Question

The number of dimensionless groups formed using Buckingham's \(\pi\)-theorem is:

• A. Equal to number of variables
• B. Equal to number of fundamental dimensions
• C. Variables minus fundamental dimensions
• D. Always 3

Hint:

Buckingham's \(\pi\)-theorem helps simplify complex physical problems by creating dimensionless groups, reducing the number of variables that need to be analyzed.

Answer 1

The Correct Option is C

Buckingham's \(\pi\)-theorem is a key concept in dimensional analysis, and it provides a method to reduce the number of variables in a physical problem by creating dimensionless groups. The number of dimensionless groups formed, denoted by \(\pi\)-groups, is determined by the number of variables in the problem and the number of fundamental dimensions involved.
The theorem states that if there are \(n\) variables and \(k\) fundamental dimensions, then the number of dimensionless groups is given by: \[ {Number of dimensionless groups} = n - k \] where:
- \(n\) is the total number of variables in the problem, and
- \(k\) is the number of fundamental dimensions involved in the analysis.
Thus, the number of dimensionless groups formed is equal to the number of variables minus the number of fundamental dimensions.
Therefore, the correct answer is: Variables minus fundamental dimensions.