Chapter 5 · Section 3Textbook page 101

Archimedes' Principle ആർക്കിമിഡീസ് തത്വം

When you lower a stone into a jar filled to the brim, some water spills out. Weigh that spilled water – and you get exactly the buoyant force! This beautiful rule, found over 2200 years ago by Archimedes of Syracuse, ties together everything from pages 5 and 6.

📖 What you need (pages 2, 5 and 6 of this guide)

An object put fully into a liquid pushes aside (displaces) its own volume of liquid (ആദേശം ചെയ്ത ജലം, displaced water).
Buoyant force = W₁ − W₂. It grows with the density of the liquid and the volume of the object.
Mass of water = density × volume: 1 cm³ of water has a mass of 1 g; weight = mass × 9.8.

1. Activity – the overflow jar (Fig 5.6, Table 5.5)

Take a stone and a glass slab. Find the buoyant force on each in water. Lower each one into the water of an overflow jar (a jar with a spout, filled until water just starts to flow out) and collect the overflowing water (കവിഞ്ഞൊഴുകിയ ജലം) in a beaker. Find the weight of the overflowing (displaced) water.

🎬 Fig 5.6 step by step – stone in the overflow jar

🧪 Your own trials – complete Table 5.5

object lower the balance %

Table 5.5 – sample readings (your real readings will differ slightly)
Object വസ്തു	Weight in air W₁ (N)	Weight in water W₂ (N)	Buoyant force W₁ − W₂ (N)	Weight of displaced water (N)

The scale under the beaker is set to zero with the empty beaker on it (“tare”), so it shows only the weight of the water collected. Sample objects (approx.): stone 300 g (≈ 2600 kg/m³), glass slab 200 g (≈ 2500 kg/m³).

Q. കല്ല്, ഗ്ലാസ് സ്ലാബ് എന്നിവയ്ക്ക് ജലത്തിൽ അനുഭവപ്പെട്ട പ്ലവക്ഷമബലവും അവ ഓരോന്നും ആദേശം ചെയ്ത ജലത്തിന്റെ ഭാരവും തമ്മിലുള്ള ബന്ധം എന്തായിരിക്കും? – What is the relation between the buoyant force on the stone and the glass slab and the weight of the water each one displaced? Write your conclusion in the science diary.

They are equal: stone – buoyant force 1.13 N, displaced water 1.13 N; glass slab – 0.78 N and 0.78 N. Why: the object takes the place of exactly that much water. That water used to be held up by the water around it with an upward force equal to its own weight – and now the same upward force acts on the object instead.

The scientist who first found this relation was Archimedes (ആർക്കിമിഡീസ്).

⭐ Key idea (textbook page 101) – Archimedes' principle (ആർക്കിമിഡീസ് തത്വം)

ഒരു വസ്തു ദ്രവത്തിൽ പൂർണ്ണമായോ ഭാഗികമായോ മുങ്ങിയിരിക്കുമ്പോൾ ആ വസ്തുവിൽ ദ്രവം പ്രയോഗിക്കുന്ന പ്ലവക്ഷമബലം വസ്തു ആദേശം ചെയ്യുന്ന ദ്രവത്തിന്റെ ഭാരത്തിന് തുല്യമായിരിക്കും. ഇതാണ് ആർക്കിമിഡീസ് തത്വം.

When an object is fully or partly immersed in a fluid, the buoyant force exerted by the fluid on it is equal to the weight of the fluid displaced by the object. This is Archimedes' principle.

Archimedes' principle as a formula

Weight of displaced fluid = (mass of displaced fluid) × g = (density of the fluid × volume displaced) × g. So

Buoyant force F = ρ_fluid × V_displaced × g

This one line explains page 6: a denser fluid (bigger ρ) or a bigger volume under the fluid (bigger V) both mean a heavier amount of fluid pushed aside – a bigger buoyant force. And once the object is completely under, V stops changing, so depth does not matter. (Page 4 showed the same result from the pressure difference between the bottom and top faces.)

🧮 Archimedes calculator – check the stone

volume under the liquid cm³ liquid

2. Archimedes and the golden crown

📚 Archimedes (ആർക്കിമിഡീസ്)

Archimedes was born in 287 BC in Syracuse, a port city in southern Italy (on the island of Sicily). A popular story is told about him.

King Hiero had a crown made of gold. He suspected that the goldsmith had mixed a cheaper metal into it, and asked Archimedes to find out – without damaging the crown. Archimedes knew that if he could find the crown's volume, he could work out its density and compare it with that of pure gold. But how do you find the volume of a crown with such an irregular shape?

One day, as he got into a bathtub filled to the brim, he noticed the water overflowing. He realised that the volume of water an object displaces equals the volume of the object. In this way he found the crown's volume and density, and proved that the crown was adulterated. (The popular story says he ran out shouting “Eureka!” – “I have found it!”)

🎬 The crown story step by step

Numbers are for illustration (approx.): density of pure gold ≈ 19 300 kg/m³ (19.3 g/cm³), silver ≈ 10 500 kg/m³. The “crown” is 700 g gold + 300 g silver.

Think. If the crown had been pure gold, how much water would it have spilled?

The same as the gold block: 51.8 cm³. Equal mass of the same material → equal volume. The crown spilled 64.8 cm³ – more – so its density was lower (1000 g ÷ 64.8 cm³ ≈ 15.4 g/cm³ instead of 19.3 g/cm³): a lighter metal had been mixed in.

🌐 Explore online needs internet · opens in a new tab

📄 Buoyant Force and Archimedes' Principle The Physics Classroom
Why pressure pushes up harder from below, and why buoyant force equals the weight of fluid displaced.
📄 Buoyancy: Archimedes Principle NASA Glenn Research Center
NASA's short, student-friendly statement of Archimedes' principle with a worked problem.
📄 Archimedes (287 BC - 212 BC) - Biography MacTutor History of Mathematics, University of St Andrews
Life and discoveries of Archimedes of Syracuse, including the crown story.
📄 The Golden Crown Chris Rorres, Archimedes Home Page (NYU Courant)
The 'Eureka!' crown story told step by step, with Vitruvius's and Galileo's versions.
📄 Vitruvius and Archimedes Museo Galileo, Florence
Museum exhibit page on how the crown story was first written down by Vitruvius.
📄 Fact or Fiction?: Archimedes Coined the Term "Eureka!" in the Bath Scientific American
Did the bath-tub story really happen? A look at what is legend and what is physics.

3. Check yourself

🧭 Summary

Archimedes' principle: buoyant force = weight of the fluid displaced (object fully or partly immersed).
Overflow jar: stone – buoyant force 1.13 N = weight of the overflowing water 1.13 N.
Formula: F = ρ_fluid × V_displaced × g – explains both factors of page 6.
Volume of an irregular object = volume of water it displaces (Archimedes' crown).

Next: Principle of floatation – boats and ships ▶