Question

If force is proportional to square of velocity, then the dimensions of proportionality constant is

• A. [ML\(^{-1}\)T]
• B. [ML\(^{-1}\)T\(^0\)]
• C. [MLT]
• D. [M\(^0\)LT\(^{-1}\)]

Hint:

This type of force occurs in turbulent flow drag.

Answer 1

The Correct Option is B

To determine the dimensions of the proportionality constant when the force is proportional to the square of velocity, follow these steps:

1. Understanding the Relationship: Assume that the force \(F\) is proportional to the square of velocity \(v\), we can write this as: \(F = k \cdot v^2\) where \(k\) is the proportionality constant.
2. Dimensions of Force: The dimensional formula for force is: \([F] = [MLT^{-2}]\)
3. Dimensions of Velocity: Velocity has the dimensions of: \([v] = [LT^{-1}]\)
4. Dimensions of \(v^2\): Therefore, \(v^2\) will have the dimensions: \([v^2] = [L^2T^{-2}]\)
5. Determining Dimensions of the Proportionality Constant: From the equation \(F = k \cdot v^2\), equate the dimensions: \([MLT^{-2}] = [k][L^2T^{-2}]\) 
   Rearranging gives: \([k] = \frac{[MLT^{-2}]}{[L^2T^{-2}]} = [ML^{-1}T^{0}]\)
6. Conclusion: The dimensions of the proportionality constant \(k\) are \([ML^{-1}T^{0}]\).

Therefore, the correct answer is

[ML\(^{-1}\)T\(^0\)]

.