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Chapter 13: Problem 19

If you turn on the faucet in the bathroom sink, you will observe that the stream seems to narrow from the point at which it leaves the spigot to the point at which it hits the bottom of the sink. Why does this occur?

Short Answer

Expert verified
Answer: The stream of water narrows because of the conservation of mass and the effect of gravity. As the water accelerates due to gravity, its velocity increases, causing the cross-sectional area of the stream to decrease to maintain constant mass flow.

Step by step solution

01

1. Understand the basic setup

We need to analyze the flow of water from the faucet to the sink to understand why the stream narrows. The water flows through the spigot, becoming narrower as it falls to the sink bottom.
02

2. Introduce the conservation of mass

The conservation of mass states that the mass of an object remains constant over time, no matter how it may change in shape or volume. For our water stream, the mass of water entering the spigot must equal the mass of water exiting and hitting the sink bottom.
03

3. Apply the continuity equation to the water stream

The continuity equation is a fundamental principle in fluid dynamics that relates the velocity and cross-sectional area of the flow at different points along the stream. The equation states that the product of the cross-sectional area (A) and the velocity (v) at any point along the stream is constant: \\ A_1 v_1 = A_2 v_2 , \\ where A_1 and v_1 represent the area and velocity at the beginning of the stream (near the faucet), and A_2 and v_2 represent the area and velocity near the sink bottom.
04

4. Explain the effect of gravity on the water stream

Due to gravity, water accelerates as it falls from the faucet to the sink bottom. An increase in velocity (from v_1 to v_2) would cause the velocity near the sink bottom to be higher than near the faucet.
05

5. Derive the relationship between cross-sectional areas

Since the product of area and velocity is constant (from the continuity equation), if the velocity increases, the cross-sectional area must decrease to maintain the equation balance. Therefore, the cross-sectional area near the sink (A_2) would be smaller than the area near the faucet (A_1): A_2 < A_1.
06

6. Conclude the explanation

The stream of water from the faucet to the sink bottom narrows as it falls because, as gravity accelerates the water, the velocity increases. To maintain constant mass flow due to the conservation of mass, the cross-sectional area of the stream must decrease. Consequently, the water stream appears to narrow from the faucet to the point where it hits the sink.

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Most popular questions from this chapter

The atmosphere of Mars exerts a pressure of only 600\. Pa on the surface and has a density of only \(0.0200 \mathrm{~kg} / \mathrm{m}^{3}\). a) What is the thickness of the Martian atmosphere, assuming the boundary between atmosphere and outer space to be the point where atmospheric pressure drops to \(0.0100 \%\) of its yalue at surface level? b) What is the atmospheric pressure at the bottom of Mars's Hellas Planitia canyon, at a depth of \(7.00 \mathrm{~km} ?\) c) What is the atmospheric pressure at the top of Mars's Olympus Mons volcano, at a height of \(27.0 \mathrm{~km} ?\) d) Compare the relative change in air pressure, \(\Delta p / p\), between these two points on Mars and between the equivalent extremes on Earth-the Dead Sea shore, at \(400 . \mathrm{m}\) below sea level, and Mount Everest, at an altitude of \(8850 \mathrm{~m}\).

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Blood pressure is usually reported in millimeters of mercury (mmHg) or the height of a column of mercury producing the same pressure value. Typical values for an adult human are \(130 / 80\); the first value is the systolic pressure, during the contraction of the ventricles of the heart, and the second is the diastolic pressure, during the contraction of the auricles of the heart. The head of an adult male giraffe is \(6.0 \mathrm{~m}\) above the ground; the giraffe's heart is \(2.0 \mathrm{~m}\) above the ground. What is the minimum systolic pressure (in \(\mathrm{mmHg}\) ) required at the heart to drive blood to the head (neglect the additional pressure required to overcome the effects of viscosity)? The density of giraffe blood is \(1.00 \mathrm{~g} / \mathrm{cm}^{3},\) and that of mercury is \(13.6 \mathrm{~g} / \mathrm{cm}^{3}\)

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