Floating and sinking Knowledge-Based Study

Study notes

1. The Origin of the Buoyant Force

   When an object is immersed in a fluid, the pressure exerted by the fluid increases with depth due to the weight of the fluid column above. Because the fluid pressure at the bottom surface of the object is greater than the pressure at the top surface, an upward net force is created. This upward force exerted by a fluid on an immersed body is called the buoyant force (Fb).

   Archimedes' Principle

   Archimedes' principle states that any object completely or partially submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the object:

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

   • $\rho_f$ is the density of the fluid.
   • $V_{\text{sub}}$ is the volume of the submerged part of the object (which equals the volume of the displaced fluid).
   • $g$ is the acceleration due to gravity.

   Comparing Forces and Densities

   The behavior of an object placed in a fluid depends on the net vertical force acting on it. The two opposing vertical forces are the object's downward gravitational weight (W) and the upward buoyant force (Fb).

   The weight of an object with total volume $V_o$ and average density $\rho_o$ is given by:

   $$W = \rho_o V_o g$$

   Three Equilibrium and Motion States

   1. Sinking ($\rho_o > \rho_f$): When the object is completely submerged, the maximum buoyant force is less than the object's weight ($F_b < W$). A net downward force accelerates the object downward until it rests on the bottom.
   2. Neutral Buoyancy / Suspended ($\rho_o = \rho_f$): When the average density of the object equals the density of the fluid, the buoyant force equals the weight ($F_b = W$) when fully submerged. The object remains in equilibrium at any depth within the fluid.
   3. Floating ($\rho_o < \rho_f$): When completely submerged, the buoyant force exceeds the object's weight ($F_b > W$). The resulting net upward force causes the object to rise to the surface until it emerges partially, reducing $V_{\text{sub}}$ until $F_b = W$.

   The Submerged Fraction of a Floating Body

   For an object floating in static equilibrium at the surface of a liquid, the upward buoyant force precisely balances the downward gravitational force:

   $$F_b = W$$

   $$\rho_f V_{\text{sub}} g = \rho_o V_o g$$

   Simplifying this relation yields the submerged volume fraction:

   $$\frac{V_{\text{sub}}}{V_o} = \frac{\rho_o}{\rho_f}$$

   This indicates that the fraction of the object's total volume that is submerged is equal to the ratio of the object's average density to the fluid's density.

   Apparent Weight

   When a fully submerged object is supported by a scale or string, the reading recorded is the apparent weight (Wapp):

   $$W_{\text{app}} = W - F_b = (\rho_o - \rho_f) V_o g$$

   If the object is denser than the fluid, it sinks unless supported, displaying an apparent weight that is strictly less than its true weight in a vacuum or air.