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A large reservoir open to the atmosphere holds water to height H = 5 m above the centreline of a horizontal exit pipe at its bottom. The pipe's first section has radius r1 = 2 cm with velocity v1 and carries a manometer where water rises to height h above the pipe centreline. The water exits the second section of radius r2 = 1 cm with velocity v2. Assume incompressible, frictionless, steady flow and that the reservoir level is essentially constant. Find (a) the exit speed v2, (b) the speed v1 in the first section, (c) the manometer height h, and (d) the volume flow rate. (Take g = 10 m/s².)

1. (a) 10 m/s, (b) 2.5 m/s, (c) 4.69 m, (d) 3.14 L/s
2. (a) 10 m/s, (b) 2.5 m/s, (c) 5 m, (d) 1.57 L/s
3. (a) 5 m/s, (b) 1.25 m/s, (c) 4.69 m, (d) 3.14 L/s
4. (a) 10 m/s, (b) 5 m/s, (c) 4 m, (d) 6.28 L/s

Correct answer: (a) 10 m/s, (b) 2.5 m/s, (c) 4.69 m, (d) 3.14 L/s

Solution

Torricelli gives the exit speed from the head H. Continuity relates v1 to v2 by area ratio. The manometer reads the gauge pressure head in section 1, which by Bernoulli equals H minus the velocity head there. Flow rate is area times velocity (same throughout).

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