Explanation:

Dimensional analysis is a mathematical method that is used to check the dimensional consistency of physical equations. It involves the analysis of the dimensions of physical quantities involved in an equation to determine whether the equation is correct or not. The fundamental principle of dimensional analysis is that physical equations must be dimensionally consistent. This means that the dimensions of physical quantities on both sides of the equation must be the same.

Limitations of Dimensional Analysis:

While dimensional analysis is a powerful tool for verifying the correctness of physical equations, it has some limitations. One of the main limitations is that it cannot be applied to equations involving more than three physical quantities. This is because the number of independent physical quantities that can be used to describe a physical system is limited to three. These quantities are usually chosen as mass, length, and time.

Explanation for Limitation:

The reason why dimensional analysis cannot be applied to equations involving more than three physical quantities is that it becomes difficult to express the dimensions of all the quantities in terms of the three fundamental quantities of mass, length, and time. This is because each additional physical quantity adds another dimension to the equation, which makes it more complex.

For example, consider an equation that involves four physical quantities, such as force, velocity, time, and temperature. The dimensions of these quantities are:

- Force: [M][L][T]^-2
- Velocity: [L][T]^-1
- Time: [T]
- Temperature: [θ]

To apply dimensional analysis to this equation, we would need to express the dimensions of all the quantities in terms of the fundamental quantities of mass, length, and time. However, this is not possible because temperature is not related to any of these three fundamental quantities. Therefore, dimensional analysis cannot be applied to equations involving more than three physical quantities.

Conclusion:

In conclusion, dimensional analysis is a powerful tool for verifying the correctness of physical equations. However, it has some limitations, and one of these limitations is that it cannot be applied to equations involving more than three physical quantities. This is because the number of independent physical quantities that can be used to describe a physical system is limited to three, and each additional physical quantity adds another dimension to the equation, making it more complex.

Bcoz in dimensional analysis there are only 3 fundamental quantities.