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

To determine the dimensions of \( \frac{A}{B} \), we need to understand the given expression and analyze the dimensional formulae involved. The expression provided is:

\(\left( \frac{P + \frac{A^2}{B}}{\frac{1}{2} \rho v^2} \right) = \text{constant}\)

This implies that the expression \(\frac{P + \frac{A^2}{B}}{\frac{1}{2} \rho v^2}\) is dimensionless, as a constant has no dimensions.

1. The dimensions of pressure \( P \) are \( ML^{-1}T^{-2} \).
2. The dimensions of density \( \rho \) are \( ML^{-3} \).
3. The dimensions of speed \( v \) are \( LT^{-1} \).

First, let's find the dimensions of the denominator:

The term \( \frac{1}{2} \rho v^2 \) has the dimensions of:

\(\rho v^2 = \left(ML^{-3}\right) \left(L^2T^{-2}\right) = ML^{-1}T^{-2}\)

Since the overall expression is dimensionless, the dimensions of the numerator \( P + \frac{A^2}{B} \) must also be \( ML^{-1}T^{-2} \).

Thus, we equate the dimensions of the numerator to ensure they match:

• \(P\) already has dimensions \( ML^{-1}T^{-2} \).
• To ensure \( P + \frac{A^2}{B} \) has these dimensions, \(rac{A^2}{B}\) must also be \( ML^{-1}T^{-2} \\)

If \(\frac{A^2}{B}\) has dimensions \( ML^{-1}T^{-2} \), then:

\(\left[\frac{A^2}{B}\right] = \frac{\left[A^2\right]}{[B]} = ML^{-1}T^{-2}\)

Therefore, \( [A^2] = [B] \cdot ML^{-1}T^{-2} \).

Thus, the dimensions of \( \frac{A}{B} \) are:

\(\left[\frac{A}{B}\right] = \left[\frac{\sqrt{B} \cdot ML^{-1}T^{-2}}{B}\right] = ML^{-1}T^{-4}\)

Hence, the correct answer is:

\( ML^{-1}T^{-4} \)

This confirms that the correct option is:

\( ML^{-1}T^{-4} \)