understand that the upthrust acting on an object in a fluid is due to a difference in hydrostatic pressure

1. Learning Objectives & Syllabus Mapping (Syllabus 4.2)

  • State that the upthrust acting on an object immersed in a fluid arises from a difference in hydrostatic pressure on its surfaces.

  • Derive mathematically that the upthrust equals the weight of the displaced fluid (Archimedes’ principle).

  • Apply the principle to predict whether an object will float, sink or remain neutrally buoyant.

  • Design, carry out and analyse a simple experiment to verify the relationship \(FB=\rho{\text{fluid}} g V\).

  • Identify and correct common misconceptions about buoyancy.

Link‑back to specific Syllabus 4.2 sub‑points

Syllabus Sub‑pointCovered In
State that upthrust arises from a pressure differenceSection 2 (Key Concepts)
Derive \(F_B=\rho g V\)Section 3 (Derivation)
Predict floating, sinking or suspensionSection 6 (Applications)
Design & analyse experimentSection 5 (Experimental Verification)
Address misconceptionsSection 7 (Misconceptions)

Key take‑away: All required 4.2 outcomes are explicitly mapped, making it easy for teachers and examiners to see coverage.

2. Key Concepts & Formulae

ConceptDefinition / Formula
Hydrostatic pressurePressure in a fluid at rest increases linearly with depth:

\(p = \rho\,g\,h\)

(ρ = fluid density, g = 9.81 m s⁻², h = depth).

Direction of pressureActs perpendicular to any surface in contact with the fluid.
Resultant pressure force on a flat surface\(F = pA\) (where A is the area).
Upthrust (buoyant force)Net upward force on a submerged body:

\(FB = \rho{\text{fluid}}\,g\,V\)

(equal to the weight of the displaced fluid).

Archimedes’ principle“A body immersed in a fluid experiences an upward force equal to the weight of the fluid it displaces.”

Cross‑topic reminder (4.1 & 4.3): You will need the definition of pressure (Topic 4.1) and the concept of density (Topic 4.3) when solving quantitative buoyancy problems.

Key take‑away: Upthrust originates from the pressure gradient; the simple formula \(F_B=\rho g V\) holds when the fluid density is uniform.

3. Derivation of Upthrust from the Pressure Difference

Assumptions

  • The fluid has a constant density ρ (i.e., it is homogeneous and incompressible).

  • The object is a rectangular block of uniform cross‑sectional area A and height h, fully submerged.

  1. Depth of the top surface: h₁.
    Pressure on the top: \(p{\text{top}} = \rho g h1\).

  2. Depth of the bottom surface: h₂ = h₁ + h.
    Pressure on the bottom: \(p{\text{bottom}} = \rho g (h1 + h)\).

  3. Force on the top (downward): \(F{\text{top}} = p{\text{top}}A = \rho g h_1 A\).

  4. Force on the bottom (upward): \(F{\text{bottom}} = p{\text{bottom}}A = \rho g (h_1 + h)A\).

  5. Net upward force (upthrust):

    \[

    FB = F{\text{bottom}} - F_{\text{top}} = \rho g h A.

    \]

  6. Since the displaced volume \(V = Ah\), the result becomes

    \[

    \boxed{F_B = \rho g V}

    \]

    which is precisely the weight of the displaced fluid.

Sanity check: The mass of the displaced fluid is \(m{\text{fluid}} = \rho V\); its weight is \(m{\text{fluid}}g = \rho g V\), confirming the derived expression.

Key take‑away: The upward resultant comes from the higher pressure on the deeper bottom surface compared with the top surface; the difference equals the weight of the displaced fluid.

4. Generalisation to Arbitrary Shapes

For any submerged body (any shape, any orientation) the net buoyant force is obtained by integrating the pressure over the entire surface:

\[

\mathbf{F}B = \oint{\text{surface}} p\,\mathbf{\hat n}\,\mathrm{d}A,

\]

where \(\mathbf{\hat n}\) is the outward unit normal. Because the pressure varies linearly with depth, the integral always reduces to

\[

\boxed{FB = \rho{\text{fluid}}\,g\,V},

\]

independent of shape. The only restriction is the assumption of a uniform fluid density; for a stratified fluid the integral must be evaluated with the local ρ(z), and the simple \(F_B=\rho g V\) does not hold.

Key take‑away: Shape does not affect the magnitude of upthrust; only the displaced volume matters (provided ρ is constant).

