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GATE 2016 ME (ME2) – Question 51

Fluid Mechanics · Fluid properties, fluid statics, forces on submerged bodies, stability of floating bodies · 2 marks · Multiple choice

Consider a frictionless, massless and leak-proof plug blocking a rectangular hole of dimensions 2𝑅× 𝐿 at the bottom of an open tank as shown in the figure. The head of the plug has the shape of a semi-cylinder of radius R. The tank is filled with a liquid of density ρ up to the tip of the plug. The gravitational acceleration is g. Neglect the effect of the atmospheric pressure. The force F required to hold the plug in its position is

Diagram for GATE 2016 ME (ME2) question 51
  1. $2\rho R^2gL(1-\pi/4)$
  2. $2\rho R^2gL(1+\pi/4)$
  3. $\pi R^2\rho gL$
  4. $\pi\rho R^2gL/2$

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Show answer and explanation

Correct answer: (A) $2\rho R^2gL(1-\pi/4)$

Explanation

Horizontal pressure components cancel by symmetry. The downward vertical resultant equals the weight of the imaginary fluid between the curved plug and the free-surface plane. Its cross-sectional area is2R²−πR²/2. Multiply byρgL to obtain **F=2ρgR²L(1−π/4), A**, directed upward to balance the fluid force.