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What we found:

Two identical open containers are filled with water to the same height \( H \). Container \( A \) has a small hole at a depth \( h \) below the surface, and Container \( B \) has a small hole at a depth \( 2h \) below the surface. If \( v_A \) and \( v_B \) are the exit speeds of the water, what is the ratio \( v_B / v_A \)?

A U-tube is filled with water and is being accelerated horizontally to the right with a constant acceleration \( a \). Which of the following describes the behavior of the water levels in the two vertical arms of the U-tube?

An open container is filled with an ideal fluid. If a small hole is poked in the side of the container at a depth \( h \) below the surface, the speed of the fluid exiting the hole is \( v \). If the container is then pressurized so that the air above the fluid is at \( 2 P_{\text{atm}} \) while the outside remains at \( P_{\text{atm}} \), how does the new exit speed \( v’ \) compare to \( v \)?

An airtight container is completely filled with an ideal, incompressible fluid. If the external pressure on the container is increased by \( \Delta P \), how does the pressure change at a point located at the very bottom of the container?

An ideal fluid is defined as being incompressible and having no viscosity. What are the physical implications of these two properties during fluid flow?

A U-tube with a uniform cross-section is open to the atmosphere and filled with water. A piston is placed in the left arm and pushed down with a constant force \( F \). If the right arm remains open, what is the difference in height \( \Delta h \) between the water levels in the two arms?

A solid block of wood floats in an open tank of water on Earth, with exactly \( 60\% \) of its volume submerged. If this entire setup were transported to the Moon, where the acceleration due to gravity is \( \dfrac{g}{6} \), what percentage of the block’s volume would be submerged?

A U-tube open to the atmosphere at both ends is partially filled with water of density \( \rho_w \). Oil of density \( \rho_o \) (where \( \rho_o < \rho_w \)) is poured into the left arm until it forms a column of height \( L \). The interface between the oil and water in the left arm is found to be a vertical distance \( d \) below the water surface in the right arm. Which of the following expressions correctly represents the density of the oil?

A student uses a mercury barometer to measure atmospheric pressure. If the student replaces the original tube with one that has \( 2 \) times the cross-sectional area, how will the height \( h \) of the mercury column change?

A technician applies a force of \( 150 \) \( \text{N} \) to a small syringe with a plunger area of \( 2.0 \times 10^{-4} \) \( \text{m}^{2} \). The syringe is connected to a larger cylinder with an area of \( 8.0 \times 10^{-3} \) \( \text{m}^{2} \). What is the gauge pressure transmitted throughout the fluid?

In a hydraulic system, Piston A has an area of \( 0.002 \) \( \text{m}^2 \) and Piston B has an area of \( 0.1 \) \( \text{m}^2 \). If a \( 40 \) \( \text{N} \) force is applied to Piston A, what is the maximum mass (using \( g = 10 \) \( \text{m/s}^2 \)) that can be supported by Piston B at the same vertical height?

Three containers have the same base area \( A \) and are filled with water to the same height \( h \), but they have different shapes: one is cylindrical, one widens toward the top, and one narrows toward the top. Which of the following is true regarding the gauge pressure \( P \) at the bottom and the total force \( F \) exerted by the fluid on the base?

A hydraulic jack is used to lift a heavy load. If the input piston is pushed down a distance \( d_1 \), and the output piston moves up a distance \( d_2 \), which of the following expressions must be true for an ideal, incompressible fluid?

A tube contains a volume of water and a volume of mercury as shown in the figure above. Both ends of the tube are open. Points A and B are at the same level and points C and D are at the same level. The distances between points A and B and the surface of the liquids are indicated: the vertical distance from the water surface to point A is \( 0.15 \) \( \text{m} \), and the vertical distance from the mercury surface to point B is \( 0.01 \) \( \text{m} \). Which of the following correctly relates the gauge pressures at the points shown? The density of the water is \( 1{,}000 \) \( \text{kg/m}^3 \) and the density of the mercury is \( 13{,}600 \) \( \text{kg/m}^3 \).

A vertical pipe has a constant cross-sectional area. As an ideal fluid flows upward through this pipe, how do the speed and pressure change?

Air flows over the top of a curved airplane wing faster than it flows underneath. Using Bernoulli’s principle, which statement best explains the resulting aerodynamic lift?

