Two-Body with external force - energy confusion

Setup

Imagine a two-body system of masses under a classical mechanics model. The separation and mass-ratio doesn't matter for this example. Presume they are initially stationary.

Now suppose that we hold one of the masses, m1, permanently stationary with an external force opposing the gravitational force due to the second mass. This external force changes over time such that it always balances the gravitational force on m1.

Take the frame-of-reference centred on m1 - the stationary mass.

Problem

1. If we think of the two bodies as as system, then we would say that the centre-of-mass will accelerate from stationary, and will be displaced in the same direction as the external force. Therefore, positive work is being done on the system by the external force (energy is added to the system).
2. However, consider the bodies individually. For m1, it experiences no net force. Nor is it displaced. Therefore, we must conclude that no work is done on m1. m2 only experiences a gravitational force, and so is accelerated from stationary in the direction of the force. Therefore work is done on m2.

But, if energy is transferred to the system from an external source (as suggested from 1), which of the two objects received this energy?

It can't have been m1, since it was stationary. But how m2 gain this external energy when the external force acted on m1?

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1 Answer

The system of the two masses does not experience work due to the external force. You are using the wrong displacement. You should use the displacement of the "point of application" of the external force to calculate work for a given force, even if that force is applied to some system. In particular, don't use the center-of-mass displacement.

For your problem, if you do choose to view the situation as a single system composed of two bodies, the displacement of the point of application of the force is zero, and hence no work. The external force does not add energy into this system. The kinetic energy of m2 comes from the stored gravitational potential energy of the system.

Or, if you want to forgo the system treatment, then most people would consider the kinetic energy of m2 coming from the (external) gravitational work done on it by m1 rather than from potential energy.

In the treatment above, I'm using the work-energy theorem written in the form $$W_\text{net,ext} = \Delta K + \Delta U,$$ where the left-hand side is the sum of works done by external forces and I'm being careful to use the displacement of the point of application. The change in potential energy $\Delta U$ arises only from internal interactions, and thus exists only when you consider the two objects as a system. This is not the only way to treat such systems, but it is the one that is most consistent with other areas of physics, like thermodynamics.

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Thanks for your reply, BSM. I don't understand your comment: "The change in potential energy $\Delta U$ arises only from internal interactions, and thus exists only when you consider the two objects as a system." Isn't the potential energy of a system increased by an external force - such as lifting an object above the earth's surface (with the object & Earth defining the system)? – jamesthenabignumber Apr 20 '14 at 19:19
Yes, your statement is correct. Changes in internal energy can arise from external interactions, but for there to be the possibility of potential energy in the first place, there must be an internal interaction between the objects within the system (e.g., gravitational interaction). – BMS Apr 20 '14 at 20:28