Question

A drinking fountain shoots water about 12 cm up in the air from a nozzle of diameter 0.60 cm (Fig. 13–67). The pump at the base of the unit (1.1 m below the nozzle) pushes water into a 1.2-cm-diameter supply pipe that goes up to the nozzle. What gauge pressure does the pump have to provide? Ignore the viscosity; your answer will therefore be an underestimate.
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Accepted Answer

Step 1: Identify the key principles involved. This problem involves fluid dynamics and can be solved using Bernoulli's equation, which relates pressure, velocity, and height in a fluid system. Additionally, the continuity equation will help relate the velocities in the supply pipe and the nozzle. Step 2: Write Bernoulli's equation for the system. The equation is:

P1 + 12&#x3C1;$$v1^2$$ + &#x3C1;gh1 = P2 + 12&#x3C1;$$v2^2$$ + &#x3C1;gh2

Here, &#x3C1; is the density of water, g is the acceleration due to gravity, and h is the height. Subscripts 1 and 2 refer to the supply pipe and the nozzle, respectively. Step 3: Use the continuity equation to relate the velocities in the supply pipe and the nozzle. The continuity equation is:

A1v1 = A2v2

Here, A is the cross-sectional area of the pipe or nozzle. Calculate the areas using the formula for the area of a circle:

A = &#x3C0;4d2

where d is the diameter. Step 4: Solve for the velocity at the nozzle. The water is shot 12 cm into the air, so use the kinematic equation to find the velocity at the nozzle:

v2 = 2gh

where h is the height of 12 cm (converted to meters). Step 5: Substitute the known values into Bernoulli's equation. Set the pressure at the nozzle to atmospheric pressure (gauge pressure is zero at the nozzle). Solve for the gauge pressure at the pump, P1, considering the height difference of 1.1 m and the velocities calculated using the continuity equation and kinematics.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Bernoulli's Principle

Hydrostatic Pressure

Continuity Equation