The water supply of a building is fed through a main pipe 6.00 cm in diameter. A 2.00-cm -diamet

step 1 Question number 68, the height of the facet from the main pipe is given is 2 meter and the main

The water supply of a building is fed through a main pipe...

The water supply of a building is fed through a main pipe 6.00 $\mathrm{cm}$ in diameter. A $2.00-\mathrm{cm}$ -diameter faucet tap, located 2.00 $\mathrm{m}$ above the main pipe, is observed to fill a $25.0-\mathrm{L}$ container in 30.0 $\mathrm{s}$ . (a) What is the speed at which the water leaves the faucet? (b) What is the gauge pres- sure in the 6 -cm main pipe? (Assume the faucet is the only "leak" in the building.)

The water supply of a building is fed through a main pipe 6.00 $\mathrm{cm}$ in diameter. A $2.00-\mathrm{cm}$ -diameter faucet tap, located 2.00 $\mathrm{m}$ above the main pipe, is observed to fill a
$25.0-\mathrm{L}$ container in 30.0 $\mathrm{s}$ . (a) What is the speed at which the water leaves the faucet? (b) What is the gauge pres- sure in the 6 -cm main pipe? (Assume the faucet is the only "leak" in the building.)

Key Concepts

Continuity Equation

The continuity equation is a statement of the conservation of mass in fluid flow. It indicates that for an incompressible fluid, the product of the cross?sectional area and the fluid velocity remains constant along a streamline. This principle is used to relate the flow speeds in pipes or openings of different sizes by ensuring that the mass (or volume, for incompressible fluids) entering a system equals the mass exiting it.

Bernoulli’s Equation

Bernoulli’s equation is a fundamental principle in fluid dynamics that relates the pressure, velocity, and elevation in a flowing fluid. It asserts that the sum of the kinetic energy per unit volume, the potential energy per unit volume, and the pressure remains constant along a streamline under steady, incompressible, and non-viscous flow conditions. This relationship is crucial for determining pressure differences in fluid systems where the flow speed and heights change.

Volumetric Flow Rate

Volumetric flow rate quantifies the volume of fluid that passes through a given cross-sectional area per unit time. It is defined as the product of the cross-sectional area of the pipe or nozzle and the fluid’s velocity. This concept is used to calculate how much fluid is delivered or removed from a system, and it is vital in relating the dimensions of a flow passage to the speed of the fluid.

Gauge Pressure

Gauge pressure represents the pressure in a system relative to atmospheric pressure. It is the additional pressure over and above the ambient atmospheric pressure that is used to drive fluid through pipes and devices. In fluid dynamics problems, determining the gauge pressure is often necessary to understand how pressure differences contribute to fluid acceleration, changes in kinetic energy, and overcoming losses such as elevation changes.

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