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Logs sometimes float vertically in a lake because one end has become water-logged and denser than the other. What is the average density of a uniform-diameter log that floats with 20.0$\%$ of its length above water?

$\rho_{\mathrm{L}}=800 \mathrm{kg} \mathrm{m}^{-3}$

Physics 101 Mechanics

Chapter 11

Fluid Statics

Fluid Mechanics

Cornell University

University of Michigan - Ann Arbor

Simon Fraser University

Lectures

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we're told that log sometimes float vertically in a lake because one end has become waterlogged and denser than the other. You want to figure out what the average density of a uniform diameter law that floats with 20% of its length above the water is. So I've drawn a little schematic here. So here's our log. Floating vertically, it has a uniform diameter all the way down. And so we have 20% of its length, your point to L above the water and which means that 80% is below water. So what we need to figure out is we need to figure out what the density of this log is, so we need to figure out what it's masses. And so we need that we can figure out what the buoyancy forces and because it's an equilibrium, the wait times he must equal the Clancy force. So we have to write that down here. The weight equals the buoyancy force, and the buoyancy force is the volume of the log submerged times, the density of the water, times gravity. And we also know that the so that is us. The wake here is the density of the log times of total volume of the law of gravity. So then we can, um you know that from these two expressions you have that density of the log kinds of total volume of the log. He was the density of water times the volume of the logs that submerged. We know that, um, the volume of the log submerged for the volume of the log is its cross sectional area kinds of totally and the volume that submerged in a cross sectional area times, you know, 0.8 times its total length. So a and L cancel out on both sides and we're left with, um that the density of the log is your 0.8 times the density of water and if we use the density of water or fresh water is 1000 kilograms. For me to cube, we get that. The density of the log is 800 kilograms per cubic meter

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