Physics Fluid Dynamics PMDC Conceptual Practice
PMDC Verified Question 35 of 37
The efflux speed \( v \) of a liquid leaving a tube depends purely on the change in pressure \( \Delta P \) and the liquid's density \( \rho \). The empirical relationship is given by \( v = k (\Delta P)^n (\rho)^m \), where \( k \) is a dimensionless constant. By analyzing dimensions, what is the exact value of \( n \)?
A
\( -1/2 \)
B
\( 1/2 \)
C
\( 1 \)
D
\( 2 \)
Tap any option to test your recall and reveal the step-by-step Propolis autopsy.

Propolis Cognitive Error Autopsy

Official Correct Choice:
Option B: \( 1/2 \)
Concept:

Dimensional analysis ensures that both sides of any valid physical equation possess the exact same fundamental units (Mass, Length, Time).

Formula:

$$ [v] = [\Delta P]^n [\rho]^m $$

Solution:

  • Speed: \( [v] = L T^{-1} \).
  • Pressure: \( [\Delta P] = M L^{-1} T^{-2} \).
  • Density: \( [\rho] = M L^{-3} \).
  • Set up the equation: \( L T^{-1} = (M L^{-1} T^{-2})^n (M L^{-3})^m \).
  • Distribute exponents: \( L^1 T^{-1} = M^{n+m} L^{-n-3m} T^{-2n} \).
  • Match the powers of T: \( -1 = -2n \implies n = 1/2 \).
  • (Optional verification) Match M: \( 0 = n + m \implies m = -1/2 \).
  • This confirms the formula relates to \( v \propto \sqrt{\Delta P / \rho} \).


Why other options are incorrect:

Any value other than 1/2 mathematically fails to balance the exponent of Time (T) on both sides of the dimensional equation.

Quality & Fidelity Assurance: Every question on BeambePrep is rigorously curated against the official PMDC syllabus with zero filler, zero out-of-syllabus content, and zero typos. When an authentic past paper originally contained a historical mistake or ambiguity from the examining board (such as UHS or NUMS), BeambePrep faithfully reflects the original paper while detailing the nuance and scientific consensus in the autopsy above.

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