An ideal fluid flows through a horizontal pipe that constricts from a diameter \( D \) to a diameter \( \dfrac{D}{2} \). If the fluid speed in the wider section is \( v_{1} \) and the pressure is \( P_{1} \), what is the pressure \( P_{2} \) in the narrower section in terms of the fluid density \( \rho \)?

A horizontal water pipe branches into three smaller identical pipes. The main pipe has a cross-sectional area \( A \) and the water flows at a speed \( v \). If the water in each of the three branches flows at a speed of \( \dfrac{4}{3} v \), what is the cross-sectional area of each branch?

A sphere of mass \( M \) and volume \( V \) is suspended from a spring scale and completely submerged in a fluid of density \( \rho_f \). What is the reading on the spring scale?

A person uses a straw to drink water from a cup. Which of the following best describes the physics of how the water moves up the straw?

A student wants to determine the density of an unknown liquid. They have a block of known density \( \rho_b \) and volume \( V \). When the block is placed in the liquid, it sinks and rests on the bottom. A spring scale attached to the block reads a tension \( T \). Which formula should the student use to find the liquid density \( \rho_L \)?

Water is squirted horizontally from a large syringe. If the force applied to the plunger is doubled while keeping the plunger area \( A \) and the nozzle area \( a \) constant (where \( A \gg a \)), by what factor does the exit velocity of the water increase?

An incompressible fluid flows through a pipe. At Point \( 1 \), the gauge pressure is \( P_g \). At Point \( 2 \), the pipe is at the same height but the fluid speed is higher. How does the absolute pressure at Point \( 2 \) compare to Point \( 1 \)?

An ideal fluid flows through a pipe that both narrows in cross-sectional area and increases in height. Which of the following statements must be true regarding the pressure change as the fluid moves into the narrower, higher section?

A horizontal pipe carrying an ideal fluid narrows from a cross-sectional area of \( A \) to \( \tfrac{1}{2}A \). How do the speed \( v \) and the pressure \( P \) of the fluid change as it enters the narrow section?

Water flows through a horizontal pipe that branches into two identical pipes, each with cross-sectional area equal to \( \tfrac{1}{2} \) the cross-sectional area of the original pipe. How does the speed of the water in one of the smaller pipes compare to the speed in the original pipe?

Water exits a small hole at the bottom of a large, open cylindrical tank. If the height of the water level above the hole is \( H \), the exit speed is \( v \). If the tank is replaced by one with double the cross-sectional area but the same water height \( H \), what is the new exit speed?

A tube contains a volume of water and a volume of mercury as shown in the figure above. Both ends of the tube are open. Points \( A \) and \( B \) are at the same level and points \( C \) and \( D \) are at the same level. The distance between points \( A \) and \( B \) and the surface of the liquids are shown. Which of the following correctly relates the gauge pressures at the points shown? The density of the water is \( 1{,}000 \) \( \text{kg/m}^3 \) and the density of the mercury is \( 13{,}600 \) \( \text{kg/m}^3 \).

How does liquid rise in a straw when you sip, even though you’re not applying any force to the liquid?

Why do small objects get sucked into a vacuum’s intake, even though you can feel air rushing out the back?

\( 2 \) identical pipes carry the same flow—\( 1 \) level, \( 1 \) sloped. Which outlet has higher pressure, and why?

Standing near a passing train, are you pushed away or pulled toward it?

Explain what would happen if you blew between \( 2 \) hanging sheets of paper.

Why does a water stream from a faucet narrow as it falls?

Why does a shower curtain blow “in” while the water is running?

Explain how a roof might “blow off” during heavy winds.

A student designs an experiment to determine the density of an unknown fluid. The student pours the fluid into a graduated cylinder and attaches an object to a force probe. The object has a density greater than the density of the fluid. The student partially submerges the object into the fluid and records both the volume of fluid displaced in the graduated cylinder and the reading on the force probe. The student then submerges the object further and, at each trial, records the new values of displaced volume and force probe reading until the object is fully submerged. The student constructs a graph of force probe reading (vertical axis) as a function of volume of fluid displaced (horizontal axis). Which of the following statements correctly describes how a feature of this graph is related to the density of the fluid?

