JEE Advanced 2021 Paper 1 Q16 Physics Fluids & Surface Tension Pressure in Fluids Hard

JEE Advanced 2021 Paper 1 · Q16 · Pressure in Fluids

A cylindrical tube, with its base as shown in the figure, is filled with water. It is moving down with a constant acceleration a along a fixed inclined plane with angle θ = 45°. P₁ and P₂ are pressures at points 1 and 2, respectively, located at the base of the tube. Let β = (P₁ − P₂)/(ρgd), where ρ is density of water, d is the inner diameter of the tube and g is the acceleration due to gravity. Which of the following statement(s) is(are) correct?

[Figure: A right-triangular incline at 45°. A vertical cylindrical tube of inner diameter d, filled with water, sits on the incline. Its base lies along the inclined surface; point 1 is at the downhill (lower) edge of the base and point 2 at the uphill (upper) edge of the base. Acceleration vector a points down the incline; g points vertically down.]

  1. A. β = 0 when a = g/√2
  2. B. β > 0 when a = g/√2
  3. C. β = (√2 − 1)/√2 when a = g/2
  4. D. β = 1/√2 when a = g/2
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Correct answer:A, C

Solution

In the tube's non-inertial frame, effective gravity along the base of the tube gives (P₁ − P₂)·dA = ρ(d·dA)·√2·(g sin 45° − a) ⇒ P₁ − P₂ = ρd(g − a√2). Thus β = (P₁ − P₂)/(ρgd) = 1 − (a√2/g). For a = g/√2: β = 1 − 1 = 0 ⇒ (A) correct, (B) wrong. For a = g/2: β = 1 − (√2/2) = (√2 − 1)/√2 ⇒ (C) correct, (D) wrong (1/√2 = √2/2 ≠ (√2 − 1)/√2).

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