Buckingham theorem: The theorem states that if there are `n` variable involved contain physical phenomenon and if these variable contain `m` primary dimensions (e.g. M, L, T) then the variable quantities can be expressed in terms of an equation containing (n - m) dimensionless groups of parameters.

Variables in a Physical Phenomenon

In a physical phenomenon, there are often multiple variables that are involved and can affect the outcome of the phenomenon. These variables can be quantities such as length, time, mass, temperature, etc. Each of these variables can be expressed in terms of fundamental dimensions, which are the basic dimensions used to describe physical quantities.

Fundamental Dimensions

Fundamental dimensions are the basic dimensions that cannot be expressed in terms of other dimensions. They are usually represented by symbols such as L for length, T for time, M for mass, etc. For example, the dimension of velocity can be expressed as [LT^-1], where L represents length and T represents time.

Number of Variables

Let's say there are 'n' variables involved in a physical phenomenon. These variables can be expressed in terms of 'm' fundamental dimensions. This means that each variable can be written as a product or combination of the fundamental dimensions, raised to certain powers.

Arranging Variables into Dimensionless Terms

When the variables can be expressed in terms of fundamental dimensions, they can also be rearranged into dimensionless terms. A dimensionless term is a term that does not have any units or dimensions associated with it. It is obtained by dividing a physical quantity by a suitable combination of fundamental dimensions.

Number of Dimensionless Terms

According to the given question, if there are 'n' variables in a physical phenomenon and they can be expressed in terms of 'm' fundamental dimensions, then the variables can be arranged into (n - m) dimensionless terms. This means that the number of dimensionless terms will be equal to the difference between the number of variables and the number of fundamental dimensions.

Explanation of the Correct Answer

The correct answer to the question is option 'C', (n - m) dimensionless terms. This means that if there are 'n' variables in a physical phenomenon and they can be expressed in terms of 'm' fundamental dimensions, then the variables can be arranged into (n - m) dimensionless terms.

In other words, the number of dimensionless terms will be equal to the difference between the number of variables and the number of fundamental dimensions. This is because each variable can be expressed in terms of the fundamental dimensions, and by rearranging them, we can obtain dimensionless terms.

By arranging the variables into dimensionless terms, it becomes easier to analyze and compare different physical phenomena, as the dimensionless terms do not have any units or dimensions associated with them. They are purely numerical values that can be used to understand the behavior and relationships between the variables in the phenomenon.