russ_watters said:
I don't agree either:
If the volume of air displaced by both systems is the same, the buoyant force provided by the air must be the same. In both cases, you have, sitting on the scale, a mass of air (in the ping pong ball), a mass of ping pong ball and a mass of water. And it is buoyed by displacing a volume of air equal to the volume of water and volume of the ping pong ball in both cases.
I think you are wrong with regard to buoyancy. Buoyancy is not a fundamental force. It's a consequence of a pressure gradient (and pressure in turn is not a fundamental force). It can be defeated.
However, that's not the question here. The question is, to paraphrase a year old song, "What does the
scale say?" In that regard, I'm most definitely wrong. I'll look at two extremes, a scale whose trays have tiny little bumps of negligible area that raise the flask a tiny bit above the tray, versus a perfectly flat tray combined with a flask with a perfectly flat bottom so no air can get underneath and cause buoyancy.
I'll assume a flask with thin vertical walls and a thin bottom. In the first case, the scale will detect the atmospheric pressure at the bottom of the flask plus the apparent weight of the flask, including buoyancy: W_{\text{tot}} = AP_{\text{bottom}} + (mg - A(P_{\text{bottom}} - P_{\text{top}})), where A is the area of the interior of the flask, and P_{\text{bottom}} and P_{\text{top}} are the atmospheric pressure at the bottom and top of the flask. After canceling common terms, this reduces to W_{\text{tot}} = mg + AP_{\text{top}}. In the second case, the scale will detect the weight of the contents of the flask plus the weight of atmosphere acting on the top of the flask: W_{\text{tot}} = mg + AP_{\text{top}}. That's the same result as the first case! Whether or not buoyancy acts on the flask and it's contents is irrelevant. What is relevant is the mass of the contents of the flask and the pressure at the top.
This contribution from atmospheric pressure is important in comparing what a scale can sense with regard to an intact versus crushed ping pong ball. The top of the liquid will be slightly higher in the case of the intact ping pong ball suspended within the water compared to a ping pong ball floating atop an equal amount of water. So what does the
scale say?
The trivial explanation is to assume that density and gravity don't change that much over the sub-centimeter level. With this assumption, the hydrostatic equilibrium condition, \frac{dp}{dz}=-\rho g, says that pressure decreases linearly with increased height. The non-trivial explanation, which involves the lapse rate for a non-ideal gas, simplifies to the trivial explanation in the case of centimeter-level changes. (I've wasted too much time playing with the math.) This pressure change is important.
Bottom line: If the pressure inside the ping pong ball is close to local atmospheric pressure, there is essentially no difference in the weight sensed by the scale between a submerged versus floating ping pong ball.
