consider the set of tubes shown in the figure. each tube contains an unknown fluid that is less dense than the water it floats on top of. rank the four unknown fluids from least dense to most dense.

unknown fluids floating on water in manifold

to rank items as equivalent, overlap them.
view available hint(s)

least dense
most dense
the correct ranking cannot be determined.

Question

consider the set of tubes shown in the figure. each tube contains an unknown fluid that is less dense than the water it floats on top of. rank the four unknown fluids from least dense to most dense.

unknown fluids floating on water in manifold

to rank items as equivalent, overlap them.
view available hint(s)

least dense
most dense
the correct ranking cannot be determined.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Hydrostatic Pressure" ], "new_concepts": [ "U-Tube Manometer", "Fluid Density Ranking" ], "current_concepts": [ "Hydrostatic Pressure", "U-Tube Manometer", "Fluid Density Ranking" ] } ## Analyze hydrostatic pressure at the water interfaces $$ \begin{aligned} &P_{\text{interface}} = P_0 + ho_{\text{fluid}} g h_{\text{fluid}} = P_0 + ho_{\text{water}} g h_{\text{water\_displaced}} \ & ho_{\text{fluid}} h_{\text{fluid}} = ho_{\text{water}} h_{\text{water\_displaced}} \ & ho_{\text{fluid}} = ho_{\text{water}} \frac{h_{\text{water\_displaced}}}{h_{\text{fluid}}} \end{aligned} $$ ## Compare the height ratios for each tube $$ \begin{aligned} &\text{For A: } h_{\text{fluid}} \text{ is large, } h_{\text{water\_displaced}} \text{ is moderate} \implies \frac{h_{\text{water\_displaced}}}{h_{\text{fluid}}} \approx 0.5 \ &\text{For B: } h_{\text{fluid}} \text{ is moderate, } h_{\text{water\_displaced}} \text{ is moderate} \implies \frac{h_{\text{water\_displaced}}}{h_{\text{fluid}}} \approx 0.7 \ &\text{For C: } h_{\text{fluid}} \text{ is large, } h_{\text{water\_displaced}} \text{ is very small} \implies \frac{h_{\text{water\_displaced}}}{h_{\text{fluid}}} \approx 0.1 \ &\text{For D: } h_{\text{fluid}} \text{ is small, } h_{\text{water\_displaced}} \text{ is moderate} \implies \frac{h_{\text{water\_displaced}}}{h_{\text{fluid}}} \approx 0.9 \end{aligned} $$ ## Rank the densities from least dense to most dense $$ ho_C < ho_A < ho_B < ho_D $$ Least dense to most dense: C, A, B, D { "subject": "Natural Science", "question_type": "Free Response", "knowledge_point": [ "Natural Science", "Physics", "Hydrostatic Pressure" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze hydrostatic pressure at the water interfaces

Compare the height ratios for each tube

Rank the densities from least dense to most dense

Answer: