From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- source/know/concept/metacentric-height/index.md | 179 ++++++++++++++++++++++ source/know/concept/metacentric-height/sketch.png | Bin 0 -> 71440 bytes 2 files changed, 179 insertions(+) create mode 100644 source/know/concept/metacentric-height/index.md create mode 100644 source/know/concept/metacentric-height/sketch.png (limited to 'source/know/concept/metacentric-height') diff --git a/source/know/concept/metacentric-height/index.md b/source/know/concept/metacentric-height/index.md new file mode 100644 index 0000000..7588196 --- /dev/null +++ b/source/know/concept/metacentric-height/index.md @@ -0,0 +1,179 @@ +--- +title: "Metacentric height" +date: 2022-03-11 +categories: +- Physics +- Fluid mechanics +layout: "concept" +--- + +Consider an object with center of mass $G$, +floating in a large body of liquid whose surface is flat at $z = 0$. +For our purposes, it is easiest to use a coordinate system +whose origin is at the area centroid +of the object's cross-section through the liquid's surface, namely: + +$$\begin{aligned} + (x_0, y_0) + \equiv \frac{1}{A_{wl}} \iint_{wl} (x, y) \dd{A} +\end{aligned}$$ + +Where $A_{wl}$ is the cross-sectional area +enclosed by the "waterline" around the "boat". +Note that the boat's center of mass $G$ +does not coincide with the origin in general, +as is illustrated in the following sketch +of our choice of coordinate system: + + + + + +Here, $B$ is the **center of buoyancy**, equal to +the center of mass of the volume of water displaced by the boat +as per [Archimedes' principle](/know/concept/archimedes-principle/). +At equilibrium, the forces of buoyancy $\vb{F}_B$ and gravity $\vb{F}_G$ +have equal magnitudes in opposite directions, +and $B$ is directly above or below $G$, +or in other words, $x_B = x_G$ and $y_B = y_G$, +which are calculated as follows: + +$$\begin{aligned} + (x_G, y_G, z_G) + &\equiv \frac{1}{V_{boat}} \iiint_{boat} (x, y, z) \dd{V} + \\ + (x_B, y_B, z_B) + &\equiv \frac{1}{V_{disp}} \iiint_{disp} (x, y, z) \dd{V} +\end{aligned}$$ + +Where $V_{boat}$ is the volume of the whole boat, +and $V_{disp}$ is the volume of liquid it displaces. + +Whether a given equilibrium is *stable* is more complicated. +Suppose the ship is tilted by a small angle $\theta$ around the $x$-axis, +in which case the old waterline, previously in the $z = 0$ plane, +gets shifted to a new plane, namely: + +$$\begin{aligned} + z + = \sin(\theta) \: y + \approx \theta y +\end{aligned}$$ + +Then $V_{disp}$ changes by $\Delta V_{disp}$, which is estimated below. +If a point of the old waterline is raised by $z$, +then the displaced liquid underneath it is reduced proportionally, +hence the sign: + +$$\begin{aligned} + \Delta V_{disp} + \approx - \iint_{wl} z \dd{A} + \approx - \theta \iint_{wl} y \dd{A} + = 0 +\end{aligned}$$ + +So $V_{disp}$ is unchanged, at least to first order in $\theta$. +However, the *shape* of the displaced volume may have changed significantly. +Therefore, the shift of the position of the buoyancy center from $B$ to $B'$ +involves a correction $\Delta y_B$ in addition to the rotation by $\theta$: + +$$\begin{aligned} + y_B' + = y_B - \theta z_B + \Delta y_B +\end{aligned}$$ + +We find $\Delta y_B$ by calculating the virtual buoyancy center of the shape difference: +on the side of the boat that has been lifted by the rotation, +the center of buoyancy is "pushed" away due to the reduced displacement there, +and vice versa on the other side. Consequently: + +$$\begin{aligned} + \Delta y_B + = - \frac{1}{V_{disp}} \iint_{wl} y z \dd{A} + \approx - \frac{\theta}{V_{disp}} \iint_{wl} y^2 \dd{A} + = - \frac{\theta I}{V_{disp}} +\end{aligned}$$ + +Where we have defined the so-called **area moment** $I$ of the waterline as follows: + +$$\begin{aligned} + \boxed{ + I + \equiv \iint_{wl} y^2 \dd{A} + } +\end{aligned}$$ + +Now that we have an expression for $\Delta y_B$, +the new center's position $y_B'$ is found to be: + +$$\begin{aligned} + y_B' + = y_B - \theta \Big( z_B + \frac{I}{V_{disp}} \Big) + \approx y_B - \sin(\theta) \: \Big( z_B + \frac{I}{V_{disp}} \Big) +\end{aligned}$$ + +This looks like a rotation by $\theta$ around a so-called **metacenter** $M$, +with a height $z_M$ known as the **metacentric height**, defined as: + +$$\begin{aligned} + \boxed{ + z_M + \equiv z_B + \frac{I}{V_{disp}} + } +\end{aligned}$$ + +Meanwhile, the position of $M$ is defined such that it lies +on the line between the old centers $G$ and $B$. +Our calculation of $y_B'$ has shown that the new $B'$ always lies below $M$. + +After the rotation, the boat is not in equilibrium anymore, +because the new $G'$ is not directly above or below $B'$. +The force of gravity then causes a torque $\vb{T}$ given by: + +$$\begin{aligned} + \vb{T} + = (\vb{r}_G' - \vb{r}_B') \cross m \vb{g} +\end{aligned}$$ + +Where $\vb{g}$ points downwards. +Since the rotation was around the $x$-axis, +we are only interested in the $x$-component $T_x$, which becomes: + +$$\begin{aligned} + T_x + = - (y_G' - y_B') m \mathrm{g} + = - \big((y_G - \theta z_G) - (y_B - \theta z_M)\big) m \mathrm{g} +\end{aligned}$$ + +With $y_G' = y_G - \theta z_G$ being a simple rotation of $G$. +At the initial equilibrium $y_G = y_B$, so: + +$$\begin{aligned} + T_x + = \theta (z_G - z_M) m \mathrm{g} +\end{aligned}$$ + +If $z_M < z_G$, then $T_x$ has the same sign as $\theta$, +so $\vb{T}$ further destabilizes the boat. +But if $z_M > z_G$, then $\vb{T}$ counteracts the rotation, +and the boat returns to the original equilibrium, +leading us to the following stability condition: + +$$\begin{aligned} + \boxed{ + z_M > z_G + } +\end{aligned}$$ + +In other words, for a given boat design (or general shape) +$z_G$ and $z_M$ can be calculated, +and as long as they satisfy the above inequality, +it will float stably in water (or any other fluid, +although the buoyancy depends significantly on the density). + + + +## References +1. B. Lautrup, + *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition, + CRC Press. diff --git a/source/know/concept/metacentric-height/sketch.png b/source/know/concept/metacentric-height/sketch.png new file mode 100644 index 0000000..1ee1d04 Binary files /dev/null and b/source/know/concept/metacentric-height/sketch.png differ -- cgit v1.2.3