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Consider a system on the level ground (let this be the datum line). Only gravitational force(force 1) and a vertically upward force(force 2) are acting on it and there is no heat transfer. When the upward force, force 2, is greater than gravitational force, force 1, there is a net-upward force which causes the system to accelerate up. After a certain distance, force 1 and force 2 become equal and the system starts moving at a constant velocity. The net-upward force multiplied by the distance traveled gives the work done on the system by the net-force which is equal to the increase in kinetic and potential energy of the system during acceleration.

Now, when it is moving at a constant velocity, as there is no net force acting on the system, the net-work would be zero. But still, the potential energy is increasing with the kinetic energy remaining constant. How is the energy in the system increasing without any work being done on it?

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The effect of (Newtonian) gravity can be included in a description by means of both

  • a potential energy $U_g$, or
  • a force $F_g$,

and you should not include them both at the same time when applying the principle of energy conservation.

That means that, at constant speed, you consider that either

  • the net force $F_{\mathrm{res}} = F+F_g$ is zero, and then its work, $W_{F_{\mathrm{res}}}$, is also zero and that's consistent with constant energy; or

  • there's a force $F=-F_g$ acting on the body, whose work is increasing the body's energy (that happens to be of potential, not of kinetic type): $\Delta E = W_F = \int F \mathrm{d}s = - \int F_g \mathrm{d}s = -W_{F_g} \equiv \Delta U_g$.

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  • $\begingroup$ Thank you @stafusa. So in the energy equation if we consider the work done by the upward directed force and the gravitational force then there is ono potential energy to be considered. TThis helped me get a better understanding of gradients of the scalar potentials and the different potential energy as a whole.That is the work done by the potential forces. $\endgroup$ – GRANZER Dec 19 '17 at 14:07

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