Determine whether an object floats based on density data

IGCSE Physics 0625 – Unit 1: Density & Floatation

1. Syllabus Mapping (Section 1)

2. Learning Objectives (AO1‑AO3)

• AO1 – Knowledge: Define density, state ρ = m/V, explain hydrostatic pressure (Δp = ρgΔh) and the buoyant force, describe the forces acting on a body in a fluid.
• AO2 – Application: Use density data to decide whether a solid will float or sink in a given liquid and whether one liquid will float on another.
• AO3 – Analysis: Carry out practical density determinations (mass, volume by displacement), calculate densities, propagate uncertainties and evaluate experimental error.

3. Core Concepts

3.1 Density (Syllabus 1.4)

• Density (ρ) = mass ÷ volume.
• Formula:
  \$\rho = \frac{m}{V}\$
• Units: kg · m⁻³ or g · cm⁻³ (1 g · cm⁻³ = 1000 kg · m⁻³).
• Regular solids: volume from geometric formulae (e.g. V = l × w × h for a rectangular block).
• Irregular solids: volume by water‑displacement (Archimedes’ principle).
  

  1. Record initial water volume V₁ in a graduated cylinder.
  2. Submerge the dry object (no air bubbles) and record the new level V₂.
  3. Displaced volume V_disp = V₂ − V₁.

3.2 Pressure in Fluids (Syllabus 1.8)

• Hydrostatic pressure increases linearly with depth:
  \$\Delta p = \rho_{\text{fluid}}\,g\,\Delta h\$
• At depth h the pressure on a horizontal surface is p = p₀ + ρg h (p₀ = atmospheric pressure).
• Because pressure acts on all sides of a submerged object, the net upward force (buoyancy) equals the pressure difference between the bottom and top surfaces multiplied by the area.
• Integrating the pressure over the whole submerged surface gives the buoyant force:
  \$FB = \rho{\text{fluid}}\,g\,V_{\text{disp}}\$

3.3 Forces & Equilibrium (Syllabus 1.5)

• Weight (W): W = mg, acts vertically downwards through the centre of gravity.
• Buoyant force (F_B): Acts vertically upwards through the centre of buoyancy (the centre of the displaced volume).
• Resultant force: Vector sum of all forces. For a body at rest in a fluid the resultant must be zero (static equilibrium).
• Condition for floating:
  \$FB \ge W \quad\Longleftrightarrow\quad \rho{\text{obj}} \le \rho_{\text{fluid}}\$
• If ρobj = ρfluid the object experiences neutral buoyancy (remains suspended anywhere in the fluid).

3.4 Buoyant Force – Summary Equation

Combining §§3.2 and 3.3 gives the practical formula used in the exam:

\$FB = \rho{\text{fluid}}\,g\,V_{\text{disp}}\$

where V_disp is the volume of fluid displaced (equal to the submerged volume of the object).

4. Decision‑Making Procedure for Floatation (AO2)

1. Determine the object’s density.

   • Regular shape: calculate V from dimensions, then ρ = m/V.
   • Irregular shape: use the displacement method (see 3.1) to obtain Vdisp, then ρ = m/Vdisp.

2. Identify the density of the fluid. Typical values:

   • Water (20 °C) ≈ 1.00 g · cm⁻³
   • Vegetable oil ≈ 0.92 g · cm⁻³
   • Mercury ≈ 13.6 g · cm⁻³
   • Alcohol (ethanol) ≈ 0.79 g · cm⁻³

3. Compare densities.

   • ρobj < ρfluid → object floats (partially or wholly submerged).
   • ρobj > ρfluid → object sinks.
   • ρobj = ρfluid → neutral buoyancy.

4. State the conclusion. Include the numerical comparison and, where relevant, the percentage difference:

   \$\%\,\text{difference} = \frac{|\rho{\text{obj}}-\rho{\text{fluid}}|}{\rho_{\text{fluid}}}\times100\%\$

5. Liquid on Liquid Floatation (Syllabus 1.4)

• A liquid will form a layer on top of another liquid if its density is lower.
• Examples:

  • Vegetable oil (0.92 g · cm⁻³) floats on water (1.00 g · cm⁻³).
  • Mercury (13.6 g · cm⁻³) sinks beneath water.

