Archimedes' principle Knowledge-Based Study

Study notes

1. Pressure Gradient in Static Fluids

   When an object is immersed in a static fluid, it experiences pressure from all sides due to collisions with fluid molecules. In a uniform gravitational field, fluid pressure increases linearly with depth according to the relation:

   $P = P_0 + \rho_f g h$

   where $P_0$ is the surface pressure, $\rho_f$ is the fluid density, $g$ is the acceleration due to gravity, and $h$ is the depth below the surface.

   Because the bottom surface of an immersed object lies at a greater depth than its top surface, the upward pressure exerted on the bottom is always greater than the downward pressure on the top. The horizontal forces acting on opposite sides cancel out due to symmetry at equal depths. The net vertical force resulting from this pressure difference is directed upward and is known as the buoyant force ($F_b$).

   Statement of the Principle

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

   Mathematical Formulation

   If an object displaces a volume of fluid $V_{\text{disp}}$ (which equals the submerged volume of the object, $V_{\text{sub}}$), the mass of the displaced fluid is:

   $m_f = \rho_f V_{\text{sub}}$

   The weight of this displaced fluid corresponds directly to the buoyant force:

   $$F_b = m_f g = \rho_f V_{\text{sub}} g$$

   Key points to consider:

   • The buoyant force depends entirely on the density of the fluid ($\rho_f$) and the submerged volume ($V_{\text{sub}}$), not on the material composition or total mass of the submerged body.
   • For a fully submerged object, $V_{\text{sub}} = V_{\text{object}}$, meaning the buoyant force remains constant regardless of depth, assuming an incompressible fluid where $\rho_f$ is uniform.

   Conditions for Vertical Equilibrium

   The net vertical force on an immersed object of mass $m$ and total volume $V$ is determined by the competition between its weight ($W = mg = \rho_{\text{obj}} V g$) and the buoyant force ($F_b = \rho_f V_{\text{sub}} g$):

   • Sinking ($\rho_{\text{obj}} > \rho_f$): When the average density of the object exceeds the fluid density, the maximum possible buoyant force (when fully submerged) is less than the object's weight. The object accelerates downward until it rests on the bottom.
   • Neutral Buoyancy ($\rho_{\text{obj}} = \rho_f$): When the object density equals the fluid density, $F_b = W$ when fully submerged. The object remains stationary at any submerged depth in neutral equilibrium.
   • Floating ($\rho_{\text{obj}} < \rho_f$): When the object's density is less than that of the fluid, it rises until partially emerged. At equilibrium, the weight equals the buoyant force:

   $$\rho_{\text{obj}} V g = \rho_f V_{\text{sub}} g \implies \frac{V_{\text{sub}}}{V} = \frac{\rho_{\text{obj}}}{\rho_f}$$

   This fraction represents the submerged volume ratio of a freely floating body.

   Apparent Weight of a Submerged Object

   When an object is weighed while immersed in a fluid, a scale supports only the uncompensated portion of its weight. This measured quantity is termed the apparent weight ($W_{\text{app}}$):

   $$W_{\text{app}} = W - F_b = (\rho_{\text{obj}} - \rho_f) V g$$

   If the true weight in air and apparent weight in a liquid of known density are measured, the density of an irregularly shaped solid can be determined without directly measuring its volume:

   $$\rho_{\text{obj}} = \rho_f \left( \frac{W}{W - W_{\text{app}}} \right)$$

   This application forms the classical basis of hydrostatic weighing used for identifying material purity and specific gravity.