### Archimedes' Principle - **Statement:** Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. - **Buoyant Force ($F_B$):** The upward force exerted by a fluid that opposes the weight of an immersed object. - **Formula:** $F_B = \rho_f V_{disp} g$ - $\rho_f$: density of the fluid (kg/m$^3$) - $V_{disp}$: volume of fluid displaced by the object (m$^3$) - $g$: acceleration due to gravity (9.8 m/s$^2$) - **Key Idea:** $V_{disp}$ is equal to the volume of the immersed part of the object. If fully submerged, $V_{disp} = V_{object}$. ### Fluid Density & Specific Gravity - **Density ($\rho$):** Mass per unit volume. - $\rho = m/V$ (kg/m$^3$) - **Specific Gravity (SG):** Ratio of the density of a substance to the density of a reference substance (usually water at $4^\circ C$, $\rho_{water} = 1000 \text{ kg/m}^3$). - $SG = \rho_{substance} / \rho_{water}$ - A dimensionless quantity. If $SG > 1$, the object is denser than water; if $SG ### Floating and Sinking - **Floating:** An object floats if the buoyant force equals the object's weight ($F_B = W_{object}$). - This occurs when $\rho_{object} \rho_f$. - The apparent weight of a submerged object is $W_{apparent} = W_{object} - F_B$. - **Neutrally Buoyant:** An object remains suspended at any level within the fluid if its density is equal to the fluid's density ($\rho_{object} = \rho_f$). ### Pressure in Fluids - **Pressure (P):** Force per unit area. - $P = F/A$ (Pascals, Pa, or N/m$^2$) - **Absolute Pressure ($P_{abs}$):** Total pressure at a certain depth in a fluid. - $P_{abs} = P_{atm} + \rho_f g h$ - $P_{atm}$: atmospheric pressure (at surface, $\approx 1.013 \times 10^5 \text{ Pa}$) - $h$: depth below the surface of the fluid (m) - **Gauge Pressure ($P_{gauge}$):** The difference between the absolute pressure and the atmospheric pressure. - $P_{gauge} = P_{abs} - P_{atm} = \rho_f g h$ - **Pascal's Principle:** A pressure change applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of the container. - Used in hydraulic systems: $F_1/A_1 = F_2/A_2$
