I kind of got stumped when trying to figure out what is, most likely, a simple formula.

I'm looking for a formula to determine how much energy, in joules per second (or watts, same difference), is expended to keep an object suspended against the force of gravity. I imagine the result would be in joules per second, but honestly am not certain, since every formula I'm finding discussing units of energy are representative of their ability to do work, namely force applied over distance, but there's no distance in this case.

For example, lets say there is a machine with no mechanical advantage or disadvantage that serves no purpose other than to keep a 1 kg object suspended, I.E. NOT moving towards the earth, but not moving away from it either, a net velocity of zero. This machine obviously has to generate a force equal and opposite to the force of gravity exerted upon the 1 kg object, but what I'm having a hard time determining is what the formula would be for the energy that has to be expended to keep this machine operating. In effect, how many Joules per second, or how many Watts, would this machine require to continue operating. It's, as far as I can tell, not doing any work since there's no motion involved so that's where the formulas for energy as defined by its ability to do work are stumping me.

Note - Yes, I realize the object could simply be set on a table. For arguments sake I'm asking about the energy that would be expended if something were actively trying to hold an object off the ground, similar to how a human might hold a weight off the ground, but without all the messy issues with mechanical advantage and disadvantages and the many processes involved with an actual person doing something like this.

I appreciate whatever formulas, explanations or alternative units of measurement folks might provide.

The answer for a complicated machine is the same as for a table. 0 watts.

This is treated in every elementary mechanics textbook as the formula for the work done. No movement, no work.

Since a machine may go about a job in an efficient or inefficient manner, the machine may require more than 0 watts to operate, but all of that is wasted. For example, in the Millikan oil drop which electrostatically levitates an oil drop, when the drop is no moving, 0 work is done and in joules per second 0 work is 0 watts. But some power might have been used by the electrical device depending on design and implementation and losses unrelated to the oil drop's levitation.

Simple Research:

You might find it interesting to read about;

Negative work, mechanical energy, and power.

http://www.glenbrook.k12.il.us/gbssci/phys...ergy/u5l1a.html

http://www.glenbrook.k12.il.us/gbssci/phys...ergy/u5l2b.html

http://www.glenbrook.k12.il.us/gbssci/phys...ergy/u5l1e.html

I was wondering about the same thing and of course found that P = 0 but:
Where does a jet engine lets say that is levitating a L.E.M lose all the efficiency?. So I derived the power for the case below:
If the mass were a spacecraft suspended out from the Earth's atmosphere and not in orbit (tangent velocity zero) it would require to propel some material in order to levitate and not fall back. By doing so, lets say it exerts a force F = Mg in a time dt on the propellant (the mass of which we assume will not change the crafts mass M significantly). Now the displacement of this propellant if integrated ( integrate with all initial conditions set to zero a=dv/dt where v=dv/dt) will give:

l=1/2 * a * t^2 where a = F/m (m=propellant mass)
but,
W = Integral(F*dl) = 1/2 * a * t^2 * F

P=dW/dt
finally:
P= F*v where v is the the final speed of the propellant exiting the engine.
Is this correct?

If this was done by light say a laser, the power required is the tremendous amount of

1kg*9.8m/sec * 300000000m/s = 3GigaJoules/second (hence the most inefficient way to do this)

if it is sitting in a table the propellant that is the table has a mass equal to infinity (or is it Earths mass?!?!) so its acceleration is zero and velocity also. So P = 0Watts.
Now, I know that it doesn't apply this way. In the case of the table the gravitational force is balanced by the electric repulsion force between the two materials, but It seems to make sense, it shouldn't!!

So, the case of the spacecraft lies in between of the two above.

Can anyone tell me if I'm correct?