Archimedes’ Principle and Buoyancy

The Principle: 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.

Buoyant Force ($F_B$): $$ F_B = m_{fluid} g = \rho_{fluid} V_{displaced} g $$

• $\rho_{fluid}$: Density of the fluid (not the object!)
• $V_{displaced}$: Volume of the submerged part of the object.

Floating vs. Sinking

• Sinking Object ($\rho_{obj} > \rho_{fluid}$): The object rests on the bottom. Normal force exists. $$ F_{net} = F_B + N - mg = 0 $$
• Floating Object ($\rho_{obj} < \rho_{fluid}$): The object floats on the surface. $$ F_B = F_g \Rightarrow \rho_{fluid} V_{displaced} g = \rho_{obj} V_{total} g $$
  • Fraction Submerged: $\frac{V_{submerged}}{V_{total}} = \frac{\rho_{obj}}{\rho_{fluid}}$

💡 Common Applications

• Steel Ships: Solid steel sinks, but a steel ship is shaped to enclose a large volume of air. This increases the total volume displaced ($V_{displaced}$) without significantly increasing mass, making the average density of the ship less than water.
• Hydrometers: Instruments used to measure the density of liquids (like battery acid or antifreeze). The depth to which the hydrometer sinks indicates the fluid density.

⚠️ Key Confusing Point: Don't confuse $V_{object}$ with $V_{displaced}$.

• If fully submerged, $V_{displaced} = V_{object}$.
• If floating, $V_{displaced} < V_{object}$.