An incompressible fluid flows steadily through a horizontal cylindrical pipe. | PGMN Solutions
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An incompressible fluid flows steadily through a horizontal cylindrical pipe. The pipe has radius 3R at a point A and 1.5 R at a point B - further along the flow direction at the same level. If the velocity at point A is v, then that at B is (outside pressure is 1 atm )
(A) 0.5 v
(B) v
(C) 2 v
(D) 4 v
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Step-by-Step Solution to Solve This Question 🚀
📌 Chapter: Mechanical Properties of Fluids
📌 Topic: Equation of Continuity
🔹 Step 1: The equation of continuity is a principle that applies to the steady flow of incompressible fluids. It states that the mass flow rate must remain constant throughout the pipe.
🔹 Step 2: Since the fluid is incompressible and flows steadily, the product of the cross-sectional area and the velocity of the fluid must be the same at all points in the pipe.
🔹 Step 3: Point A has a larger radius than point B, which means the cross-sectional area at A is larger than at B. Therefore, to maintain the same flow rate, the velocity at B must be greater than at A.
🔹 Step 4: Specifically, the radius at point A is three times R, and at point B it is one and a half times R. The area is proportional to the square of the radius. So, the area at B is one-fourth that at A. Hence, to keep the flow rate constant, the velocity at B must be four times that at A.
✅ Final Answer: (D) 4 v 🚀
🔥 Pro Tip: When dealing with fluids in pipes of varying diameters, always apply the equation of continuity—smaller cross-section means higher velocity if the fluid is incompressible.
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