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# Basic Physics Lesson 6: Volume and Density

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## Introduction

In one of the earlier lessons (Basic Physics : lesson-4), in this series, we learned about the entity called the mass of a body which is defined as the amount of material in it. Now it is time to understand and learn two other basic physical entities known as volume and density. These three entities that are mass, volume, and density characterise and define the physical aspect of a body and from the value of these, we can infer a lot about a particular material. We will also try to understand what is the relationship between these physical entities that is mass, volume and density.

## What do we understand by volume?

Volume is the space occupied by a body. For bodies having cubicle or rectangular structures, it is very easy to calculate volume by multiplying their length, width and height in accordance with the formula for it. For example, a Rubik cube having dimensions of 10 cm x 10 cm x 10 cm has a volume of 1000 cm3. The unit cm3 is commonly represented as cc and is actually the short form of centimetre cube. The graduated beakers, flasks, glasses are generally marked in cc only. So when you use them for taking a particular quantity of liquid medicine or use them to measure a volume then that is in cc only. Let us find some more volumes in other measuring units. A room is 18 feet x 12 feet x 10 feet. Its volume is 2160 ft3. Another example is a big building which is 200 metres long, 100 metres wide and 50 metres high. Its volume is 1000000 m3.

For other symmetric bodies like a cylindrical body or conical body, there are simple formulas to find the volume. For example, the volume of a sphere having a radius of r is 4πr3/3 while for a cylinder having radius r and height h, it comes to πr2h. Here π is a constant equal to about 3.14. It is the same π that we get when we divide the circumference of a circle by its diameter.

Another example is that of a cone having base radius r and height h. It has a volume of πr2h/3. These formulas are very helpful in engineering and architecture. For example, if one wants to find out the volume of the material in a thick hollow pipe of length L having inner and outer diameters as d1 and d2 respectively then one can subtract the volume of the inner cylinder from the volume of the outer cylinder and get the result. You can try it yourself using the formula for the volume of the cylinder and I am sure you can find out the volume of material in the hollow pipe as π(d22 - d12)L/4. If you are not able to do it then find how to arrive at it at 'Note-1' at the end of this article.

Volume is a very important measurement as it helps us to make and calculate many things quickly in our life. For example, a warehouse having a floor dimension of 50 meters by 100 meters with a height of 8 meters will have a volume of 40000 m3. Now, suppose we are getting a large number of packages of size 80 cm x 50 cm x 50 cm to store them in it and we want to calculate as how many packages we will be able to store here. What we would do is divide the volume of the warehouse by the volume of one package and that gives us the number. Please remember that whenever we do such calculations we have to keep the units same. So, let us do it. We would convert the dimensions of a warehouse to centimetre and after that divide, it's volume by that of one package.

So, the maximum number of packages that we can keep in this warehouse is -

50*100*100*100*8*100 / 80*50*50 = 200000.

Voila! We can store 200000 packages in that.

## How to find out volume of irregular shapes?

If a body is having an irregular shape say a piece of a small stone then there is no simple formula or calculation to find out the volume of the stone. In such cases, we have to take help of some other method and in Physics we have a simple solution for calculating that complex appearing volume of the irregularly shaped stone. For doing it we have to take a graduated beaker or mug and put some water in it. Now read the level of the water. Say it shows 400 cc. Now immerse the stone fully in it so that it sinks to the bottom. The water level would rise as the stone has displaced some water equal to its volume. Now note the new level. Let us say it is 435 cc. Just subtract and you get the volume of stone as 35 cc. Very simple. School children do it in to find the volume of irregularly shaped bodies.

One thing that we have to take care in such methods is that if the stone is a weaker one with its material falling here and there and chipping out and starts breaking in flakes and absorbs water or swells or develops fissure then this method would not work and the volume calculated will be erratic and misleading. Our assumption in using such methods is based on the fact that generally water would not alter the stone and also would not be absorbed by it. In scientific experiments, such precautions are a matter of common sense and should always be used with great care and concern.

At the same time let us say that there are some mineral stones which absorb water and also swell slightly. In such a case, our calculations will be affected by two things - the amount of water absorbed and the amount of increase in the volume of the stone due to swelling. It is interesting to note that these two effects are in opposite directions and may annul each other to some degree reducing the error in our observations and results.

## Importance of volume of displacement

There is another interesting thing about the volume that when a body is immersed in liquid then its apparent weight is reduced. The weight of a ship floating in the ocean is less than its weight in the ship workshop in the port facility. Mass of a body does not change and is a fixed thing but due to the forces of buoyancy, the weight appears to be reduced considerably. All ship and barge designings are based on this fact only. For example, a ship that is just launched into the waters sinks down till the weight of the water it displaces is just equal to its own weight. After this, if we load the ship with cargo then it will sink more displacing more water, and the buoyant forces will also increase and match the new weight of ship which is increased due to the cargo loading.

