add notes on buoyancy

mechanical/mmme1048_fluid_mechanics.md

@@ -466,3 +466,41 @@ M_{OO} &= F_py_p = \int_{area}\! \rho gh^2 \,\mathrm{d}A \\
 \\
 y_p = \frac{M_{OO}}{F_p}
 \end{align*}
+
+## Buoyancy
+
+### Archimedes Principle
+
+> The resultant upwards force (buoyancy force) on a body wholly or partially immersed in a fluid is
+> equal to the weight of the displaced fluid.
+
+When an object is in equilibrium the forces acting on it balance.
+For a floating object, the upwards force equals the weight:
+
+$$mg = \rho Vg$$
+
+Where $\rho$ is the density of the fluid, and $V$ is the volume of displaced fluid.
+
+### Immersed Bodies
+
+As pressure increases with depth, the fluid exerts a resultant upward force on a body.
+There is no horizontal component of the buoyancy force because the vertiscal projection of the body
+is the same in both directions.
+
+### Rise, Sink, or Float?
+
+- $F_B = W$ \rightarrow equilirbrium (floating)
+- $F_B > W$ \rightarrow body rises
+- $F_B < W$ \rightarrow body sinks
+
+### Centre of Buoyancy
+
+Buoyancy force acts through the centre of gravity of the volume of fluid displaced.
+This is known as the centre of buoyancy.
+The centre of buoyancy does not in general correspond to the centre of gravity of the body.
+
+If the fluid density is constant the centre of gravity of the displaced fluid is at the centroid of
+the immersed volume.
+
+![](./images/vimscrot-2021-12-21T15:08:22,285753421+00:00.png)
+