Question

In dimensional analysis, according to Buckingham's \(\pi\)-theorem, if n is the total number of variables and m is the number of independent dimensions, then the maximum number of independent dimensionless \(\pi\)-groups will be

• A. m - n
• B. mn
• C. m + n
• D. n - m

Hint:

Remember the formula simply as "Pi groups = Variables - Dimensions" or \(k = n - m\). The goal of dimensional analysis is to *reduce* the number of variables you have to work with, so the answer must be a subtraction, not an addition or multiplication.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The question asks for the statement of Buckingham's \(\pi\)-theorem, which is a fundamental theorem in dimensional analysis. The theorem provides a method for reducing the number of variables in a physical problem by forming dimensionless groups.
Step 2: Statement of Buckingham's \(\pi\)-Theorem:
Buckingham's \(\pi\)-theorem states that if a physical phenomenon is described by a dimensionally homogeneous equation involving \(n\) physical variables, and if these variables can be expressed in terms of \(m\) fundamental (or independent) dimensions (such as mass [M], length [L], time [T], temperature [\(\Theta\)]), then the relationship can be rewritten in terms of \(k\) independent dimensionless groups (called \(\pi\)-groups), where: \[ k = n - m \] The number of independent dimensions, \(m\), is also referred to as the rank of the dimensional matrix.
Step 3: Detailed Analysis of Options:
- (A) m - n: Incorrect. This is the negative of the correct expression.
- (B) mn: Incorrect. This is multiplication, not subtraction.
- (C) m + n: Incorrect. This is addition, not subtraction.
- (D) n - m: Correct. This is the precise statement of the theorem. The number of dimensionless \(\pi\)-groups is the total number of variables minus the number of fundamental dimensions.
Step 4: Why This is Correct:
Option (D) is the correct mathematical statement of Buckingham's \(\pi\)-theorem. It defines how to determine the number of dimensionless parameters that govern a physical problem.