# Derivations of Units With Dimensional Analysis

When the study of physics started, the first thing they figured out is the units. The units are the primary requirement to study an aspect of a physical quantity. Because you must have to measure the various components of the system to be studied and the measurement can be done only if you have the units. For example, if you want to construct a dam, you will have to first study the quantity of water it will withhold in future and the quantity will be measured in litres.

So you can see how important these general terms are in the world of physics. I have also mentioned a term, "physical quantity". So what is the physical quantity?

A physical property that can be quantified or measured is known as the physical quantity. Mass, length, volume, pressure, velocity are some examples of the physical quantity.

The physical quantities are of the following two types:

- Basic physical quantities. There seven basic physical quantities in physics as given below:

- Length(Measured in metre, kilometre, feet etc.)
- Mass(Measured in gram, kilogram etc.)
- Time(Measured in seconds, minutes, hours)
- Electric Current(Measured in ampere)
- Thermodynamic Temperature(Measured in Celsius, Kelvin, Fahrenheit)
- Amount of substance(Measured in Mole or molecule)
- Luminous intensity(Candela)

- Derived physical quantities

The derived physical quantities are the quantities that are derived from the basic quantities mentioned above. These are generated through a formula and contains a mathematical link between one or more of the seven basic physical quantities. For example, the volume, speed, acceleration are the derived physical quantities.

We know that the formula for volume is

Volume = Length x Breadth x Height = metre³

The length, breadth, and height are measured in meter (a basic physical quantity) and there is an interlink(the multiplication) between them all. So we can say that the volume is a derived physical quantity.

Similarly, Velocity = Displacement/Time = Metre/second

We have learned about physical quantities. Now we will discuss how the units of the derived physical quantities are defined using dimensional analysis.

I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind.

— Lord Kelvin## What is Dimensional Analysis

The units of a physical quantity are defined by the section of physics known as "Dimensional Analysis" It ascertains some rules and methods that are used to make the units for measuring a certain physical quantity. The dimensional analysis is done using the formula or scientific equation used to calculate the quantity. For example, the formula of pressure is:

Pressure = Force/per unit area

So the units of pressure will be derived using the units of Force and area; its sub-quantities.

Before we move further, we will have to keep some basic rules in mind that are used in the method of dimensional analysis. Here are the rules:

**The units of physical property are defined by keeping in view all the physical quantities directly or inversely affecting the equation.**

For example, the pressure is directly proportional to Force exerted

and inversely proportional to the area on which force exerted

so the units of pressure will be defined using the units of both the force and area

**The physical quantities can be multiplied and divided mathematically**

In the previous example of pressure, the Pressure = Force÷Area

Now we will split the force and area in their basic quantities

Force = Mass x Acceleration = Mass x Velocity/Time = Mass x Length/Time^{2}

Area = Length x Breadth = Length x Length = Length^{2}

Now we will get back to our initial equation Pressure = Force ÷ Area

Splitting all the physical quantities in terms of basic physical quantities;

Pressure = Mass x Length/Time^{2 }÷ Length^{2}

As per the rule these physical quantities can be multiplied and divided mathematically, so doing some mathematical manipulation we will get the conclusion that

Pressure = Mass x length ÷ Time^{2}

Now we will move on to the next rule of our dimensional analysis

**The physical quantities cannot be added or subtracted mathematically.****it means that If the equation of the derived physical quantity contains a subtraction or addition symbol linking two physical quantities then it will have no effect on the units of the quantity.**

For example, the equation for the units of the area of green coloured highlighted portion in the following image can be calculated with the equation

Area = b^{2 }− a^{2}

Where b and a are lengths of the sides of each square shown in the figure.

Splitting the terms in their basic physical quantities:

Required area = Length^{2 }− Length^{2}

= Length^{2}

So we can say that the subtraction has not any effect on the units of the physical quantity. Similarly, the addition will also have no effect on the same.

We have learned the fundamental things required for dimensional analysis. Now we will apply the same to find out the units of a real-life physical quantity.

I am going to take the example of momentum for defining the units.

We know that,

Momentum = Mass x Velocity

Splitting in basic quantities:-

Momentum = Mass x Length/Time

In physics, the Mass is represented with the symbol M, length with L, Time with T and momentum with P

Substituting the respective symbols we get the equation:-

P = MxL/T

Now we will have to substitute the units of each of these basic physical quantities in the equation.

As we know that the SI system unit of length is metre, mass kilogram and time seconds. Substituting these values in the equation we get:-

P = Kilogram metre/second

or it may be written as:

Kilogram metre second^{−1 }

or

Kg m s^{−1}

which is the unit of momentum.

The units of all the basic physical quantities must be taken in the same unit system(SI, MKS, FPS or CGS etc.) while making the units of a derived physical quantity

A universal constant is not considered in dimensional analysis. For example, the area of a circle is calculated with the equation:

Area = πr^{2 }where the r represents the radius of the circle and π is a universal constant having the value 22/7

Here the π will have no effect on the units of the area and the units will be

Area = L^{2 }

Area = metre^{2}

This way the units of various physical quantities are defined in Physics.

## Which of the following has an effect on dimensional analysis?

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.

**© 2019 Sourav Rana**

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