Floating 3

The block is submerged to a depth of $h$ , as indicated on the Figure.

The center of mass of the block is at an elevation of $a/\sqrt{2}-h$ from the water (red dot on the Figure). The center of the mass of the displaced water is at depth of $h/3$ below the water level (blue dot). The weight of the block and the weight of the displaced water are equal to each other, resulting in $W=gch^2 \rho_w=gca^2 \rho$ , where $g$ is the acceleration of gravity. Therefore $h^2=\rho a^2$ , if we use $\rho_w=1$ .

Imagine that the block is tilted by a small angle $\alpha$ , and the displaced water water does not change its shape. In this situation the weight vector of the block does not line up with the upward force due to the displaced water. There will be a torque of $\tau_1=W \ell$ acting on the block, where $\ell=\alpha (\frac{a}{\sqrt{2}}-\frac{2}{3}h)$ is the distance between the two parallel an opposite forces. We get

$\tau_1=W \ell=\alpha (\frac{a}{\sqrt{2}}-\frac{2}{3}h)gch^2$

This torque is counter-acted by a wedge-shaped piece of water that flows from one side of the block to the other side, so that the water surface remains horizontal. For small $\alpha$ , the height of the water at a distance $x$ from the center line is $\alpha x$ . The weight of a slice of water of width $dx$ is $dW=\rho_w g c dx (\alpha x)$ and the torque is $d\tau=x \rho_w g c dx (\alpha x)$ . The torque due to this water amount is

$\tau_2= 2g \alpha c \int_{0}^{h}x^2 dx = \frac{2}{3} g \alpha c h^3$ ,

where the factor 2 stands for the left and the right side from the center line and we assumed that the density of water is $\rho_w=1$ . For the configuration to be stable one needs $\tau_2>\tau_1$ . At the at the critical density the two torques are equal

$\left(\frac{a}{\sqrt{2}} -\frac{2}{3}h\right) h^2 \alpha c g = \frac{2}{3}h^3 \alpha c g$ or, using $h^2=\rho a^2$ ,

$\frac{1}{\sqrt{2}} -\frac{2}{3}\sqrt{\rho} = \frac{2}{3} \sqrt{\rho}$

The solution is $\rho=9/32= 0.28125$ .

Comments There is a huge literature on the stability of floating bodies, because the topic has an immediate and critical application to the stability of boats and ships. Most, if not all, of the discussions first establish the concept of "metacenter", and derive the rest from there. In my solution I did not use that concept.

1. A slightly mathematical but very detailed and precise treatment is in a 1991 paper by E.M. Gilbert that is available here .

2. Nick Berry's "How things float" WEB page has a detailed discussion and also numerical simulations about the topic. Lots of graphical illustrations! However, there is no analytical derivation of the magic number $\rho=9/32$ , instead we have a statement: ""It's in interesting exercise in geometry to work out the centroids of the underwater sections under the various conditions of differing number of potential corners under the surface (maybe a future article?), but here, without any explanation of the derivations, are the results..."

3. The Floating block simulator by Fu-Kwun Hwang, Fremont Teng and Loo Kang Wee allows you to adjust many parameters, and it also displays the time evolution in the position-velocity space. If you want to speed up the process leading to the equilibrium, click on "Display", "Drag coefficient" and adjust the value to 0.6.

4. I have 3 other problems related to floating objects, 1 here and 2 here and 3 here .

The short answer is that between density ratio of 9/32 and 23/32, the log will rest at 45° angle.

Therefore,

Answer=9/32=0.281