Total gravitational potential energy of four objects

  • Thread starter nhmllr
  • Start date
  • #1
nhmllr
185
1

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
 

Answers and Replies

  • #2
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
38,455
7,965
Imagine removing each object to infinity, one at a time. How much energy is needed for the first, how much for the second...?
 
  • #3
voko
6,054
391
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?
 
  • #4
dx
Homework Helper
Gold Member
2,119
41
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.
 
  • #5
ehild
Homework Helper
15,543
1,915

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
 
  • #6
nhmllr
185
1
Ahhhh I see. Both explanations made a lot of sense. Thanks!
 

Suggested for: Total gravitational potential energy of four objects

Replies
7
Views
369
Replies
2
Views
643
  • Last Post
Replies
2
Views
370
Replies
7
Views
324
Replies
1
Views
341
  • Last Post
Replies
1
Views
244
Replies
1
Views
365
  • Last Post
Replies
4
Views
451
Replies
2
Views
514
Replies
11
Views
505
Top