Question

An empty steel pipeline with massless endcaps has an outer diameter, $D$, and thickness, $t$. The density of steel is $7850\ \mathrm{kg/m^3}$. The critical $D/t$ ratio at which the pipeline starts floating in seawater of density $1025\ \mathrm{kg/m^3}$ is _________ (rounded to two decimal places).

Hint:

For thin-walled sealed pipes, set the mass of steel shell equal to the displaced water of the {outer} cylinder to get the floating criterion.

Answer 1

- Neutral floatation $\Rightarrow$ weight of steel shell $=$ buoyant force on outer volume. For unit length, with $R=D/2$ and inner radius $r=R-t$,
\[ \rho_s\,\pi (R^2-r^2) = \rho_w\,\pi R^2 \;\Rightarrow\; \rho_s\,[2Rt - t^2] = \rho_w\,R^2. \] - Divide by $t^2$ and set $x=\dfrac{R}{t}$:  $\rho_w x^2 - 2\rho_s x + \rho_s = 0$.
- With $\rho_s=7850,\ \rho_w=1025$, the valid root ($x>1$) is $x=14.7996$.
- Hence $D/t = 2x = 29.5992 \approx \mathbf{29.60}$ (two decimals).