1. Dimensional analysis is a technique used in physics to derive relationships between different physical quantities by identifying their fundamental dimensions and units. 2. Key physical quantities like length, mass, time, charge, temperature have fundamental dimensions, while other quantities can be expressed as relationships between fundamental dimensions. 3. As an example, the dimensional analysis technique is demonstrated to derive the formula for the time period of a simple pendulum, which relates time period to the length of the pendulum and acceleration due to gravity. 4. The method involves identifying the parameters that influence the quantity of interest, writing a proportionality relationship between these parameters, and equating the dimensions on both sides to arrive at the formula.

1. DIMENSIONAL
ANALYSIS
Kazi Abu Rousan
CHEENTA.COM

2. Dimensional Analysis
A line has a length equals L and The square has a 2 lengths & so the important measure is Area( 𝑳 𝟐
).
Notice the power of L in each of them

3. Quantities In Physics
In Physics there are certain quantities which are elementary and all other quantities can be derived from them.
Those quantities are called Fundamental Quantities.
The one’s which are derived from the fundamental one’s are called Derived Quantities.
One such fundamental one is Length represented by L and it’s unit is meter or centemeter.
MASSrepresented by M and unit is kg or gram Time represented by T and unit is second or Hour
Fundamental Quantity Symbol Unit(S.I)
Length L Meter
Mass M Kilogram
Time T Second
Electric Current A Ampere
Temperature K Kelvin
If a quantity is given by Q = YR, Then the Symbol of Q can also
be given by ,
Symbol of Q([Q])=Symbol of Y([Y]) χ Symbol of R([R])
If the quantities are created using addition like
Q = Y ± R, then
Symbol of Q([Q])=Symbol of Y([Y]) ± Symbol of R([R])
In which case Q, Y and R are same type of unit so same symbol.

4. Dimension in Physics
Consider the Area of a Rectangle. We know Area = Length χ Width . So, The symbol
of Area will be ,
Symbol of Area = Symbol of Length χ Symbol of Width = L χ L = 𝐿2
So we say that the Dimension of Area is 2 with respect to Length and it is
written as Dimension of Area [A]= 𝐿2
Volume = Area χ Height
So,
[Volume] = L χ L χ L = 𝑳 𝟑
velocity = 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑇𝑖𝑚𝑒 𝑡𝑜 𝑚𝑎𝑘𝑒 𝑡ℎ𝑎𝑡 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
Dimension of Velocity=
𝑳
𝑻
= 𝑳𝑻−𝟏
Accleration =
𝐹𝑖𝑛𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 −𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑇𝑖𝑚𝑒 𝑡𝑜 𝑚𝑎𝑘𝑒 𝑡ℎ𝑎𝑡 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
Dimension of Acceleration =
𝑳𝑻−𝟏
𝑻
= L𝑻−𝟐

5. Now Let’s solve a real world problem as in the end we physicists what to understand this physical world
and try to predict the future.
Find a formula to calculate the Time period of this pendulum in the video.
Time Period is the time The ball needed to travel from it’s initial
position to that same position again.
As Time period is actually a time. Hence [Time Period] = T
Steps to Find the Formula:
1. Find all parameter upon which the quantity of interest depends.
2. Write a pseudo equation and write the dimensions of all
quantities on both side.
3. Just solve some linear equation.
Note: Any Constant doesn’t have any dimension.
Example: Just the number 5. It has no dimension. [5]=𝒚 𝟎
= 𝟏

6. 1. Find all parameter upon which the quantity of interest depends.
2. Write a pseudo and write the dimensions of all quantities on both side.
3. Just solve some linear equation.
As pendulum has a string with certain length, It will depends upon It. As the earth is always pulling all
things, It will depend upon that force or rather acceleration.
As acceleration =
𝑭𝒐𝒓𝒄𝒆
𝒎𝒂𝒔𝒔
= g (acceleration due to gravity, which is a constant).
Now as we are finding Time Period(T), we can write,
T = some combination of acceleration and Length of the string.
Dimension of T and acceleration and length are different. So, They cannot be created using addition and subtraction . So,
The only way left is by Multiplication and Division.
T = 𝑳 𝒂 𝒈 𝒃 , This is the pseudo Equation. Here a , b and are non zero constants. We actually don’t know how T depends
upon L and g.
Now let’s write the dimensions on both sides.
[T] = [ ][ 𝑳 𝒂
] 𝒈 𝒃
or T = 1× 𝑳 𝒂
× (𝑳𝑻−𝟐
) 𝒃
or T = (𝑳 𝒂+𝒃
× 𝑻−𝟐𝒃
)
a+b = 0 and -2b=1 a =
𝟏
𝟐
and b =
−𝟏
𝟐

7. Finally the formula is then, T = ×
𝑳
𝒈
Now what about that constant ?
After measuring Time Period experimentally we find for a certain pendulum,
T = 1.56 sec for a length of 61cm.
Putting this values we get = 6.2527 , If we do real calculations we get = 2𝝅 which is very close to what we get.
So Finally our formula is ,
T = 2𝝅
𝑳
𝒈
This method is too powerful to ask that it’s just a simple tool.
This method can find simple relations just by knowing a single
experimental data.
In the time of 2nd world war, A physicist named G.I.Taylor
calculated the Blast-Radius of the first ever atomic bomb(Trinity
Test) using this method, which was like a shock to U.S military.
The formula is E = 𝝆 𝟎 𝑹 𝟓
𝒕−𝟐 Images of Trinity Test