Personally, I’d just use some simple mathematics.
As we all know, the volume of the ball of radius r is equal to (1/3)r(πr2); i.e., (4/3)πr3. You might want to choose your best micrometer to measure “r” from your lead shot. Naturally, there may be variations in the lead shot, so I would measure a few - 50 or so - then figure the average.
You can approximate the volume of a bulb of imprecise shape by assuming the volume of several subsections and using a simple length cubed formula (Boyle’s model). Of course you are thinking that the greater the number of partitions - the more accurate your answer. To strike a balance on the number of subsections to calculate, you’ll probably want to apply Clarke’s 2nd law.
Naturally, for the brand of epoxy you use, you will have already logged and analysed the cured mass under different curing conditions. Everybody does this, right? Just refer to your tabulated data of mass, weight and density the mix and curing environment you intend for your bulb. Extrapolate a weight for any given volume. Of course I know some folk like to formularize the results and program the same into their TI-30X IIS scientic calculator, or similar. My 7 year-old thinks this is too old fashioned, and prefers to build a little java-script routine on his PDA. But I like to just graph the output for easy reference during the further calculations below.
So now we have a fair approximation of the volume within our theoretical mold, and we know the average volume of our lead shot. Now the electronic configuration of lead is [Xe]4f145d106s26p2 (the forum doesn’t display “to the power of” but you all know which numbers are which). I know you are thinking that, if we could measure this, we might be able to use it as a means of identifying impurities through the existence of other elements in our lead shot - which of course will impact on the weight of the same. But we need some equipment which is just not found in the typical home laboratory, and I do like to keep things simple - don’t you?
So instead, we’ll just assume that our lead has a density of 11.3 kg/dm3. Feel free to use a different value if you have determined the actual density of your lead shot.
Now I’m assuming you’ve got a handle on the target weight you want to achieve for your finished bulb. Fortunately, as you’ve already calculated the volume of the theoretical bulb above, you can use a similar approach to calculate the surface area using (of course you know this), good old 2(pi r 2) + (2 pi r)* h. [thats “r to the power of 2” in the first set of paratheses]. You can use this to estimate a weight for the paint you want to use on the finished bulb. Why not calculate the wetted surface and displacement while you are at it - just for fun?
Ah ha! You are obviously thinking - “Now when do we get to Keppler’s Problem?” Well you are right - we get to it now. Of course we need to work out how densely we can pack all those little lead shot spheres into our bulb shape.
Don’t be fooled by that sweet 74.048% answer. Despite the fact that you probably heard that evey mathematicial believes, and every physicist knows this to be the answer - personally I have my doubts. If you think you can acheive the close latic spherical packing necessary to achieve this volume ratio, go for it - but I suspect your bulb will turn out somewhat lighter than anticipated. I think I’d be a little more prudent and assume a standard cubic latice (pi/6) and just go with 52%. It’s easier to take weight off the bulb than to add it after all. I knew you’d agree. (OK smartar#, 52.35987755982933 if you must! But don’t you think you are overdoing things? Be realistic. puleeease!)
So now we have all the variables in hand, or a means by which we can calculate them. It’s obviously a straight-forward matter to work out how many lead shot spheres will fit into our bulb-mold, what they will weigh, what the residual volume of epoxy will weigh (cured) and what your surface paint (if used) will weigh. And the sum is the weight of the bulb.
I asked one of my nerdy-type friends whether he could formularize all of the above. He said…
- Get 6 ozs of lead shot (assuming this is the bulb weight you want).
- Roll it into a ball with some plasticine, Playdoh, or similar. The plasticine serves as a proxy for the weight of the epoxy. It’s proxy epoxy! Or maybe it’s eproxy.
- Remove material from the lead shot/plastercine mix until you get your 6 oz (plus or minus) mix.
- Shape it by hand into the desired bulb shape.
- Cover it with ceram wrap and use it to make your mold. If it’s too squishy, then use it as the model against which you make a more rigid male mold.
- Get some kids to clean the plastercine off the lead shot when finished.
Now I haven’t tried this myself - prefering to use the simple mathematical approach - buy my friend seems to think it’s worth a shot.