Three immiscible

PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Fluids · Fluids

Problem

Three immiscible, incompressible liquids are stacked inside a vertical tube of uniform internal cross-sectional area 5.0×10−4 m25.0\times10^{-4}\,\text{m}^2. From top to bottom the liquids are silicone oil (density 9.5×102 kg m−39.5\times10^{2}\,\text{kg}\,\text{m}^{-3}, thickness 0.040 m0.040\,\text{m}), water (density 1.00×103 kg m−31.00\times10^{3}\,\text{kg}\,\text{m}^{-3}, thickness 0.060 m0.060\,\text{m}), and mercury (density 1.36×104 kg m−31.36\times10^{4}\,\text{kg}\,\text{m}^{-3}, thickness 0.020 m0.020\,\text{m}). The tube is sealed at the top by a frictionless, massless piston that supports a metal disk of mass 1.5 kg1.5\,\text{kg}. Atmospheric pressure outside the tube is 1.01×105 Pa1.01\times10^{5}\,\text{Pa}. The tube’s bottom is rigid and closed. (a) Determine the pressure exerted on the upper surface of the silicone oil by the metal disk. Account for all forces acting on the piston–disk system. (b) Calculate the gauge pressure at (i) the oil–water interface, (ii) the water–mercury interface, and (iii) the bottom of the mercury layer. (c) Compute the absolute pressure at the bottom of the mercury layer. (d) Draw a carefully labeled pressure–depth graph for the entire fluid stack beginning at the top of the oil and ending at the bottom of the mercury. Indicate the numerical value of the slope in each region and explain why the slope changes at each interface.

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