Project ForgeA frictionless circular piston of area 10⁻² m² and mass 100 kg sinks into a cylindrical container of the same area filled with water of density 1000 kg/m³. The container has a hole of area 10⁻³ m² at the bottom, open to the atmosphere, located 0.5 m below the piston-water interface. Assuming no leakage from the edges of the piston and incompressible water (g=10 m/s²), the magnitude of the piston velocity (in m/s) at the instant shown is __________ (correct to three decimal places).
Explanation
By continuity, the water exit velocity through the hole relates to the piston (=water-surface) velocity by V2 = (A/a)V1 = 10V1. Applying Bernoulli between the piston-water interface (pressure = Patm+Mg/A, elevation h=0.5 m above the hole, velocity V1) and the hole exit (pressure = Patm, elevation 0, velocity V2): Mg/A + ρgh + ½ρV1² = ½ρV2². Mg/A = (100×10)/0.01 = 100,000 Pa; ρgh = 1000×10×0.5 = 5,000 Pa. So 105,000 = ½×1000×(V2²−V1²) = 500×(100V1²−V1²) = 49,500V1². V1² ≈ 2.1212, V1 ≈ 1.456 m/s.
Source: GATE ME 2018 Session 2 (Set 2) (2018)
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