Classwork Series and Exercises {Physics- SS1}: Density and Relative Density

Introduction

Equal volumes of different substances have different masses or weights. For example 1 m³ of lead weighs differently from 1 m³ of wood. Also 1 m³ of water has a weight different from that of sulphuric acid or palm oil of the same volume. This is due to differences of in the quality of the substances known as density.

The principle of density was discovered by the Greek Scientist named Archimedes over 2000 years ago.

Density (ρ) is defined as the mass per unit volume of the material.

To calculate the density (usually represented by the Greek letter “rho“) of an object, take the mass (m) and divide by the volume (v):

rho = m / v

The SI unit of density is kilogram per cubic meter (kg/m³) Using Density .It is also frequently represented in the cgs unit of grams per cubic centimeter (g/cm³). For example the density of water at 4⁰C is approximately 1gcm^-3.

 1g/cm³ = 1 x 10^-3kg/10^-6m³ = 10³ kgm^-3

To convert density from cgs to S .I. unit we merely multiply the cgs density by 10³. Thus water has a density of 1 gcm^-3 or 10^– kgm^-3.

Table below shows the density values of some solids and liquids:

Material	Density
Aluminum	2.7 x 103
Copper	8.9 x 103
Bamboo wood	0.4 x 103
Gold	19.3 x 103
Glass	2.6 x 103
Lead	11.3 x 103
Platinum	21.5 x 103
Iron	7.9 x 103
Steal (variable)	7.8 x 103
Ice (at 00)	0.92 x 103
Water (at 40)	1 x 103
Mercury	13.6 x 103
Sand (variable)	2.6 x 103
Methylated spirit	0.8 x 103
Paraffin wax	0.9 x 103
Zinc	7.1 x 103

One of the most common uses of density is in how different materials interact when mixed together. Wood floats in water because it has a lower density, while an anchor sinks because the metal has a higher density. Helium balloons float because the density of the helium is lower than the density of the air.

To measure the density of a liquid
To find the mass:
Weigh an empty graduated cylinder, and then weigh the graduated cylinder after pouring water into it.
Then subtract the two readings.

To find the volume simply note the level of water in the graduated cylinder.

Relative Density or Specific Gravity

A concept related to density is the specific gravity (or, even more appropriate, relative density) of a material, which is the ratio of the material’s density to the density of water. An object with a specific gravity less than 1 will float in water, while a specific gravity greater than 1 means it will sink.

Relative density is defined as: R.d. = mass (or weight) of a substance/mass (or weight) of equal volume of water, it is also defined as R.d. = density of substance/density of water, note that relative density has no unit, the relative density of mercury is 13.6 but its density 13.6 x 10³ kgm^-3. In the cgs unit where the density of water is 1 gcm^-3, the density of a substance is numerically equal to its relative density.

Material	Specific Gravity
Balsa Wood	0.2
Oak Wood	0.75
Ethanol	0.75
Water	1
Table Salt	2.17
Aluminium	2.7
Iron	7.87
Copper	8.96
Lead	11.35
Mercury	13.56
Depleted Uranium	19.1
Gold	19.3
Osmium	22.59

Measurement of Density and Relative Density

1     A body that can sink in water e.g. lead

• Regular solid

The density of a regular solid is obtained by measuring its mass (for example with a balance) and its volume by measuring its dimension.

• Irregular solid (e.g. A piece of stone)

The mass is obtained as for regular solids but its volume is measured by immersing the object completely in a measuring cylinder containing water. The difference in the level of water before and after the immersion of the solid gives the volume of the solid. Then Density = Mass/Volume.

