Dimensional Analysis - Meaning, Examples, FAQs

What is Dimensional Analysis?

Dimensional analysis is a method in physics used to study physical quantities by expressing them in terms of basic dimensions like mass (M), length (L), and time (T).

In this method, every physical quantity is written as a dimensional formula, which shows how that quantity depends on fundamental dimensions.

For example,

• Velocity $\rightarrow\left[L T^{-1}\right]$
• Force $\rightarrow\left[M L T^{-2}\right]$

What is Meant by Dimension?

In physics, any physical quantity can be expressed in terms of fundamental units, and the representation of a physical quantity in terms of fundamental units is called the dimension of the physical quantity.

Fundamental Dimensions and Units

The SI units corresponding to fundamental dimensions are called fundamental units.

What Is Principle of Homogeneity Of Dimensions:

The principle of homogeneity of dimensions says that “ In any physical mathematical equation the dimensions of each term appearing in the equation are the same on each side of that equation”. This is called the principle of homogeneity.

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Application of Dimensional Analysis:

Application of dimensional analysis in various fields are:

1. Dimensional Analysis is applicable in the physics field for uses like unit conversion, and checking the correctness of equations.
2. In thermodynamics for deriving dimensionless numbers like Reynold's number.
3. In environmental science by helping researchers develop equations to predict natural occurrences like cyclones.

Uses Of Dimensional Analysis Class 11

The most widely uses of dimensional analysis are mentioned as:

1. The validity of a physical equation can be checked using dimensional analysis.
2. Dimensional analysis is used to determine the dimensions of any unknown variable’s dimension in a given physical equation.
3. Units can be converted from one system to another using dimensional analysis.

Problem Solving Strategy of Dimensional Analysis

1. Identify the physical quantity to be found.
2. List the quantities on which it depends.
3. Assume a relation:

$
X=k a^\alpha b^\beta c^\gamma
$

4. Write the dimensional formulae of all quantities.
5. Apply the principle of dimensional homogeneity
(Dimensions of LHS = Dimensions of RHS).
6. Equate powers of $M, L, T$ on both sides.
7. Solve the equations to find the powers.
8. Write the final formula (numerical constant cannot be found).

Limitations Of Dimensional Analysis

Some of the limitations of dimensional analysis are:

1. Cannot determine dimensionless quantities or constants
2. It depends on known variables only during the analysis.
3. Not able to determine the addition and multiplication of terms with the same dimensions

Examples of Dimensional Analysis

Example 1: Dimensional Formula of Velocity
Velocity $v=\frac{\text { distance }}{\text { time }}$

$
[v]=\frac{[L]}{[T]}=\left[L T^{-1}\right]
$

Example 2: Dimensional Formula of Force
Force $F=m a$

$
[F]=[M]\left[L T^{-2}\right]=\left[M L T^{-2}\right]
$

Example 3: Checking Correctness of an Equation
Given equation:

$
s=u t+\frac{1}{2} a t^2
$

Dimensions of LHS:

$
[s]=[L]
$

Dimensions of RHS:

$
\begin{aligned}
{[u t] } & =\left[L T^{-1}\right][T]=[L] \\
{\left[a t^2\right] } & =\left[L T^{-2}\right]\left[T^2\right]=[L]
\end{aligned}
$

Since dimensions of LHS = RHS, the equation is dimensionally correct.