Mass density (ρ): equivalent definitions used in hydraulics Select the correct comprehensive statement(s) for the mass density of a liquid ρ.
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Aρ = m / V (mass per unit volume)
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Bρ = W / (g * V) (weight-based definition)
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Cρ = γ / g (specific weight divided by g)
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Dall the above
Answer
Correct Answer: all the above
Explanation
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:
Start with ρ = m/V.Use W = mg ⇒ ρ = (W/g)/V = W/(gV).Use γ = ρg ⇒ ρ = γ/g.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