ARCHIMEDES' PRINCIPLE NUMERICAL CHALLENGE
Can You Outsmart Upthrust, Relative Density and Floatation?
A heavy iron block seems lighter in water. A wooden block floats. A ship made of steel stays on the water, while a small iron nail sinks.
What is the science behind these mysteries?
Welcome to SpeedUpScience!
Understand the formula. Apply the concept. Master the numerical.
1. Formula Station: Learn Before You Solve
Before attempting the numericals, revise these important formulas. Keep this section open while solving the questions.
A. Density
SI unit of density: kg/m³
Useful conversion:
B. Relative Density
Relative density is a ratio and has no unit.
For water at approximately 4°C:
C. Upthrust and Archimedes' Principle
Use consistent SI units in the last formula: density in kg/m³, volume in m³, and g in m/s².
D. Apparent Weight
E. Volume of an Immersed Solid
For a completely immersed solid, the volume of liquid displaced equals the volume of the solid.
F. Floatation
For a freely floating body at rest:
G. Relative Density Using Weights
For a solid completely immersed in water, when weights are measured in the same force unit:
The last formula assumes the same solid is completely immersed in both liquids and the measurements are made under comparable conditions.
2. Level 1: Warm-Up Round
Start with straightforward questions to build your confidence. Try each problem yourself before revealing the solution.
Find the Density of Copper
The density of copper is 8.83 g/cm³. Express it in kg/m³.
Given:
Density = 8.83 g/cm³
Formula:
1 g/cm³ = 1000 kg/m³
Calculation:
Density = 8.83 × 1000
Answer: 8830 kg/m³
Relative Density of Iron
The density of iron is 7.8 × 10³ kg/m³. Calculate its relative density. Take the density of water as 1000 kg/m³.
Formula:
R.D. = Density of iron ÷ Density of water
Substitution:
R.D. = 7800 ÷ 1000
Answer: Relative density = 7.8
Relative density has no unit.
Find the Upthrust
A solid weighs 50 gf in air and 44 gf when completely immersed in water. Calculate the upthrust.
Formula:
Upthrust = Weight in air − Apparent weight in water
Calculation:
Upthrust = 50 − 44
Answer: 6 gf
The liquid exerts an upward buoyant force of 6 gf.
Apparent Weight in Water
A body weighs 20 N in air. The upthrust acting on it in water is 4 N. Find its apparent weight.
Formula:
Apparent weight = Actual weight − Upthrust
Calculation:
Apparent weight = 20 − 4 = 16 N
Answer: 16 N
3. Level 2: Build Your Concept
Now use Archimedes' principle to find volumes, densities and relative densities.
Find the Volume of a Solid
A solid weighs 30 gf in air and 26 gf when completely immersed in a liquid of relative density 0.8. Find the volume of the solid.
Take g = 10 m/s² and density of water = 1000 kg/m³.
Step 1: Find the loss in weight.
Loss in weight = 30 − 26 = 4 gf
Since 1 gf = 0.01 N when g = 10 m/s²:
Upthrust = 4 × 0.01 = 0.04 N
Step 2: Find the density of the liquid.
Density = 0.8 × 1000 = 800 kg/m³
Step 3: Apply Archimedes' principle.
Upthrust = Density × Volume × g
Volume = Upthrust ÷ (Density × g)
Volume = 0.04 ÷ (800 × 10)
Volume = 0.000005 m³
Since 1 m³ = 10⁶ cm³:
Answer: Volume = 5 cm³
Find the Relative Density of a Solid
A solid weighs 32 gf in air and 28.8 gf when completely immersed in water. Calculate its relative density.
Step 1: Find the loss in weight.
Loss in weight = 32 − 28.8 = 3.2 gf
Step 2: Apply the formula.
R.D. = Weight in air ÷ Loss in weight
R.D. = 32 ÷ 3.2
Answer: Relative density = 10
Find the Density of a Stone
A stone has a mass of 113 g. When completely immersed in water in a measuring cylinder, the water level rises from 30 mL to 40 mL. Calculate the relative density of the stone.
Step 1: Find the volume of the stone.
Volume = Final reading − Initial reading
Volume = 40 − 30 = 10 cm³
Step 2: Calculate density.
Density = Mass ÷ Volume
Density = 113 ÷ 10 = 11.3 g/cm³
Step 3: Calculate relative density.
R.D. = 11.3 ÷ 1
Answer: Relative density = 11.3
Calculate the Volume of a Wooden Block
A wooden block has a mass of 24 kg. It floats on water, and the volume of water displaced is 0.024 m³. Verify whether the block floats in equilibrium.
Take density of water = 1000 kg/m³ and g = 10 m/s².
Step 1: Calculate the weight of the block.
Weight = mg = 24 × 10 = 240 N
Step 2: Calculate the upthrust.
Upthrust = Density × Volume displaced × g
Upthrust = 1000 × 0.024 × 10 = 240 N
Since the upthrust equals the weight, the forces balance.
Answer: The block can float in equilibrium.
4. Level 3: Application Challenge
These questions require more than direct substitution. Read carefully and identify the physical principle before calculating.
Will the Block Float or Sink?
A block has a mass of 13.5 kg and a volume of 15 × 10⁻³ m³. Determine whether it will float or sink in water. Take density of water = 1000 kg/m³ and g = 10 m/s².
Step 1: Calculate the density of the block.
Density = Mass ÷ Volume
Density = 13.5 ÷ (15 × 10⁻³)
Density = 900 kg/m³
Step 2: Compare densities.
Density of block = 900 kg/m³
Density of water = 1000 kg/m³
The block is less dense than water.
Answer: The block will float.
Find the Fraction of a Wooden Block Above Water
A wooden block floats on water with two-fifths of its volume above the surface. Find the density of the wood.
