Any object in a liquid whether floating or submerged experiences an upward force from the liquid; the force is known as upthrust force. Upthrust force is also known as buoyant force and is denoted by letter ‘u’.

**Archimedes’ principle**

**Floating and Sinking**

Archimedes, a Greek scientist carried out first experiments to measure upthrust on an object in liquid in the third century. Archimedes principle states that ‘When a body is wholly or partially immersed in a fluid (liquid/ gas), it experiences an upthrust equal to the **weight** of the displaced fluid”.

**Floating and Sinking**

**Experiment: **To demonstrate Archimedes principle

**Procedure**

- Pour
**water**into an overflow can (eureka can) until it starts to flow out then wait until it stops dripping - Tie a suitable solid body securely and suspend it on a spring balance. Determine
**weight**in air. - Lower the body slowly into the overflow can while still attached to the spring balance then read off its
**weight**when fully submerged. - Weigh the displaced
**water**collected in a beaker.

**Record your readings as follows;**

**Weight** of body in air = W1

**Weight** of empty beaker = W3

**Weight** of beaker and displaced liquid = W4

Upthrust of the body = W1 – W2

**Weight** of displaced water = W4 – W3

**Discussion**

The upthrust on the solid body will be found to be equal to the **weight** of displaced water therefore demonstrating the Archimedes principle.

**Example**

A block of metal of volume 60 cm3 weighs 4.80 N in air. Determine its **weight** when fully submerged in a liquid of density 1,200 kgm^{-3}.

**Solution**

Volume of liquid displaced = 60 cm_{3} = 6.0 × 10-5 m_{3}. Weight of the displaced liquid = volume × density × gravity = v × ρ × g

= 6.0 × 10-5 × 1200 × 10

= 0.72 N

**Upthrust** = weight of the liquid displaced.

Weight of the block in the liquid = (4.80 – 0.72) = 4.08 N.

**Floating objects**

Objects that float in a liquid are less dense than the liquid in which they float. We have to determine the relationship between the weight of the displaced liquid and the weight of the body.

**Experiment:** to demonstrate the law of floatation

**Procedure**

- Weigh the block in air and record its weight as W1.
- Put water into the overflow can (eureka can) up to the level of the spout.
- Collect displaced water in a beaker. Record the weight of the beaker first in air and record as W2. Weigh both the beaker and the displaced water and record as W3.
- Record the same procedure with kerosene and record your results as shown below.

**Discussion**

The weight of the displaced liquid is equal to the weight of the block in air.

This is consistent with the law of floatation which states that “A body displaces its own weight of the liquid in which it floats”.

Mathematically, the following relation can be deduced Weight = volume × density × gravity = v × ρ × g, therefore W = v d × ρ × gwhere vdis the volume of displaced liquid.

**NOTE** – Floatation is a special case of Archimedes principle.

This is because a floating body sinks until the upthrust equals the weight of the body.

**Example**

A wooden block of dimensions 3 cm × 3 cm × 4 cm floats vertically in methylated spirit with 4 cm of its length in the spirit.

Calculate the weight of the block. (Density of methylated spirit = 8.0 × 102 kgm^{-3}).

**Solution**

Volume of the spirit displaced = (3 × 3 × 4) = 36 cm^{3} = 3.6 × 10-5 m^{3}

Weight of the block =v d × ρ × g = (3.6 × 10-5) × 8.0 × 102 × 10 = 2.88 × 10-1 N.

**Relative density**

We have established the relative density as the ratio of the density of a substance to the density of water. Since by the law of floatation an object displaces a fluid equal to its own weight hence the following mathematical expressions can be established.

Relative density = density of substance / density of water.

= weight of substance / weight of equal volume of water

= mass of substance / mass of equal volume of water

Applying Archimedes principle, the relative density‘d’;

d = weight of substance in air / upthrust in water or d = W / u

Since upthrust is given by (W2 – W1)where W2 – weight in air, W2– weight when submerged. Hence d = W / u = W / W2 – W1, the actual density, ρ of an object can be obtained as follows ρ of an object = d × 1,000 kgm-3.

**Relative density of a floating body**

**Experiment:** To determine the relative density of a cork

**Procedure**

- Select a sinker which is heavy enough to make the cork to sink.
- Attach the cork and the sinker as follows.
- Record the results obtained as follows

Weight of the sinker in water = W1

Weight of the sinker in water and cork in air = W2

Weight of the sinker and cork in water = W3

Weight of the cork in air = W2 – W1

Upthrust on the cork = W2 – W3

The relative density of the cork in air is determined as follows;

d = weight of the cork in air / upthrust on the cork.

**Applications of Archimedes principle and relative density**

**Ships**– steel which is used to make ships is 6-7 times dense than water but a ship is able to float on water because it is designed to displace more water than its volume.

Load lines called plimsoll marks are marked on the side to indicate the maximum load at different seasons to avoid overloading.

**Submarines**– they are made of steel and consists of ballast tanks which contain water when they have to sink and filled with air when they have to float.

This makes the submarines to balance their weight and be able to rise upwards.

**Balloons**– when they are filled with helium gas balloons become lighter and the upthrust on the balloon becomes greater than their weight therefore becoming able to rise upwards.**Hydrometers**– they are used to measure the relative densities of liquids quickly and conveniently. Various types of hydrometers are made to measure different ranges of different densities i.e. lactometer – for measuring milk water (range 1.015 – 1.045), battery acid tester – used to test the charge in a lead-acid battery.

**Examples**

- A solid of mass 1.0 kg is suspended using a thread and then submerged in water.

If the tension on the thread is 5.0 N, determine the relative density of the solid.

**Solution**

Mass of solid = 1.0 kg

Weight of solid W = mg = 10 N

Tension on the string (T) = 5 N

Upthrust on solid (u) = W – T = 10 – 5 = 5

Relative density (d) = W / u = 10 / 5 = 2.

- A balloon made up of a fabric weighing 80 N has a volume of 1.0 × 107 cm. the balloon is filled with
**hydrogen**of density 0.9 kgm-3.

Calculate the greatest weight in addition to that of the **hydrogen** and the fabric, which the balloon can carry in air of average density 1.25 kgm^{-3}.

**Solution**

**Upthrust** = weight of the air displaced

= volume of air × density × gravity

= (1.0 × 107 × 106) × (1.25 × 10)

= 10 × 1.25 × 10 = 125 N

Weight of **hydrogen** = 10 × 0.09 × 10 = 9 N

Total weight of hydrogen and fabric = 80 + 9 = 89 N

Total additional weight to be lifted = 125 – 89 = 36 N.

- A material of density 8.5 gcm
^{-3}is attached to a piece of**wood**of mass 100g and density 0.2 gcm^{-3}.

Calculate the volume of material X which must be attached to the piece of **wood** so that the two just submerge beneath a liquid of density 1.2 gcm^{-3}.

**Solution**

Let the volume of the material be V cm^{3}

The mass of the material be 8.5 V grams

Volume of **wood** = 100 g / 0.2 g/cm = 500 cm^{3}.

In order to have an average density of 1.2 gcm^{-3} = total mass / total volume

Therefore (100 + 8.5V) / (500 + V) = 1.2 gcm^{-3}

Hence V = 68.5 cm^{3}.

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