Gravitational Potential Energy & Mass Change: Andrew's Question

• B
• andrew s 1905
In summary: Therefore the total mass of the system increases by ##W/c^2##.In summary, the mass of a system consisting of two initially together masses M and m, which are separated by work, will increase by an amount equal to the work done divided by the speed of light squared. This is in accordance with the equation e=mc^2 and is analogous to chemical or nuclear binding. The overall mass of the system will increase due to the increase in gravitational binding energy. However, the treatment of the gravitational field and energy source must be considered in order to fully understand the change in mass.
andrew s 1905
TL;DR Summary
Does the work done in separating two masses increase their mass in accordance with e=mc^2
If I start with two, otherwise isolated, masses M and m initially together and do work to separate them then the work done, I assume, goes into the gravitational binding energy between them. Will the system of mass M and m have increased in mass due to this in accordance with e=mc^2?

I believe yes as it is broadly analogous to chemical or nuclear binding.

If I am wrong please provide some pointers in the right direction.

Thanks Andrew
PS this is not a homework question I am 70 and not formally studying it's for personal interest.

I think for this question to make sense you need to be far enough away from the masses for them to be treated as a single point mass at all times. Otherwise you can't really characterise the gravitational field in terms of "the mass", since it depends on the mass distribution even in Newtonian gravity.

But then you run into the problem of where the energy came from to move the masses apart. Was it already there, e.g. in the form of rocket fuel? Or was it supplied from outside, e.g. by shining a laser on to a solar sail on one of the masses? In the first case I would not expect the overall mass to change because no energy was added to the system, but in the second I would expect it to change because energy was supplied.

Thanks, it was intended to be very simplified and I did intend the energy to be supplied from outside the system. Regards Andrew

Last edited:
andrew s 1905 said:
Will the system of mass M and m have increased in mass due to this in accordance with e=mc^2?
Assuming we are working with point masses, the mass of the system is greater than ##M+m## and it increased by an amount ##W/c^2##, where ##W## is the work you did separating them.

Mister T said:
Assuming we are working with point masses, the mass of the system is greater than ##M+m## and it increased by an amount ##W/c^2##, where ##W## is the work you did separating them.
If we are treating gravitational potential energy has having a mass equivalent then...

The mass of a gravitationally bound system will be less than the sum ##M+m##. But after being separated by the applied work, the energy deficit ("binding energy") is reduced by an increment of ##W/c^2##.

1. What is gravitational potential energy?

Gravitational potential energy is the energy an object has due to its position in a gravitational field. It is the energy that is stored in an object as a result of its height above the ground.

2. How is gravitational potential energy calculated?

The formula for calculating gravitational potential energy is GPE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object above the ground.

3. What is the relationship between gravitational potential energy and mass?

The gravitational potential energy of an object is directly proportional to its mass. This means that as the mass of an object increases, its gravitational potential energy also increases.

4. Can gravitational potential energy change due to a change in mass?

Yes, gravitational potential energy can change due to a change in mass. As the mass of an object increases or decreases, its gravitational potential energy also changes accordingly.

5. How does a change in height affect gravitational potential energy?

A change in height also affects gravitational potential energy. As an object is raised to a higher height, its gravitational potential energy increases, and as it is lowered to a lower height, its gravitational potential energy decreases.

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