GATE 2018 ME (ME2) – Question 50
A frictionless circular piston of area $10^{-2}\ \mathrm{m^2}$ and mass 100 kg sinks into a water-filled cylindrical container of equal area. Water density is 1000 kg/m³. A bottom hole of area $10^{-3}\ \mathrm{m^2}$ is open to atmosphere, 0.5 m below the piston as shown. There is no leakage past the piston and water is incompressible. Take g = 10 m/s². Find piston speed at the shown instant (m/s), to three decimal places.
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Correct answer: 1.400 to 1.500
Explanation
Use the quasi-steady inviscid discharge model implicit in the official interval. Neglecting piston acceleration, gauge pressure below it is $Mg/A=100000$ Pa. Continuity gives outlet speed $v_e=10v_p$. Bernoulli gives $\tfrac12(v_e^2-v_p^2)=100000/1000+10(0.5)=105$. Hence $99v_p^2=210$ and **vₚ = 1.456 m/s**. An unrestricted unsteady piston solution would need motion history.