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
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
Show 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.
Updated On: Aug 30, 2025
- m - n
- mn
- m + n
- n - m
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
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.
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.
Was this answer helpful?
0
0
Top GATE NM Fluid Mechanics and Marine Hydrodynamics Questions
- The sum of the static pressure and dynamic pressure at a point in a fluid flow is called the ________.
- Identify the range of Reynolds number (Re) for a creeping flow.
- The ratio of the magnitudes of vorticity to rate of rotation in a fluid flow is ________.
- In a laminar boundary layer, the ratio of boundary layer thickness (\( \delta \)) to the corresponding displacement thickness (\( \delta^* \)) lies between ________.
- Which one of the following wave energy spectra is formulated for two peaks?
View More Questions
Top GATE NM Fluid Mechanics Questions
- The sum of the static pressure and dynamic pressure at a point in a fluid flow is called the ________.
- Identify the range of Reynolds number (Re) for a creeping flow.
- The ratio of the magnitudes of vorticity to rate of rotation in a fluid flow is ________.
- In a laminar boundary layer, the ratio of boundary layer thickness (\( \delta \)) to the corresponding displacement thickness (\( \delta^* \)) lies between ________.
- For a freely floating body in water, which of the following degrees of freedom has/have inherent restoring force?
View More Questions
Top GATE NM Questions
- If \( \rightarrow \) denotes increasing order of intensity, then the meaning of the words [dry \( \rightarrow \) arid \( \rightarrow \) parched] is analogous to [diet \( \rightarrow \) fast \( \rightarrow \) \(\_\_\_\_\_\_\_\_\)]. Which one of the given options is appropriate to fill the blank?
- If two distinct non-zero real variables \( x \) and \( y \) are such that \( (x + y) \) is proportional to \( (x - y) \), then the value of \( \frac{x}{y} \) is:
- Consider the following sample of numbers: \[ 9, 18, 11, 14, 15, 17, 10, 69, 11, 13 \] The median of the sample is:
- The number of coins of ₹1, ₹5, and ₹10 denominations that a person has are in the ratio 5:3:13. Of the total amount, the percentage of money in ₹5 coins is:
- For positive non-zero real variables \( p \) and \( q \), if \[ \log \left( p^2 + q^2 \right) = \log p + \log q + 2 \log 3, \] then the value of \( \frac{p^4 + q^4}{p^2 q^2} \) is:
View More Questions