I am helping with a small construction project that will involve a 200 pound log, 8" diameter, 8 feet long, in a fresh water lake.

I know that the log would displace 326 pounds of water if it were submerged, but since it only weighs 200 pounds, some part of the log will be floating above the waterline.

It has been 60 years since my last calculus class, I no longer have the math to solve this: How much of the log will be above the water?

tanstaafl.

Heh.. 38.65% of it (aka. 126lbs).

But you probably really want to know what cross-section of the log would be above water. That will require a bit of calculation. My preferred way to solve it would be a simple program, iterating on the "how much" part until it hits 38.65%. Or calculus, which I have totally forgotten at this point.

But empirically, the top of the log should be about 3.3" above the water in this case. I think.

https://www.mathopenref.com/segmentareaht.html

I also received an answer to this when I posted a query on Quora.com. (If you're not familiar with that website, it's worth looking into. Lots of really smart people hang out there. Similar to another website I visit a lot where bunch of stereo geeks hang out.)

Attached is a MS-Word file with his reply in it.

tanstaafl.

Heh, pretty much the same technique to arrive at the 3.3" approximation, then. I just simplified it to working on a 2D circle rather than a 3D cylinder.

Of course, it'll ride lower when it gets waterlogged. I remember sailing out between Seattle and Vancouver, seeing the escaped logs (aka tree trunks - 3 ft diameter) floating vertically, with just a few inches above the surface.

Had to be very careful - and no night sailing - or it'd hole the boat.