• If the liquids are miscible (e.g., ethanol and water) they mix; the “floating” concept does not apply and is examined in the exam.

6. Practical Skills & Safety (AO3)

7. Common Materials – Densities (g · cm⁻³)

8. Worked Examples (AO1 / AO2)

Example 1 – Regular Solid in Water

Problem: A rectangular block has mass 150 g and dimensions 5 cm × 4 cm × 5 cm. Will it float in water?

1. Calculate volume: V = 5 × 4 × 5 = 100 cm³.
2. Density: ρ_block = 150 g / 100 cm³ = 1.50 g · cm⁻³.
3. ρ_water = 1.00 g · cm⁻³ → 1.50 > 1.00.
4. Conclusion (AO2): The block sinks.

Example 2 – Irregular Stone (Displacement Method)

Problem: An irregular stone has a mass of 250 g. When placed in a 250 mL graduated cylinder containing 120 mL of water, the level rises to 150 mL. Does the stone float?

1. Displaced volume V_disp = 150 mL − 120 mL = 30 mL = 30 cm³.
2. Density: ρ_stone = 250 g / 30 cm³ ≈ 8.33 g · cm⁻³.
3. ρ_water = 1.00 g · cm⁻³ → 8.33 > 1.00.
4. Conclusion (AO2): The stone sinks.

Example 3 – Liquid on Liquid

Problem: Will a layer of vegetable oil (ρ = 0.92 g · cm⁻³) float on water (ρ = 1.00 g · cm⁻³)?

• Because 0.92 < 1.00, the oil is less dense and will form a layer on top of the water.

Example 4 – Neutral Buoyancy

Problem: What density must an object have to remain suspended anywhere in water?

• For neutral buoyancy: ρobj = ρwater = 1.00 g · cm⁻³.

9. Practice Questions (with AO tags)

1. AO2 – A piece of cork has mass 2.5 g and volume 5.0 cm³. Determine whether it will float in water.
2. AO1/2 – What density must an object have to remain suspended (neutral buoyancy) in water?
3. AO2 – A metal alloy has density 8.0 g · cm⁻³. Will a shape made from this alloy float in oil of density 0.92 g · cm⁻³?
4. AO3 – Design a practical procedure to measure the density of an irregular solid using water displacement. List the equipment, steps, and show how you would calculate the combined uncertainty.
5. AO1/2 – Explain why a glass bottle filled with water sinks in mercury but floats in oil.

10. Suggested Classroom Activity – “Density Lab” (AO3)

1. Equipment: electronic balance (±0.01 g), graduated cylinders (10 mL & 100 mL), distilled water, vegetable oil, a selection of solid objects (metal, wood, plastic), safety goggles, lab coat, data sheet.
2. Procedure:

   1. Measure and record the mass of each solid (to 0.01 g).
   2. Determine volume:

      • Regular objects – use geometric formulae.
      • Irregular objects – apply the displacement method (record V₁ and V₂ three times, take the mean).

   3. Calculate density for each object and propagate uncertainties.
   4. Predict, using the density‑comparison rule, whether each object will:

      • float or sink in water, and
      • float or sink in oil.

   5. Test the predictions by gently placing each object in the two liquids and observing the outcome.

3. Data analysis:

   • Construct a table of ρ_obj, predicted outcome, observed outcome, and % error.
   • Plot density (y‑axis) against outcome (float = 1, sink = 0) for each liquid.
   • Discuss discrepancies – possible sources of error: air bubbles, temperature (affects ρ_water), parallax reading, surface tension.

11. Summary Checklist (AO1 / AO2)

• Define density and write ρ = m/V.
• Calculate density for regular and irregular objects (including displacement method).
• State the hydrostatic pressure formula Δp = ρgΔh and show how it leads to FB = ρgVdisp.
• Identify all forces acting on a submerged body (weight, buoyant force) and apply the equilibrium condition.
• Use the density comparison rule to predict floatation of solids and liquids.
• Carry out a practical density measurement safely, record uncertainties, and evaluate results.