Question

Find dimensions of \( \frac{A}{B} \) if \( \left( \frac{P + \frac{A^2}{B}}{\frac{1}{2} \rho v^2} \right) = \text{constant \) where \(P \) = pressure, \( \rho \) = density, \( v \) = speed.}

• A. \( ML^T^4 \)
• B. \( ML^{-1}T^{-4} \)
• C. \( ML^T^2 \)
• D. \( ML^{-1}T^{-2} \)

Hint:

When performing dimensional analysis, always break down the equation into basic units and apply consistent units for each physical quantity involved.

Answer 1

The Correct Option is B

Step 1: Understanding the problem.
We are given a dimensional equation with various physical quantities: pressure \(P\), density \( \rho \), and velocity \(v\). We need to determine the dimensions of the ratio \( \frac{A}{B} \).
Step 2: Dimensional analysis.
From the equation: \[ \left( \frac{A^2}{B} \right) = [P] = ML^{-1}T^{-2} \] Thus, \[ \frac{A^2}{B} = ML^{-1}T^{-2} \quad \Rightarrow \quad \frac{A}{B} = ML^{-1}T^{-4} \] Step 3: Conclusion.
The dimensions of \( \frac{A}{B} \) are \( ML^{-1}T^{-4} \), which corresponds to option (2).