PE = Gm^2/r0 is that the derivation of that equation is based on the energy required to separate one object from the other a distance r0. Imagine that you hold star 1 steady wh1le moving star 2 away from it - let's call this case 1. Energy is force times distance, so star 2 requires energy to move it distance r0 while star 1 is stationary (and hence star 2 doesn't gain or lose PE). The total change in PE for the system is GM^2/r_0.

you mean star 1 right?

Alternatively consider case 2 - suppose the two stars begin life touching, and then you move them apart by having each move a distance ro/2 in opposite directions. If we define x as the distance each has moved, then the gravitational force betwen them at any point is GM^2/(2x)^2, and the energy required to move each to r0/2 is (1/2)GM^2/r0. Hence each gains half the PE of case 1, but the total change in PE for the system is the same.

One thing to remember is that PE is not an absolute number - unlike KE there really is no place where PE=0. What we must be careful to talk about is the change in PE from one state to another, not an absolute value for PE in either state. It's a subtle distinction, but it's one reason why you can't "add" the PE of two different objects unless you know that they both have the same starting reference point.
Chip as usual - you're the best! But I got the following explanation for this question, saying that U=0 in the inital state. Who's right?

http://www.physicsforums.com/showthr...38#post4117838
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Old Oct 16th 2012, 02:26 PM   #4
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Originally Posted by assaftolko View Post
Chip as usual - you're the best! But I got the following explanation for this question, saying that U=0 in the inital state. Who's right?

http://www.physicsforums.com/showthr...38#post4117838
Yes I meant "star 1" - thanks for catching that. And thanks for the kudos!

And we're both right. You can arbitrarily define PE=0 any place you'd like, and then work out the change in PE as objects move about. For example it's quite common to say that the PE of a mass held h meters above the ground is PE=mgh, but implicit in this is the definition that the PE of that mass at ground level (where ever that is) is 0. But there's nothing to prevent me from defining the PE at ground level is, say, 123 N-m ( I just made that number up). In this case the PE for an object at height h above the ground is 123 N-m + mgh. In both cases the change in PE is mgh. So who's right? We both are! At the end of the day there is no place that has PE absolutely equal to zero except as defined for the purposes of helping to calculate delta PE. For example consider the equation that started this discussion: PE = -GMm/r; it's always a negative value and never zero! But it's a convenient equation and meets the criteria that as an object gains in altitude it has positive change in PE (the PE becomes less negative).

Last edited by ChipB; Oct 16th 2012 at 02:28 PM.
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Old Oct 16th 2012, 11:07 PM   #5
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got it, thanks a lot!
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