Dimensional analysis fundamentals: which statements correctly describe dimensional homogeneity and its implications?

Introduction / Context:
Dimensional analysis is vital for verifying formulas, planning experiments, and reducing variables using Buckingham Pi theorem. Ensuring dimensional homogeneity prevents unit-based mistakes and supports physical plausibility of equations.

Given Data / Assumptions:

• Fundamental dimensions: typically Mass (M), Length (L), Time (T), Temperature (Θ), etc.
• Engineering equations should be dimensionally homogeneous.
• Units are specific choices within a dimensional framework (e.g., SI or USCS).

Concept / Approach:

A dimensionally homogeneous equation has identical dimensional exponents of M, L, T, etc., on both sides of the equality. This property ensures validity regardless of unit system and is a necessary (though not sufficient) condition for correctness of a physical relation.

Step-by-Step Solution:

Verification / Alternative check:

Convert an equation from SI to USCS; if numeric coefficients adjust consistently and the relationship holds, dimensional homogeneity is demonstrated.

Why Other Options Are Wrong:

Options (a), (b), and (c) are each correct; choosing any single one would be incomplete. Therefore, “All the above” is the correct comprehensive choice.

Common Pitfalls:

Adding unlike dimensions (e.g., adding a length to a velocity); mistaking units (e.g., N vs kgf); ignoring derived dimensions (e.g., pressure as ML^-1T^-2).

Final Answer:

All the above.