Worksheet: Conservation of Energy in Moving Fluids

In this worksheet, we will practice using the Bernoulli equation to describe the relation of the pressure exerted by a fluid flow to its total energy.

Q1:

The Bernoulli equation applies to the streamlines of smoothly flowing fluids of constant density. The equation can be written in the form 𝑝+πœŒπ‘”β„Ž+12πœŒπ‘£=π‘˜οŠ¨, where π‘˜ is a constant. In the equation,𝑝 is term (i), πœŒπ‘”β„Ž is term (ii), and 12πœŒπ‘£οŠ¨ is term (iii).

Which term represents the dynamic pressure exerted by a fluid?

  • ATerm (iii)
  • BNone of the terms
  • CTerm (ii)
  • DTerm (i)

Which term represents the hydrostatic pressure exerted by a fluid?

  • ANone of the terms
  • BTerm (iii)
  • CTerm (ii)
  • DTerm (i)

Which term represents the static pressure exerted by a fluid?

  • ATerm (iii)
  • BNone of the terms
  • CTerm (i)
  • DTerm (ii)

Q2:

Water with a constant density of 1,000 kg/m3 flows smoothly along a streamline. The water enters the streamline moving at 3.33 m/s and exits the streamline moving at 2.72 m/s. The vertical displacement of the exit point of the streamline from the entry point of the streamline is 0.164 m.

What is the change in the static pressure exerted by the fluid between the entry and exit points of the streamline?

Does the static pressure exerted by the fluid increase or decrease along the streamline?

  • AIt increases.
  • BIt decreases.

Q3:

The Bernoulli equation applies to the streamlines of smoothly flowing fluids of constant density. The equation relates the density of the fluid, 𝜌; the static pressure exerted by the fluid, 𝑝; the velocity of the fluid, 𝑣; the acceleration due to gravity, 𝑔; and the vertical displacement of the fluid, β„Ž. The equation can be written in the form 𝑝+πœŒπ‘”β„Ž+12πœŒπ‘£=π‘˜οŠ¨, where π‘˜ is a constant.

The term 12πœŒπ‘£οŠ¨ in the equation represents the energy per unit of the fluid along the streamline.

  • Akinetic, volume
  • Bmechanical, length
  • Cmechanical, time
  • Dkinetic, mass
  • Ekinetic, length

The term πœŒπ‘”β„Ž in the equation represents the energy per unit of the fluid along the streamline.

  • Akinetic, length
  • Bgravitational potential, mass
  • Cgravitational potential, volume
  • Dkinetic, time
  • Emechanical, mass

Q4:

Water with a constant density of 1,000 kg/m3 flows smoothly along a streamline 𝑆, as shown in the diagram. The water passes through a pipe that narrows along its length, entering the pipe at 5.5 m/s. The center of the wider end of the pipe is 2.2 m below the center of the narrower end of the pipe. The static pressure exerted by the water where it leaves the pipe equals the static pressure exerted by the water where it enters the pipe. At what speed does the water exit the pipe?

Q5:

Water with a constant density of 1,000 kg/m3 flows smoothly along a streamline 𝑆, as shown in the diagram. The water passes through a pipe that widens along its length. The center of the wider end of the pipe is 1.4 m below the center of the narrower end of the pipe. Water enters and exits the pipe at the same speed. How much greater is the static pressure exerted by the water where the water leaves the pipe than the static pressure exerted by the water where the water enters the pipe? Answer to two significant figures.

Q6:

Water with a constant density of 1,000 kg/m3 flows smoothly along a streamline 𝑆, as shown in the diagram. The streamline 𝑆 passes at constant height along a wedge-shaped channel that has a cross-sectional area of 0.75 m2 where the water enters the channel and a cross-sectional area of 0.45 m2 where the water exits the channel. The water enters the channel moving at 1.5 m/s. How much greater is the static pressure exerted by the water where it enters the channel than where it leaves it?