Take density of water = 1000 kg/m³.
Step 1: Find the fraction submerged.
Fraction above water = 2/5
Fraction submerged = 1 − 2/5 = 3/5
Step 2: Apply the floatation formula.
Density of wood ÷ Density of water = Fraction submerged
Density of wood = (3/5) × 1000
Answer: Density of wood = 600 kg/m³
Iceberg Mystery
An iceberg floats in fresh water. The density of ice is 917 kg/m³ and the density of fresh water is 1000 kg/m³. Calculate the fraction of the iceberg's volume below the water surface.
Formula:
Fraction submerged = Density of ice ÷ Density of water
Substitution:
Fraction submerged = 917 ÷ 1000
= 0.917
Answer: 0.917 of the total volume, or 91.7%, is submerged.
The remaining 8.3% is above the water surface.
A Solid in Two Liquids
A solid weighs 2.10 N in air and 1.85 N when completely immersed in water. Find its weight when completely immersed in a liquid of relative density 1.2.
Step 1: Calculate the upthrust in water.
Upthrust in water = 2.10 − 1.85 = 0.25 N
Step 2: Calculate the upthrust in the second liquid.
Upthrust in liquid = 0.25 × 1.2 = 0.30 N
Step 3: Calculate the apparent weight.
Apparent weight = Actual weight − Upthrust
Apparent weight = 2.10 − 0.30
Answer: 1.80 N
5. Level 4: HOTS and Expert Challenge
Ready for the next level? These problems combine multiple concepts. Attempt them independently before revealing the worked solutions.
The Wooden Block Under Water
A wooden block has a volume of 200 cm³ and a density of 0.84 g/cm³. It floats in a liquid of density 1.05 g/cm³.
Calculate:
(i) The volume of the block above the liquid surface.
(ii) The force required to keep the block completely submerged.
Take g = 10 m/s².
(i) Volume above the liquid
For a floating body:
Fraction submerged = Density of body ÷ Density of liquid
Fraction submerged = 0.84 ÷ 1.05 = 0.8
Submerged volume = 0.8 × 200 = 160 cm³
Volume above surface = 200 − 160
Answer (i): 40 cm³
(ii) Force required for complete submersion
Mass of block = Density × Volume
Mass = 0.84 × 200 = 168 g = 0.168 kg
Weight of block = 0.168 × 10 = 1.68 N
For complete submersion, displaced liquid volume = 200 cm³
Convert volume: 200 cm³ = 0.0002 m³
Upthrust = 1050 × 0.0002 × 10 = 2.10 N
Required downward force = Upthrust − Weight
= 2.10 − 1.68 = 0.42 N
Answer (ii): 0.42 N downward
The Brass Piece Mystery
A brass piece weighs 175 gf in air and 150 gf when completely immersed
in water. Calculate:
(i) Its volume.
(ii) Its relative density.
Step 1: Calculate the loss in weight.
Loss in weight = 175 − 150 = 25 gf
This equals the weight of the displaced water.
Since water has density 1 g/cm³, 1 cm³ of water weighs 1 gf.
Therefore, displaced water volume = 25 cm³.
Answer (i): Volume = 25 cm³
Step 2: Calculate relative density.
R.D. = Weight in air ÷ Loss in weight
R.D. = 175 ÷ 25
Answer (ii): Relative density = 7
The Iron Cube in a Liquid
A metal cube has an edge of 5 cm and a density of 7.9 g/cm³.
It is completely immersed in a liquid of density 1.1 g/cm³.
Calculate:
(i) The weight of the cube.
(ii) The upthrust on the cube.
(iii) The tension in the supporting thread.
Take g = 10 m/s².
Step 1: Calculate the volume.
Volume = 5 × 5 × 5 = 125 cm³
Step 2: Calculate the mass.
Mass = Density × Volume
Mass = 7.9 × 125 = 987.5 g = 0.9875 kg
(i) Weight of the cube
Weight = 0.9875 × 10 = 9.875 N
Step 3: Calculate the upthrust.
Volume = 125 cm³ = 0.000125 m³
Upthrust = 1100 × 0.000125 × 10
(ii) Upthrust = 1.375 N
Step 4: Calculate the tension.
For a stationary, fully immersed cube:
Tension + Upthrust = Weight
Tension = Weight − Upthrust
Tension = 9.875 − 1.375
(iii) Tension = 8.50 N
The Relative Density Detective
A solid weighs 50 gf in air and 44 gf when completely immersed
in water. Calculate:
(i) The upthrust.
(ii) The volume of the solid.
(iii) The relative density of the solid.
(i) Upthrust
Upthrust = 50 − 44 = 6 gf
(ii) Volume
In water, 1 cm³ of displaced water weighs approximately 1 gf.
Volume of displaced water = 6 cm³
Since the solid is completely immersed, its volume is 6 cm³.
(iii) Relative density
R.D. = Weight in air ÷ Loss in weight
R.D. = 50 ÷ 6 = 8.33 approximately
Answers: (i) 6 gf; (ii) 6 cm³; (iii) 8.33
6. SpeedUpScience Final Boss: Test Yourself!
You have practised the formulas and explored worked examples. Now test your understanding independently.
Select one option for each question and click Submit Quiz. The correct options and explanations will appear automatically.
7. Your Performance Report Card
5–6 correct: Excellent! You are an Archimedes' Principle Champion.
4 correct: Very good! Practise a few more numericals.
2–3 correct: Good effort. Revise the formulas and try again.
0–1 correct: Start with Level 1 and progress step by step.
8. The Five-Step Numerical Method
Step 1: Identify the given quantities.
Step 2: Write the formula.
Step 3: Convert units if necessary.
Step 4: Substitute values and calculate carefully.
Step 5: Write the final answer with its unit.
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