# Why does the work done by an internal force differ from the work done by external force?

Let's consider the following situation. We put a body of mass $m$ at a distance $A$ from the center of Earth. We let the Earth attract the body and analyze the situation at a point $B$, closer to the Earth.

Now, the work done by the gravitational force (a conservative force, which seems to be important) is given by:

$$W = GMm\left(\frac{1}{r_B} - \frac{1}{r_A}\right)$$

This work equals the change of the kinetic energy of the system (approx., the Earth didn't budge too much) and the negative change of the potential energy of the system. The mechanical energy hasn't changed, the system is isolated.

What bothers me is this: why doesn't the work done by the gravitational force change the overall energy? It seems inconsistent to say "the work done equals the energy change, BUT not when the work is done by a conservative/internal force". Why does one work differ from another?

Energy is conserved so it can't be created or destroyed. All we can do is change energy from one form to another.

In your example we are changing the potential energy of the mass $m$ into kinetic energy. The increase in kinetic energy must be equal to the decrease otherwise energy wouldn't have been conserved.

By an external force I assume you mean some third party outside the system. To give a slightly ridiculous example this could be me standing well away from the Earth and the mass and poking the mass with a long pole to accelerate it. In this case the energy of the Earth + mass wouldn't be conserved, but also my energy wouldn't be conserved. However the energy of the Earth, the mass and me would be conserved. The distinction between internal and external forces is a bit artificial because all systems are closed and all forces are internal if you look on a big enough scale.

• Thank you for your answer. So we've come to this: work is always a transfer of one kind of energy to the other, but if the force is external, the energy of the system changes because we don't have a complete picture. If instead our system comprised also of the cause of the force, the energy would be conserved in that system. But there's another thing that bothers me still: it seems, that this reasoning derives from experience. That is, there seems to be no proof that an internal force can't increase the system's energy or a work done by an external force can't keep the energy constant. Sep 10, 2014 at 17:54
• Continuation: It seems that we explain it in that way because that's what we observe, not because that's what the theory says. Do we have to bear with it, or am I wrong here? Sep 10, 2014 at 17:59

Work done on a system increases the energy of the system.

But you have to be clear about what system you are talking about. For the system consisting of the Earth and your other object, there is no external force, and no work done on the system. The total energy is constant; it just changes from potential energy to kinetic energy.

If your system is the other object, then there is an external force, gravity due to the Earth. In this case the energy of the system is increasing.

In this simple system, the added energy has to be entirely kinetic energy. We've implicitly made a model of the object as a point particle with no internal structure. Potential energy is the energy of interaction and requires that the internal structure of the object comprises at least two objects. No internal structure means no potential energy.

The fact that the force is conservative has no bearing on the question.

In a closed system, there is nowhere for the energy to go. There is no way for the gravitational force to change the overall energy.

I would simply change

"the work done equals the energy change, BUT not when the work is done by a conservative/internal force"

to

"The work done equals the kinetic energy change".

This just means that you can convert potential to kinetic energy and back.

A non-conservative force means that the energy lost from the system (which cannot be used to return the system to its original state) is effectively being held in an independent external source or sink.

• Good comment: "Work equals the kinetic energy change." Kinetic energy and potential energy interchange into each other by doing work. Regarding conservative forces, this refers to forces that move a round-trip cycle between two forms of energy that return to their original state. When non-conservative forces act, the system cannot return to its original state. A non-conservative force problem may refer to a force that converts coherent macro-mass kinetic energy into thermal energy. If the thermal energy were to flow to the environment, this could be the external source/sink you refer to. Jun 23, 2018 at 18:23