Which of the following are dimensionally correct? (A) h=(2 T cosθ)/(ρr g) (B) v=√((p)/(ρ)) (C) (

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Which of the following are dimensionally correct? (A) h=(2 T...

Which of the following are dimensionally correct?
(A) $h=\frac{2 T \cos \theta}{\rho r g}$
(B) $v=\sqrt{\frac{p}{\rho}}$
(C) $\frac{d v}{d t}=\frac{\pi p r^{4} t}{\delta \eta l}$
(D) $T=\sqrt{\frac{m g l}{I}}$

Key Concepts

Equation Verification

Verifying equations involves checking that the dimensions on both sides of any physical equation match. This process helps to identify potential mistakes in formulation or derivation by ensuring that operations such as multiplication, division, or taking roots do not lead to incompatible units, thereby ensuring the equation could potentially represent a real physical relationship.

Units of Measurement

Understanding the base units and dimensions (such as mass, length, time) for various physical quantities is crucial. These units serve as the building blocks for more complex, derived units, and their correct application is essential when analyzing the structure and validity of equations in any physical context.

Dimensional Analysis

Dimensional analysis is a method used to verify the consistency of equations by ensuring that each term is expressed in the fundamental units of the physical quantities involved. This technique ensures that both sides of an equation have matching combinations of units, providing an initial check on the equation’s validity before further numerical or experimental validation.

Dimensional Consistency

Dimensional consistency requires that all terms in an equation, whether added, equated, or compared, share the same dimensional form. This concept is fundamental in physics and engineering because it ensures that the expression is physically meaningful and that no unit conversion errors have occurred during derivation.

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