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Date: 1991 

I have a student with questions that I cannot always answer. Please help me with the following (my student is 13 and in middle school. question 1--We say that a system which is bound by the nuclear force is le less massive due to the absence of the energy required to "separate" the system. An analogue can seemingly be made with the gravitational force, in that a gravitationally bound system should possess a smaller mass thanits (unbound) constituents. (this effect would, of course be minute in everyday circumstances, due to the weak relative strength of f the gravitational interaction, however in a case such as two black holes in orbit, the effect should become notable.) We must also, however take into account the added energy (and thereby mass) of the force itself, which of course, increases, as does the binding energy, as the system becomes more strongly bound. Which effect takes precedence; i.e. is the overall m mass of a gravitationally bound system less than, greater than, or approximately equal to the mass of its unbound constituents?

When two objects that attract one another (whether by gravitation or anything else) are far apart, let us suppose they are not moving, so the total energy is basically given by the sum of the two masses (multiplied by c^2). If they start moving they gain kinetic energy and the mass goes up. Suppose we just let them gradually accelerate towards one another from a long distance away. The total energy should remain constant (except for radiation effects - if they are charged they will radiate electromagnetically, and there will also be gravitational radiation). That is, the kinetic energy at any time is just equal to the lowering of the energy due to the attractive interaction at that distance. So the mass of the system considered as a whole (two objects moving towards each other), stays constant, because the energy is constant. If radiation is included, the radiated energy goes off in all directions, and so the system has lost that energy - ie. the total energy goes down. Eventually, if the two objects combine in a "bound state" they are close together and not moving fast, so their total energy is lowered relative to the original sum of the masses, and that energy must have disappeared in some form of radiation.

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