I need to know if this exists

[lanchester]

is there an equation that could be used to determine the energy output of a given weight being pulled by gravity?

 

[AlephZero]

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

Work = force * distance.

 

[Andrew Mason]

is there an equation that could be used to determine the energy output of a given weight being pulled by gravity?

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

$$F = \frac{GMm}{R^2}$$

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:

$$dW = \frac{GMm}{R^2}dR$$

so the power output would be:

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

where $\dot r$ is the radial speed or the speed toward the centre.

AM