I need to know if this exists

Yes, it's the same equation as the work done by any other force:

Work = force * distance.

 

I think you mean power (energy per unit time) of the weight. Since the gravitational force is:

[tex]F = \frac{GMm}{R^2}[/tex]

where R is the distance of the object from the centre of mass of the large mass (toward which the weight is gravitating), the change in energy by moving a distance dR closer to the centre would be:

[tex]dW = \frac{GMm}{R^2}dR[/tex]

so the power output would be:

[tex]P = dW/dt = \frac{GMm}{R^2}\frac{dR}{dt} = \frac{GMm}{R^2}\dot r[/tex]

where [itex]\dot r[/itex] is the radial speed or the speed toward the centre.

AM