Water of density \( \rho \) flows through the section of circular pipe shown in the figure. At Point A, where the diameter of the pipe is \( D \), the water has a pressure \( P_0 \) and velocity \( v_0 \). Point B is located a vertical distance \( H \) above Point A in a section of the pipe that has diameter \( 2D \).Which of the following expressions is equal to the pressure of the water at Point B?

A student places a wooden block of mass \( m \) in a container of water. The block floats with half of its volume above the surface of the water. The student then begins to stack small objects on top of the block until the wooden block is completely submerged but none of the objects stacked on top are submerged. What is the buoyant force acting on the block now?

Two points, \( A \) and \( B \), are in a pipe carrying a flowing ideal fluid. Point \( B \) is \( 2.0 \) \( \text{m} \) higher than point \( A \), and the fluid speed at \( B \) is twice the speed at \( A \). If the pressure at \( A \) is \( P_A \), which of the following expressions correctly represents the pressure at \( B \) \( (P_B) \)?

A Venturi tube has a pressure difference of \( 15\,000 \) \( \text{Pa} \). The entrance radius is \( 3 \) \( \text{cm} \), while the exit radius is \( 1 \) \( \text{cm} \). What are the entrance velocity, exit velocity, and flow rate if the fluid is gasoline \( (\rho = 700 \) \( \text{kg/m}^3 ) \)?

A cube of side length \( s \) rests on the bottom surface of a container of fluid as shown in the figure. The fluid is at a height \( y \) above the bottom of the tank. The fluid has density \( \rho \) and the atmospheric pressure is \( P_{\text{atm}} \).Which of the following expressions is equal to the absolute pressure exerted by the fluid on the top surface of the cube?

A cylindrical tank of water (height \( H \)) is punctured at a height \( h \) above the bottom. How far from the base of the tank will the water stream land (in terms of \( h \) and \( H \))? What must the value of \( h \) be such that the distance at which the stream lands will be equal to \( H \)?

Two paper cups are suspended by strings and hung near each other. They are separated by about \( 10 \) \( \text{cm} \). When you blow air between them, the cups are attracted to one another. Explain why this occurs.

A Venturi meter is a device used for measuring the speed of a fluid within a pipe. The drawing shows a gas flowing at a speed \( v_2 \) through a horizontal section of pipe with a cross-sectional area \( A_2 = 542 \) \( \text{cm}^2 \). The gas has a density of \( 1.35 \) \( \text{kg/m}^3 \). The Venturi meter has a cross-sectional area of \( A_1 = 215 \) \( \text{cm}^2 \) and has been substituted for a section of the larger pipe. The pressure difference between the two sections \( P_2 – P_1 = 145 \) \( \text{Pa} \).

The drawing below shows a spherical reservoir that contains \( 455,000 \) \( \text{kg} \) of water when full. The reservoir is vented to the atmosphere at the top. Assuming the reservoir is full and the diameter of the reservoir is much larger than any of the pipes on the ground.

The diagram below shows a hydraulic chamber with a spring \( (k_s = 1250 \, \text{N/m}) \) attached to the input piston and a rock of mass \( 55.2 \, \text{kg} \) resting on the output plunger. The input piston and output plunger are at about the same height, and each has negligible mass. The chamber is filled with water.

Marc’s favorite ride at Busch Gardens is the Flying Umbrella, which is lifted by a hydraulic jack. The operator activates the ride by applying a force of \( 72 \) \( \text{N} \) to a \( 30 \)\( \text{cm}^2 \) cylindrical piston, which holds the \( 20,000 \)\( \text{N} \) ride off the ground. What is the area of the piston that holds the ride?

Johnny the auto mechanic is raising a \( 1200 \) \( \text{kg} \) car on her hydraulic lift so that she can work underneath. If the area of the input piston is \( 12 \) \( \text{cm}^2 \), while the output piston has an area of \( 700 \) \( \text{cm}^2 \), what force must be exerted on the input piston to lift the car?

When the button of a trash compactor is pushed, a force of \( 350 \) \( \text{N} \) pushes down on a \( 1.3 \) \( \text{cm}^2 \) input piston, creating a force of \( 22,076 \) \( \text{N} \) to crush the trash. What is the area of the piston that crushes the trash?