The physics behind trackball inflation

[octothorpe]

Okay, it starts with a simple question:

Does the trackball weight change when you inflate it?

My team had a fairly heated argument about this, and somehow I suspect we weren't the only ones. At any rate, to avoid further misunderstandings, let this post try to quash some of the debate about trackball weight vs. inflation:

• An uninflated trackball consists of just the skin.
• An inflated trackball consists of the skin plus a large quantity of air.

Since air does have mass, this increases the mass of the trackball.

1. The density of air at sea level pressure and 20 degrees Celsius is approximately 1.2 kg per cubic meter.
2. The trackball contains about 0.55 meters of air (radius = 20 in = 0.508 m, volume of sphere = 4/3 * pi * r^3 = 0.55).
3. Mass = volume * density, so 0.55 cubic meters * 1.2 kg per cubic meter = 0.66 kg.
4. On Earth, this translates to about 1.46 pounds of additional weight.

HOWEVER, the additional weight (downward force) of the air in the trackball is completely negated by the UPWARD BUOYANT FORCE of the atmosphere on the ball assuming that the trackball is inflated to the same pressure and temperature as the surrounding air.

On buoyant force: basically, the scale doesn't measure extra weight because when you put extra air inside the trackball, you're causing it to displace the same volume of air that would otherwise be weighing down on the scale anyway. The scale doesn't care if the air is inside the trackball or outside it.

THEREFORE, the only way inflating your trackball can cause it to weigh more is if the trackball actually contains more air mass-wise than the volume of air it displaced. For this to happen, the air in the trackball must be at a greater pressure than the outside air.

Pressure and amount of gas are directly proportional thanks to the ideal gas law (pV = nRT, where p is pressure and n is amount of gas), so if your trackball is inflated to twice atmospheric pressure, it will contain twice the amount of gas and thus twice the extra mass. Since you now have 2 masses of air, the weight is twice as much (about 3 lbs) whereas the upward buoyant force (based on the volume of air displaced) remains the same at about 1.5 lbs. Therefore the net added weight (downward force) would be about 1.5 lbs.

There's a pretty good walkthrough expressing these principles with a similar situation at this site:
http://rabi.phys.virginia.edu/105/2006/ps5s.html

So, the gist of this argument is that:

• It's perfectly possible for an inflated trackball to weigh exactly the same as it did uninflated (if it's inflated to atmospheric pressure).
• The above trackball still has a greater mass, meaning it would be slightly harder to accelerate.
• It's also perfectly possible for a trackball to weigh more when inflated (if it's inflated to above atmospheric pressure).
• All the same, it's unlikely that you could get your trackball to weigh 10 pounds if the skin only weighs 7 (that would require inflating it to 3x atmospheric pressure, or 45 PSI).

Okay. Sorry to wax a bit verbose, but I wanted to create a thorough explanation to help clear up ANY doubts about this issue, since there seem to be a fair number. Hope this helps people understand the science behind the problem. I tried to be as careful as possible, double-checking and citing my work, but if I've made any mistake please correct me. Thanks.