Figure P 14.39 shows a stream of water in steady flow from a...

Figure $\mathrm{P} 14.39$ shows a stream of water in steady flow from a kitchen faucet. At the faucet, the diameter of the stream is $0.960 \mathrm{cm} .$ The stream fills a $125-\mathrm{cm}^{3}$ container in 16.3 $\mathrm{s}$ . Find the diameter of the stream 13.0 $\mathrm{cm}$ below the opening of the faucet.

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Key Concepts

Continuity Equation for Incompressible Fluids

The continuity equation is a fundamental principle in fluid dynamics which states that for an incompressible fluid, the mass flow rate remains constant along a streamline. This means that the product of the cross-sectional area and the velocity of the fluid must be constant at every point along the flow, allowing one to determine how the speed and cross-sectional area change relative to each other.

Volumetric Flow Rate

Volumetric flow rate is the volume of fluid passing through a given cross-section per unit time. It can be calculated by multiplying the fluid's cross-sectional area by its velocity. In problems involving steady flow, the volumetric flow rate is constant, which is essential when relating measurements taken at different locations along the stream.

Kinematics of Free-Fall under Gravity

When a fluid jet is in free-fall, its velocity increases as it descends under the influence of gravity. This acceleration due to gravity means that, although the volumetric flow rate remains the same, the increasing speed of the fluid causes the stream's cross-sectional area (and thus its diameter) to decrease as it falls.

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