## A Question of Balance

Here’s an interesting physics problem for you, based on the idea that the mass of a set of bodies changes if the energy of their mutual interactions changes according to Einstein’s famous formula “E=mc2“.

Four identical masses are placed at rest in pairs either side of an extremely sensitive balance in a symmetrical way such that the distance between the members of a pair is identical for each pair and the centre of mass of each pair is equally spaced from the fulcrum of the balance. In this configuration the system is in equilibrium and the balance is level.

As illustrated schematically in the graphic, one pair of weights is adjusted by displacing each weight slightly away from the centre of mass of the pair by an equal and opposite distance, thus keeping the position of the centre of mass of the pair constant. The other pair of weights is not adjusted.

Assuming that the balance is sufficiently sensitive to detect the slight change in mass associated with the gravitational interactions between the masses in each pair, does the balance move?

If it does move which pair moves up: the displaced pair or the undisturbed pair?

### 2 Responses to “A Question of Balance”

1. Chris_C Says:

While they are moving the mass goes up, but returns to nominal as they stop, so the RH (undisturbed side) goes up and then returns to the initial position. [This is of course assuming an otherwise empty Universe, with the balance in a vacuum above a uniform gravitational field in the normal manner of physics problems.] Or am I being too simplistic? Do the moving masses generate gravitational waves significant on this scale, considering that the mass increase is so small that the effects will be next to zero in any imaginable situation?

2. Martin Says:

OK, I’ll bite and give what looks like the answer the question is expecting… In moving the two balls apart you’ve done work against their mutual gravitational attraction. So the mass of the displaced pair increases, and the balance goes up on the undisplaced side.