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Dickfore
Dickfore is offline
#15
Jul31-11, 12:15 PM
P: 3,015
For me, it is just a bookkeeping convention. Let me clarify. The work-energy theorem says that:
[tex]
W_{\mathrm{tot}} = K_{f} - K_{i}
[/tex]
All the forces that act on a system can be split into two broad categories.
1) Forces whose work is independent on the path between two points, or, equivalently, is zero on any closed trajectory. These forces are called conservative. For them, one can assign a function of coordinates (up to an arbitrary additive constant), [itex]V = V(\vec{r})[/itex] such that the work done by the conservative force on the system is:
[tex]
W_{\mathrm{cons}} = V_{i} - V_{f}
[/tex]
The force is derivable from this function through the negative gradient:
[tex]
\vec{F}_{\mathrm{cons}} = -\vec{\nabla} V(\vec{r})
[/tex]

2) Forces that do not satisfy the above criterion. They are called non-conservative forces. There are several subcategories, among which I can think of the following:
a) Dissipative - the work done on any trajectory is negative;
b) Gyroscopic - the work is zero on any trajectory, because the force is always perpendicular to the velocity vector

Now, let us rewrite the work-energy theorem by splitting the work done by conservative and non-conservative forces:
[tex]
W_{\mathrm{cons}} + W_{\mathrm{noncons}} = K_{f} - K_{i}
[/tex]
[tex]
(V_{i} - V_{f}) + W_{\mathrm{noncons}} = K_{f} - K_{i}
[/tex]
[tex]
W_{\mathrm{noncons}} = (K_{f} - K_{i}) + (V_{f} - V_{i})
[/tex]
[tex]
W_{\mathrm{noncons}} = (K_{f} + V_{f}) - (K_{i} + V_{i})
[/tex]
Now, we can define a quantity:
[tex]
E = K + V
[/tex]
called total mechanical energy. The function [itex]V(\vec{r})[/itex] is called potential energy of the system in a force field. Then, the work-energy theorem simply states that the change of the total mechanical energy is equal to the work done by non-conservative forces:
[tex]
W_{\mathrm{noncons}} = E_{f} - E_{i}
[/tex]
If the work of the non-conservative forces is zero, we arrive at the conclusion that the total mechanical energy is conserved, hence the name.

So, it is a matter of convention. The only important thing is to not double count. If you attribute the potential energy to the system, then do not count the work done by the conservative forces in the work-energy theorem. If you use the work-energy theorem in its original form (with only the kinetic energy), then by all means count the work of the conservative forces. It is a simple difference of the initial and final potential energy.

According to the first option, the answer to your question is that the total mechanical energy is decreased because of the negative work done by the external force that opposes the gravitational force.

According to the second option, the answer is that the kinetic energy remains unchanged (and zero) because the external force is equal by magnitude and opposite in direction to the gravitational force. Thus the works done by these forces are exactly opposite. But, the work done by the gravitational force is equal to the difference of the initial and final potential energy.