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.

[Doug G]

Sounds pretty good to me, I don't claim to be an ball pressure-weight expert, but I have do have a degree in physics and have taught physics for 14 years and would give you an A for your presentation of the situation. You could've have got an A+, but you missed the part about the difference in composition of the gas inside the ball and the surrounding air (the compressor may add a few dirty molecules). But seriously - you summed it up nicely!!

We weighed our ball, inflated to 40" in size, and it came out to about 7.2 lbs. After reading the manual, it said it will weigh 10 pounds. I thought our scale was off or needed to be calibrated, but I think your findings support our scale's reading.

[Woody1458]

well in theory with the added weight of the skin any air on the inside would be compressed past 1 atm (~14.7 psi) (PROOF: unplugging a partially inflated ball will still deflate signaling that the interior pressure was greater then the exterior.) so any inflation would add mass, but should be negligable till ball reaches maximum volume, dictated by the material's elastic constant.

(I always love myself a good debate )

[Kevin Sevcik]

The pressure cs. diameter function is probably highly non-linear once you reach the size of the nylon and start trying to expand that. All the same we whipped out a ball pump to hook up to the trackball after it was inflated and it read something less than 2 psi. Probably 1. So there will be some additional weight. I imagine the manufacturing variances will compound the weight fluctuations from air pressure with slightly thicker, heavier balls requiring even more air to inflate. Overall, though, I wouldn't expect for than +/- .25 lb variation out to 3 SDs

[MrForbes]

The mass of the air inside the ball could make a difference in a design that "shoots" the ball....that's a significant added load....that I had not thought of!

thanks for the lesson!

[Miniflash]

i just fill it up ,weigh my self, then weigh myself while holding the ball and take the diffrence... easy math for us k.i.s.s people ... then to double chech take a string and take the circumfrence and divide by pi.... easy enough to the point of i have a 10lb 40 in diamater trackball

[mneary]

i just fill it up ,weigh my self, then weigh myself while holding the ball and take the diffrence... easy math

But you have to move the ball, too. For that you'll need to know the mass, as described above.