Total gravitational potential energy of four objects

• nhmllr
In summary, the gravitational potential energy of four masses arranged at the vertices of a tetrahedron of side length a is -6Gmm/a. This is calculated by considering the work done in bringing in each mass separately, taking into account the forces from the other masses already present. However, since each mass is counted twice, the final answer is divided by two, resulting in an answer of -6Gmm/a.
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?

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

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

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?

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.

nhmllr said:

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?

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

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

What is total gravitational potential energy?

Total gravitational potential energy is the sum of the potential energy of all objects within a system due to the force of gravity.

How is total gravitational potential energy calculated?

Total gravitational potential energy is calculated by multiplying the mass of each object by the gravitational potential energy between that object and all other objects in the system.

Why is total gravitational potential energy important?

Total gravitational potential energy is important because it helps us understand the potential energy stored in a system due to the force of gravity. This can be useful in predicting the behavior and movement of objects within the system.

What factors affect the total gravitational potential energy of four objects?

The total gravitational potential energy of four objects is affected by the masses of the objects, their distances from each other, and the universal gravitational constant.

Can the total gravitational potential energy of four objects be negative?

Yes, the total gravitational potential energy of four objects can be negative if the potential energy between the objects is negative. This can occur when the objects are moving towards each other and the force of gravity is acting in the opposite direction of their motion.

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