The great scholar, inventor, and mathematician Archimedes was able to realise this fact when he declared that the weight of a body immersed in water appears to lose weight and the decrease is equal to the weight of the water displaced by the body. When we say water displaced by the body then it means the volume displaced and that is where the importance of volume comes in picture. More the volume, more will be the displacement and reduce the apparent weight. That is the reason why a metal piece sinks in water but the same metal piece beaten and carved in a bowl floats in the water.

The invention of this law of buoyancy by the Archimedes was a great leap in understanding the floating mechanisms and is a milestone invention in the scientific progress made so far.

## Units of volume and conversions

As we saw in preceding paragraphs that volume can have a unit like cm3 or ft3 or m3 but for uniformity purposes in M.K.S. system, we keep the units as m3 only. At the same time, there are practical units also as litre or gallon. One litre is 1000 cm3. As regards gallon, we have two systems one is US gallon other is the UK or imperial gallon. One US gallon is about 3.78 litres while imperial gallon used in the UK is 4.55 litres. All this sometimes appears confusing but we can not ignore or neglect the conventional units which are in use. Incidentally in the US, one gallon is converted to 128 US fluid ounces also.

Fortunately conversion tables are available in the internet for quickly converting these quantities from one unit to another.

## What do we understand by density

Mass and volume are two entities that define a particular object - how much material it contains and how much space it occupies. For the same mass two bodies can occupy different volumes. For example, 500 gram of iron piece will occupy a smaller space as compared to 500 gram of cotton. At the same time, two same volumes of two different materials would have different masses.

Here comes the concept of density. It is defined as the mass of a unit volume of a material and is a physical characteristic of it. It is represented by d or D or Greek alphabet ρ (Rho). In mathematical terms, we can write as -

density = mass / volume

For example, if we take a volume of 30 cc of water and weigh it then its mass would be 30 gram. Now let us find its density. As per the above definition, the density of water would be 30 gram/30 cc = 1 gram/cc. Let us take an example of an iron piece of length 4 cm, width 2 cm and height 2 cm having a mass of 125 gram. Let us now try to find its density in gram/cc. The volume of this iron piece is 4 cm x 2 cm x 2 cm = 16 cm3 = 16 cc. So, density of iron = 125 gram/16 cc = 7.8 gram/cc.

Water having a density of 1 gram/cc is taken as a reference and from above details, we can very well say that iron is 7.8 times heavier than water.

So, mass and volume will change with the shape and size of the bodies but the density will not change like that and is a characteristic of that particular material. Let us just see the approximate density values of some commonly known items -

Carbon dioxide gas - 0.002 g/cc

Alcohol - 0.08 g/cc

Milk - 1.03 g/cc

Blood - 1.6 g/cc

Granite - 2.65 g/cc

Aluminium - 2.7 g/cc

Copper - 8.6 g/cc

Gold - 19.3 g/cc

Platinum - 21.4 g/cc

## Units of density and conversion

Just like mass and volume, density also has various units which can be converted to each other by using conversion tables. Some of the common units of density are gram/cc, kg/m3, kg/litre, gram/milli litre, metric ton/m3, pound/cubic feet etc. M.K.S. unit of density is kg/m3.

## Specific weight and specific gravity

Though density is a common nomenclature to denote itself but scientifically speaking, the mass per unit volume is known as specific weight. So, density and specific weight mean same thing. The densities are generally represented with reference to or in comparison to a standard reference material like water. Once density is represented like that then it becomes a unit less quantity known as specific gravity.

Specific gravity = density of a material / density of water

## Note-1

Using the formula πr2h for the volume of the the cylinder, the volume of outer cylinder is -

π(d2/2)2L

In the same way the volume of inner cylinder is -

π(d1/2)2L

By substructing the inner volume from the outer one, we get the volume of the wall material of the pipe -

π(d2/2)2L - π(d1/2)2L = π(d22 - d12)L/4

1. https://www.unitconverters.net

2. https://en.m.wikipedia.org/wiki/Density

3. https://www.britannica.com/science/Archimedes-principle

## Conclusion

Volume and density are two important physical parameters which give important information about the material as how much space it occupies, how big it is apparently, and how dense it is. These are the basic entities and their basic understanding is required by the students as well as the professionals as they are used in science studies at so many places and in so many scientific and technical projects. We would also be using them in our future lessons accordingly.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.