5. Experimental Verification & Practical Tips (AO3)

5.1. Standard Procedure

StepActionWhat is measured?
1Measure the mass \(m_{\text{block}}\) and linear dimensions of a solid block (e.g., wood or metal) to obtain its volume \(V\).Mass (g), dimensions (cm), calculate \(V\) (cm³).
2Attach the block to a spring balance with a thin, light string; record the reading in air (\(W_{\text{air}}\)).Apparent weight in air (N).
3Submerge the block completely in a water tank (no contact with the bottom) and record the new reading (\(W_{\text{water}}\)).Apparent weight in water (N).
4Calculate the experimental upthrust: \(FB^{\text{exp}} = W{\text{air}} - W_{\text{water}}\).Upthrust (N).
5Predict the theoretical upthrust using Archimedes’ principle: \(FB^{\text{theory}} = \rho{\text{fluid}} g V\) (use \(\rho_{\text{water}} = 1000\;\text{kg m}^{-3}\)).Theoretical upthrust (N).
6Compare the two values, calculate percentage difference and discuss sources of error.Percentage error, error analysis.

5.2. Practical Tip – Determining Fluid Density

  • Use a hydrometer: measure the depth to which it sinks and read the calibrated density.

  • Or, weigh a known volume of the fluid (e.g., 100 mL) on a balance; density \(\rho = \dfrac{m}{V}\).

  • Record temperature, because \(\rho\) varies with temperature (e.g., water at 20 °C ≈ 998 kg m⁻³).

Key take‑away: Accurate knowledge of the fluid density is essential for a reliable comparison between experimental and theoretical upthrust.

6. Applications: Floating, Sinking & Neutral Buoyancy

6.1. Decision Table

ConditionMathematical testResult
Floats (partly immersed)\(\rho{\text{object}} < \rho{\text{fluid}}\) or \(mg < \rho{\text{fluid}} g V{\text{disp}}\)Upthrust > weight of the immersed part → equilibrium at the surface.
Sinks (fully immersed)\(\rho{\text{object}} > \rho{\text{fluid}}\) or \(mg > \rho_{\text{fluid}} g V\)Weight exceeds upthrust → net downward force.
Neutral buoyancy\(\rho{\text{object}} = \rho{\text{fluid}}\) or \(mg = \rho_{\text{fluid}} g V\)Upthrust exactly balances weight → object can remain suspended at any depth.

6.2. Numerical Example

Block: \(V = 2.5\times10^{-4}\,\text{m}^3\); density of water \(\rho = 1000\;\text{kg m}^{-3}\); \(g = 9.81\;\text{m s}^{-2}\).

  • Theoretical upthrust:

    \[

    F_B^{\text{theory}} = \rho g V = 1000 \times 9.81 \times 2.5\times10^{-4} = 2.45\;\text{N}

    \]

  • Experimental data: spring balance reads \(3.20\;\text{N}\) in air and \(0.75\;\text{N}\) when submerged.

    Experimental upthrust: \(3.20 - 0.75 = 2.45\;\text{N}\).

  • Percentage difference: \(\dfrac{|2.45-2.45|}{2.45}\times100 = 0\%\) (excellent agreement).

Key take‑away: By comparing the object's density with the fluid’s density, you can predict its behaviour without needing to perform an experiment each time.

7. Common Misconceptions & Clarifications

  • “Buoyancy is a separate force.” It is the vector sum of all pressure forces acting on the object's surface.

  • “Only the bottom surface contributes.” Both top and bottom (and all side surfaces) experience pressure; the net upward resultant comes from the pressure gradient.

  • “Heavier objects experience more upthrust.” Upthrust depends solely on the displaced volume, not on the object's own weight.

  • “If an object floats, the fluid exerts an upward force equal to the object's weight.” Correct, but only the *displaced* portion of fluid contributes; the part of the object above the surface does not affect the buoyant force.

  • “Archimedes’ principle works for any fluid.” The simple form \(F_B=\rho g V\) assumes a uniform density. In a stratified fluid the integral form must be used.

Key take‑away: Understanding that upthrust is a pressure‑difference effect eliminates these common errors and aligns thinking with the formal definition used in the syllabus.

8. Suggested Diagrams (to be drawn by teacher or student)

  1. Rectangular block fully submerged: label depths \(h1\) and \(h2\), show pressure vectors on top and bottom faces, and draw the resultant upthrust arrow.

  2. Side view of a floating object: indicate the submerged depth, displaced volume, weight arrow (\(mg\)) and upthrust arrow (\(F_B\)).

  3. Force diagram for the experimental set‑up: spring balance, string, block, water surface, and the three forces \(W{\text{air}}\), \(W{\text{water}}\) and \(F_B\).

  4. Optional: pressure variation with depth in a fluid column (linear graph) to visualise \(p=\rho g h\).

9. Summary

The upthrust on any submerged object is a direct consequence of the hydrostatic pressure gradient in a fluid. By integrating the pressure over the object's surface, we obtain Archimedes’ principle: upthrust = weight of the displaced fluid. This principle explains why objects float, sink, or remain neutrally buoyant, and it can be verified with a straightforward spring‑balance experiment. Mastery of these ideas satisfies the Cambridge 9702 requirements for “Equilibrium of Forces”.