Mass density (ρ): equivalent definitions used in hydraulics Select the correct comprehensive statement(s) for the mass density of a liquid ρ.

Introduction / Context:
Mass density is a primary thermophysical property appearing in continuity, momentum, and energy equations. Engineers routinely move between mass-based and weight-based formulations, so recognizing equivalent forms is essential.

Given Data / Assumptions:

• Standard gravity symbol g denotes acceleration due to gravity.
• Specific weight γ = W / V = ρ * g.
• m is mass; W is weight; V is volume.

Concept / Approach:
By definition, density ρ is mass per unit volume. Since weight W = m * g, the ratio W/(gV) reduces to (mg)/(gV) = m/V = ρ. Similarly, γ = ρ * g, so ρ = γ / g. These interchangeable expressions are used depending on whether measurements are mass- or weight-based.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional analysis: [ρ] = M L^-3; W/(g V) has units (M L T^-2)/(L T^-2 * L^3) = M L^-3, and γ/g likewise yields M L^-3.

Why Other Options Are Wrong:
Each of (a), (b), (c) by itself is correct. Therefore “all the above” is the complete correct option.

Common Pitfalls:
Confusing ρ with specific weight γ; forgetting to divide by g when converting weight-based values to mass density.

Final Answer:
all the above