Buoyancy Knowledge-Based Study

Study notes

1. Hydrostatic Pressure Differences

   When an object is immersed in a stationary fluid, the fluid exerts normal forces against every part of the object's surface. Because hydrostatic pressure increases linearly with depth according to the relation $P = P_0 + \rho_{\text{f}} g h$, the pressure exerted on the bottom surface of the object is strictly greater than the pressure exerted on its top surface.

   Consider a submerged rectangular cylinder of height $h$ and cross-sectional area $A$:

   • The downward force exerted by fluid pressure at the top surface is $F_1 = P_1 A$.
   • The upward force exerted by fluid pressure at the bottom surface is $F_2 = P_2 A$.
   • Horizontal forces acting on opposite vertical sides cancel each other completely due to symmetry.

   The net vertical force resulting from these hydrostatic fluid pressures is called the buoyant force ($F_b$):

   $$F_b = F_2 - F_1 = (P_2 - P_1) A = (\rho_{\text{f}} g h) A = \rho_{\text{f}} g V$$

   where $V = A h$ is the volume of fluid displaced by the submerged object, and $\rho_{\text{f}}$ is the density of the fluid.

   Statement and Mathematical Formulation

   Archimedes' Principle states that any object completely or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object.

   Mathematically, the magnitude of the buoyant force is given by:

   $$F_b = m_{\text{disp}} g = \rho_{\text{f}} V_{\text{sub}} g$$

   where:

   • $\rho_{\text{f}}$ is the mass density of the surrounding fluid,
   • $V_{\text{sub}}$ is the submerged volume of the object (equal to the volume of displaced fluid),
   • $g$ is the acceleration due to gravity.

   Key takeaway: The buoyant force depends solely on the density of the fluid, the submerged volume of the object, and the local gravitational acceleration. It does not depend on the mass, material, or internal composition of the object itself.

   Static Equilibrium of Submerged Bodies

   The vertical motion or equilibrium of an immersed object of total volume $V$ and average density $\rho_{\text{avg}}$ is determined by comparing its gravitational weight $W = m g = \rho_{\text{avg}} V g$ with the buoyant force $F_b$:

   • Sinking: If $\rho_{\text{avg}} > \rho_{\text{f}}$, the downward gravitational force exceeds the maximum possible buoyant force (when fully submerged). The net vertical acceleration is downward, and the body sinks to the bottom.
   • Neutral Buoyancy: If $\rho_{\text{avg}} = \rho_{\text{f}}$, the weight equals the buoyant force when fully submerged ($F_b = W$). The object remains in static equilibrium at any depth within a homogeneous fluid.
   • Floating: If $\rho_{\text{avg}} < \rho_{\text{f}}$, the object rises toward the surface until it partially breaches it. At equilibrium, the weight of the object is balanced entirely by the buoyant force generated by the submerged portion $V_{\text{sub}}$.

   Submerged Fraction of a Floating Body

   For a floating body in static equilibrium, setting $F_b = W$ yields:

   $$\rho_{\text{f}} V_{\text{sub}} g = \rho_{\text{avg}} V g \implies \frac{V_{\text{sub}}}{V} = \frac{\rho_{\text{avg}}}{\rho_{\text{f}}}$$

   Thus, the fraction of an object's total volume that remains submerged is directly proportional to the ratio of its average density to the density of the fluid.