2 Relative density of a body which floats in water (e.g. cork) (using Archimedes principle)

Because the cork cannot sink in water of its own accord, we make use of a ‘sinker’, i.e. a solid that can make the cork to sink or to be completely immersed in water. The sinker is first hung by a thread from the hook of spring balance and its weight when completely immersed in water is read from the balance. The cork is then tied to the thread carrying the sinker such that it does not touch the water surface. The new weight of sinker in water and the cork in air is obtained. Finally the cork is tied near the sinker and the weight of sinker and cork, both immersed in water is obtained. The relative density of cork is then calculated as follows:

Weight of sinker in water = W₁

Weight of sinker in water and cork in air = W₂

Weight sinker and cork in water = W₃

Weight of cork in air = (W₂ – W₁)

Upthrust on cork = apparent loss in weight of cork = (W₂-W₃)

Relative Density of cork = Weight of cork in air/Weight of equal volume of water

                                            = Weight of cork in air/Upthrust on cork

                                            = W₂ – W₁/W₂ – W₃

3   Relative Density of solid in form of particles of powder (e.g. sand)

We make use of a relative density bottle and a chemical balance: The relative density bottle is first cleaned, dried and weighed empty. It is then reweighed when about one third full of sand. The relative density bottle containing sand is now filled up filled with water and reweighed. Finally the bottle is emptied, cleaned of any sand particle and filled with water alone. The weight of bottle and water is cleaned. The relative density of sand is calculated as follows: Mass of empty bottle = M₁

Mass of bottle + sand =M₂

Mass of bottle + sand + water = M₃

Mass of bottle + water only = M₄

Mass of water filling the bottle = (M₄ – M₁)

Mass of water having the same value as the sand = (M₄ – M₁) – (M₃ – M₂)

Therefore, relative density of sand = Mass of sand/mass of equal volume of water

(M₂ – M)₁ / (M₄ – M₁) – (₃ – M₂)

4 Relative density of particle that is soluble in water (e.g. salt)

The experiment for the relative density of sand is repeated but now turpentine instead of water is used. If the density of turpentine is ρ gcm^-3 then the relative density is calculated as sand and become: R.D. = (M₂ – M₁)ρ/(M₄ – M₁) – (M₃ – M₂)

5 Relative density of a liquid

• Using a relative density bottle

A chemical balance is used in all the weighting involved In this experiment. The relative density bottle is cleaned, dried and weighed when empty. It is then reweighed when filled with the liquid. The emptied of the liquid, cleaned dried and reweighed when filled with the water , the R.D of the liquid calculated as follows:]

Mass of empty bottle = M₁

Mass of bottle + liquid = M₂

Mass of bottle + water = M₃

Mass of liquid = (M₂ – M₁)

Mass of equal volume of water = (M₃ – M₁)

Relative density of liquid = M₂ – M_1/M₃ – M₁

Using Archimedes Principle

When an object is wholly immersed in a fluid, it experiences an upthrust (apparent loss of weight) which is equal to the weight of the fluid displaced by the object.

We make use of a solid e.g. a glass stopper. The solid is suspended from the hook of a spring balance and weighed in air, it is then weighed when completely immersed in the liquid. Finally it is cleaned, dried and reweighed when completely immersed in water. The relative density of the liquid is calculated as follows:

Mass of object in air = M₁

Apparent mass of object in liquid = M₂

Apparent mass of object in water = M₃

Upthrust in liquid = weight of liquid displaced by the object = M₁ – M₂

Upthrust in water = weight of water displaced by the object = M₁ – M₃

Relative density = Upthrust in fluid/Upthrust in water

                              = M₁ – M₂/M₁– M₂

EXERCISES

Lets see how much you’ve learnt, attach the following answers to the comment below

1. Density is defined as the mass per unit …………… A. width B. height C. volume D. area
2. A body weighs 0.30 N in air, 0.25 N when fully immersed in water and 0.27 when fully immersed in a liquid calculate: (a) its water weight loss (b) its relative density (c) the relative density of the liquid. A. 0.07, 1, 1.6    B.   0.05, 6, 0.6    C.   0.9, 6, 1.10    D.   0.04, 9, 0.1
3. A metal block of density 9000 kgm^-3 weighs 60 N in air. Find its weight when it is immersed in paraffin wax of density 800 kgm ^-3. (Take g = 10 ms^-2) A. 54.67 N B. 60.67   C. 64. 76   D. 96.45N
4. A relative density bottle weighs 20 g when empty, 80 g when filled with water and 100 g when filled with a liquid. Find the relative density of the liquid. A. 1.45 B.  2.33    C.  2.08    D.  1.33
5. A stone has a mass of 120 g and a volume of 20 cm³, what is the density of the stone?

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