Total gravitational potential energy of four objects

[nhmllr]

Homework Statement

Four masses m are arranged at the vertices of a tetrahedron of side length a. What is the gravitational potential energy of this arrangement?

(answer is -6Gmm/a)

Homework Equations

gravitational potential energy = -Gmm/r

The Attempt at a Solution

One mass is "a" away from another mass. So the gravitational potential energy there is -Gmm/a. But it is attracted to two other masses, so the gravitational potential energy of this one mass is -3Gmm/a. So for all four masses, the total potential energy should be -12Gmm/a, right? So why is this not the answer? Thanks

 

[haruspex]

Imagine removing each object to infinity, one at a time. How much energy is needed for the first, how much for the second...?

 

[voko]

Potential energy is a function that depends on position (relative to the gravitating masses). The answer does not have any such dependency, which means you are asked about the value of the function at some particular location. Where is it?

 

[dx]

Assemble the configuration one mass at a time, bringing them in from infinity. To bring in the first mass, no work is done. Bringing in the second mass, the work done is -Gmm/a. Now bringing in the third mass, we must consider the forces from both the masses that have been already been brought in, so the work is the sum -Gmm/a - Gmm/a = -2Gmm/a.

What is the work done in bringing in the third mass?

Once you have them all, add them up.

 

[ehild]

Homework Statement

Four masses m are arranged at the vertices of a tetrahedron of side length a. What is the gravitational potential energy of this arrangement?

(answer is -6Gmm/a)

Homework Equations

gravitational potential energy = -Gmm/r

The Attempt at a Solution

One mass is "a" away from another mass. So the gravitational potential energy there is -Gmm/a. But it is attracted to two other masses, so the gravitational potential energy of this one mass is -3Gmm/a. So for all four masses, the total potential energy should be -12Gmm/a, right? So why is this not the answer? Thanks

A point mass alone does not have potential energy. A pair of masses has, and the energy of the pairs add up. You can make 6 pairs from the four masses.

You have counted the potential energy of each mass twice. If mass 1 has potential energy from masses 2, 3, 4, it includes also the potential energy of masses 4,3,2 from mass 1. So you have to divide that -12Gmm/a by two.



ehild

 

[nhmllr]

Ahhhh I see. Both explanations made a lot of sense. Thanks!