Summary of "Buoyant force | AP Physics | Khan Academy"

Main ideas and concepts

Buoyant force arises because fluid pressure increases with depth. For an object submerged in a fluid, horizontal pressure forces cancel, but the pressure on the bottom surface is greater than on the top surface, producing a net upward force: the buoyant force.

Archimedes’ principle: the buoyant force on any object equals the weight of the fluid it displaces.

Whether an object floats, sinks, or remains neutrally buoyant depends on the comparison between the object’s weight and the buoyant force — equivalently, on the comparison of densities:

If density_object > density_fluid → object sinks (weight > buoyant force).
If density_object < density_fluid → object floats (buoyant force > weight); the object rises until the displaced fluid’s weight equals the object’s weight.
If density_object = density_fluid → neutral buoyancy: the object stays at whatever depth it is placed (neither rises nor sinks).

The fraction of an object’s volume submerged at equilibrium is rho_object / rho_fluid. This explains common examples:

Beach ball: very low density → only a small portion submerged.
Iceberg: density close to water → most of the iceberg is submerged.
Helium balloon: helium density < air density → balloon rises in air.
Submarines adjust average density by flooding or blowing tanks to sink, rise, or maintain neutral buoyancy.

Buoyant force formula

F_buoyant = rho_fluid × V_displaced × g
- For a fully submerged object, V_displaced = object volume.
- For a floating object, V_displaced is the submerged portion.

Derivation / reasoning steps (methodology)

Consider an object (e.g., a cube) submerged in a fluid.
Fluid pressure acts perpendicular to surfaces and increases with depth.
Horizontal pressure forces cancel (left vs. right, front vs. back). Vertical forces do not (bottom pressure > top pressure).
The net upward force from pressure differences is the buoyant force.
Thought experiment: replace the object with the same-shaped volume of the fluid — that fluid volume is in equilibrium, so its weight equals the buoyant force. Hence buoyant force = weight of displaced fluid (Archimedes’ principle).
Compare forces algebraically:
- Weight_object = rho_object × V_object × g
- Buoyant_force = rho_fluid × V_displaced × g
- For a fully submerged object V_displaced = V_object; sinking condition reduces to rho_object > rho_fluid.
- For floating objects, solve rho_fluid × V_submerged × g = rho_object × V_total × g to get V_submerged = (rho_object / rho_fluid) × V_total. The submerged fraction equals rho_object / rho_fluid.
Apply to practical cases (beach ball, iceberg, helium balloon, submarine) to predict behavior and required submerged fraction.

Key formulae (compact)

F_buoyant = rho_fluid × V_displaced × g
Weight = rho_object × V_object × g
Submerged fraction for equilibrium = V_submerged / V_object = rho_object / rho_fluid

Examples & applications highlighted

Beach ball vs iceberg: difference due to relative densities; iceberg floats but mostly submerged because its density is close to water.
Steel ball sinks because its density is greater than water.
Helium balloon rises because helium is less dense than air.
Submarines achieve neutral buoyancy by adjusting tank water/air to change average density.
NASA’s Neutral Buoyancy Lab uses neutral buoyancy in a pool to simulate weightless training for astronauts.