Dimensions

[Disclaimer: I just wanted to share some things which I am learning.Kindly confirm those from the textbooks.]

Hello friends,

Today we will learn about dimensions. The concept of dimensions takes very important role in Physics. For the competitive exams such as IIT JEE this chapter is very important. You had learnt about units in our last website. Now we will learn about dimensions. But before learning about dimensions you have to know the basic knowledge about units. I assume that you know about units. Now we will see about dimensions. What is dimension? Let us find out.

2.1 – Introduction to dimensions

We know about fundamental & derived quantities. Let us revise about fundamental & derived quantities. The quantities which are Independent are fundamental quantities. Also, fundamental quantities do not require help of any other physical quantities for their definition. On the other hand, the quantities which are not independent are said to be derived quantities. Derived quantities need help of other physical quantities for their definition.

Now what is dimension? When derived quantities are obtained from multiplying or dividing two or more fundamental quantities, it is written as a product of different powers of the fundamental quantities. The powers of fundamental quantities are the dimensions & we can write derived quantity in dimensional formula. What is dimensional formula? Let us find out...

2.2 – Finding dimensional formula

Now we have to learn about finding dimensional formula. Let us consider the physical quantity force. We know that the force is a derived quantity. Let us find out the dimensional formula of force. Use the following steps to find the dimensional formula of force.

STEP 1

First of all we have to write the formula of force. You know the formula of force is Mass x Acceleration. Write down the formula of force as follows.

FORCE = MASS x ACCELERATION

STEP 2

Before going to see step 2, we have to know when we have to find the dimensional formula of any physical quantity then we have to dissect it in the form of fundamental quantities. So we will dissect this mass x acceleration in the short form.

We know the formula of acceleration is Velocity / Time. So we will replace acceleration with velocity / time. Write down the complete formula of force.

FORCE = (MASS x VELOCITY) / TIME

STEP 3

Now we will dissect this formula once again. We know the formula of velocity. The formula of velocity is Length / Time. So we will replace velocity with Length / Time. Write down the complete formula of force.

FORCE = [MASS x (LENGTH / TIME)] / TIME

So, FORCE = (MASS x LENGTH) / TIME²

So, FORCE = MASS¹ x LENGTH¹ x TIME^-2

STEP 4

Note that in dimensional formula we write Mass as M, Length as L & Time as T. So the dimensional formula of force is M¹ x L¹ x T^-2. Also, when we write the dimensional formula we cannot write ‘x’ this symbol of multiplication. So the actual dimensional formula of force will be M¹L¹T^-2. Note that the dimensional formula of any physical quantity should be written in two square brackets. Write down the dimensional formula of force.

[M¹L¹T^-2]

Finding dimensional formula of any physical quantity was very easy. Isn’t it? In the following chart some quantities and their dimensional formulae are given.

2.3 – Some quantities and their dimensional formulae

Now let us see about some applications based on dimensional analysis.

2.3 – Applications based on dimensional analysis

TYPE 1: Finding unit of a physical quantity in a given system of units

How to find the unit of a physical quantity in a given system of units? For learning about it let us take an example.

Q.1 – What is the unit of density in MKS system?

Before solving this question let us revise about the system of units. Here is the chart given for system of units.

Observe it.

In MKS system, the units for Length, Mass & Time are meter, kilogram & second respectively. The dimensional formula of density is [M¹L^-3T⁰]. In MKS system L, M & T is meter, kilogram & second respectively.

Density = M¹L^-3T⁰

Density = kg¹ X m^-3

Density = kg / m³

So the unit of density in MKS system is kg / m^3.

TYPE 2: Converting physical quantities from one system to other

In this case we have to change a physical quantity from one system to the other. Here is the formula of this type of questions.

n2 = n1  X [M1 / M2]^a X [L1 / L2]^b X [T1 / T2]^c

In this formula, n1 is a numerical value of a physical quantity which is in the first system & n2 is a numerical value of a physical quantity which is in the second system. M1, L1 & T1 are mass, length & time of a physical quantity in the first system respectively. M2, L2 & T2 are mass, length & time of a physical quantity in the second system respectively & a, b & c are dimensions of Mass, length & time respectively.

Now let us take an example to clear this concept.

Q.2 – Convert one newton into dyne.

You know that newton is the unit of force in SI system & dyne is the unit of force in CGS system. In SI system the units of Mass, Length & Time are kg, m & s respectively & in MKS system the units of Mass, Length & Time are g, cm & s respectively. Here the dimensional formula of force is M¹L¹T^-2. So here a, b & c are 1, 1 & -2 respectively. By using the formula,

n2 = n1  X [M1 / M2]^a X [L1 / L2]^b X [T1 / T2]^c

1 N = 1 X [kg / g]¹ X [m / cm]¹ X [s / s]^-2

1 N = 1 X [10³ g / g] X [10² cm / cm] X [1]^-2

1 N = 1 X 10³ X 10² X 1

1 N = 10⁵ dyne.

So 1 N = 10⁵ dyne.

It was very easy, wasn’t it? Now I think your concept based on dimensions is complete. Thank you.

2.4 – Thank you

So friends thank you for watching my website. Best of luck for your exams… See you next time.

Your friend,

Aditya P