Buoyancy and Archimedes' Principle Notes

Object's Volume and Fluid Density

  • Consider an object of volume VV suspended in a fluid with density ρ\rho (blue fluid).

  • The pressure of the fluid creates forces on the object throughout its boundary.

  • Pressure increases with depth, hence:- Forces at the bottom are greater than at the top due to the pressure gradient.

Buoyancy Force

  • The net upward force on the object is called the buoyancy force, denoted as FBF_B.

  • This upward force arises because the sideways pressure forces cancel out.

Calculating Buoyancy Force

  • Understanding buoyancy requires knowing the magnitude of the buoyancy force:- Imagine a balloon filled with the same fluid (like water in a swimming pool).

    • When the balloon is submerged, it neither sinks nor floats, indicating equilibrium.

  • The buoyancy force matches the weight of the displaced fluid, leading to:

    FB=ρgVF_B = \rho g V
    where ρ\rho is the fluid’s density, gg is the gravitational force, and VV is the volume of the fluid displaced.

Displacement and Buoyancy Force Independence

  • The buoyancy force remains unchanged regardless of the object’s interior content; it is determined by the external fluid’s density and volume of the object submerged.

  • Important to recall that the density used for calculations is that of the fluid, not the object itself.

Conditions for Sinking or Floating

  • An object will float if the buoyancy force exceeds its weight:- If F_B > W (weight of the object).

  • Alternatively, this can be expressed as:

    \frac{F_B}{W} > 1.

  • We calculate the buoyancy force using:

    FB=ρfluidgVFB = \rho{fluid} g V

    The weight of the object is:

    W=ρobjectgVW = \rho_{object} g V.

  • Thus, the ratio becomes:

    FBW=ρfluidgVρobjectgV=ρfluidρobject\frac{FB}{W}=\frac{\rho{fluid} g V}{\rho{object} g V}=\frac{\rho{fluid}}{\rho_{object}}

  • This shows that if \rho{object}<\rho{fluid} ho{object} < \rho{fluid}</code>objectwillfloat.</p></li></ul><h5id="35acec8e−a7dc−42ce−8b8f−d90c1d83d1fb"data−toc−id="35acec8e−a7dc−42ce−8b8f−d90c1d83d1fb"collapsed="false"seolevelmigrated="true">ExamplesofDensityinBuoyancy</h5><ul><li><p><strong>Corkvs.Rock</strong>:−Corkfloatsbecauseitsdensityislessthanthatofwater.</p><ul><li><p>Rocksinksasitsdensityisgreaterthanthatofwater.</p></li></ul></li></ul><h5id="719e763f−8262−4772−bf9d−54521c95ff74"data−toc−id="719e763f−8262−4772−bf9d−54521c95ff74"collapsed="false"seolevelmigrated="true">IcebergsandSeawater</h5><ul><li><p>Exampleofbuoyancyeffectwithicebergs:−Seawaterdensity</code> object will float.</p></li></ul><h5 id="35acec8e-a7dc-42ce-8b8f-d90c1d83d1fb" data-toc-id="35acec8e-a7dc-42ce-8b8f-d90c1d83d1fb" collapsed="false" seolevelmigrated="true">Examples of Density in Buoyancy</h5><ul><li><p><strong>Cork vs. Rock</strong>:- Cork floats because its density is less than that of water.</p><ul><li><p>Rock sinks as its density is greater than that of water.</p></li></ul></li></ul><h5 id="719e763f-8262-4772-bf9d-54521c95ff74" data-toc-id="719e763f-8262-4772-bf9d-54521c95ff74" collapsed="false" seolevelmigrated="true">Icebergs and Seawater</h5><ul><li><p>Example of buoyancy effect with icebergs:- Seawater density= 1030 kg/m^3,Icedensity, Ice density= 920 kg/m^3.</p></li><li><p>Proportionsubmergedforanicebergcanbedeterminedbytheratiooftheirdensities:</p></li></ul><p>SubmergedVolume/TotalVolume=P(Density)ice/P(Density)water</p><ul><li><p>Result:</p><p>.</p></li><li><p>Proportion submerged for an iceberg can be determined by the ratio of their densities:</p></li></ul><p>Submerged Volume / Total Volume = P (Density) ice / P (Density) water</p><ul><li><p>Result:</p><p>\frac{920}{1030} \approx 0.89$$ or 89% of the iceberg is submerged.

Archimedes' Principle

  • The idea that the buoyancy force equals the weight of the displaced fluid is known as Archimedes' principle.

  • Important for understanding buoyancy and performing related calculations.