Q7:

A fluid with constant density flows smoothly through a cuboid pipe with a cross-sectional area shown in the diagram as π‘¦π‘§οŠ§οŠ§. The fluid then flows through a second cuboid pipe with a cross-sectional area of π‘¦π‘§οŠ¨οŠ¨. In a time Δ𝑑, the fluid travels π‘₯ meters along the wider pipe and travels π‘₯ meters along the narrower pipe.

Which of the following is the ratio of the work that can be done perpendicularly to the flow direction by the fluid over a distance of π‘₯ meters along the wider pipe to the work that can be done perpendicularly to the flow direction by the fluid over a distance of π‘₯ meters along the narrower pipe?

  • Aπ‘¦π‘§π‘¦π‘§οŠ¨οŠ¨οŠ§οŠ§
  • Bπ‘¦π‘§π‘¦π‘§οŠ§οŠ§οŠ¨οŠ¨
  • Cπ‘₯π‘₯
  • D1
  • Eπ‘₯π‘₯

Which of the following is the ratio of the work that can be done perpendicularly to the flow direction by the fluid over a length of π‘₯ meters of the wider pipe to the work that can be done perpendicularly to the flow direction by the fluid over a length of π‘₯ meters of the narrower pipe?

  • A1
  • Bπ‘₯π‘₯
  • Cπ‘₯π‘₯
  • Dπ‘¦π‘§π‘¦π‘§οŠ§οŠ§οŠ¨οŠ¨
  • Eπ‘¦π‘§π‘¦π‘§οŠ¨οŠ¨οŠ§οŠ§

Which of the following is the ratio of the static pressure exerted by the fluid per unit length of the wider pipe to the pressure exerted by the fluid per unit length of the narrower pipe? Answer in terms of the cross-sectional areas of the pipes, where 𝐴 is the wider pipe’s cross-sectional area and 𝐴 is the narrower pipe’s cross-sectional area.

  • A𝐴𝐴
  • B𝐴𝐴
  • C(𝐴+𝐴)
  • D(𝐴+𝐴)2
  • E(π΄βˆ’π΄)2

Q8:

Water with a constant density of 1,000 kg/m3 flows smoothly along a streamline. The static pressure exerted by the water decreases by 3,520 Pa along the streamline. The water enters the streamline moving at 2.33 m/s and exits the streamline moving at 2.45 m/s. What is the vertical displacement of the exit point of the streamline from the entry point of the streamline?

Q9:

Water with a constant density of 1,000 kg/m3 flows smoothly along a streamline. The water enters the streamline moving at 1.33 m/s and exits the streamline moving at 1.12 m/s. The vertical displacement of the exit point of the streamline from the entry point of the streamline is 0.131 m. The change in static pressure, Δ𝑝, exerted by the fluid between the entry and exit points of the streamline has different values on Earth and on the Moon. Find the difference in Δ𝑝 on Earth and on the Moon. Use 1.62 m/s2 for the gravitational acceleration at the Moon’s surface.

Q10:

A liquid with a constant density flows smoothly along a streamline. The liquid enters the streamline moving at 2.22 m/s and exits it moving at 3.66 m/s. The vertical displacement of the streamline’s exit point from its entry point is βˆ’0.185 m. The static pressure exerted by the liquid at the streamline’s exit point is 6.58 kPa lower than the static pressure exerted by it at the streamline’s entry point. What is the density of the liquid?

Q11:

Water with a constant density of 1,000 kg/m3 flows smoothly along a streamline S, as shown in the diagram. The streamline S passes along a wedge-shaped channel that has a cross-sectional area of 2.24 m2 where the water enters the channel and a cross-sectional area of 1.18 m2 where the water exits the channel, 1.26 m vertically above the streamline’s entry point. The water enters the channel moving at 4.25 m/s. How much greater is the static pressure exerted by the water where the water enters the channel than the static pressure exerted by the water where the water leaves the channel?

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