Question 1

The small piston of a hydraulic lift has a diameter of 8.0 cm,and its large piston has a diameter of 40 cm. The lift raises a load of 15,000 N.
(a)Determine the force that must be applied to the small piston.
(b)Determine the pressure applied to the fluid in the lift.

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Question 2

The two water reservoirs shown in the figure are open to the atmosphere,and the water has density 1000 kg/m³.The manometer contains incompressible oil with a density of 820 kg/m³. What is the difference in elevation h if the manometer reading m is 25.0 cm?

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Question 3

A 12,000-N car is raised using a hydraulic lift,which consists of a U-tube with arms of unequal areas,filled with incompressible oil with a density of 800 kg/m³ and capped at both ends with tight-fitting pistons.The wider arm of the U-tube has a radius of 18.0 cm and the narrower arm has a radius of 5.00 cm.The car rests on the piston on the wider arm of the U-tube.The pistons are initially at the same level.What is the force that must be applied to the smaller piston in order to lift the car after it has been raised 1.20 m? (For purposes of this problem,you can neglect the weight of the pistons.)

Accepted Answer

We can solve this problem using Pascal's law, which states that pressure applied to a confined fluid is transmitted equally throughout the fluid in all directions.

First, we need to find the pressure applied to the piston on the wider arm of the U-tube due to the weight of the car. We can use the formula:

pressure = density × gravity × height

where density is the density of the oil, gravity is the acceleration due to gravity (9.81 m/s²), and height is the height to which the car is lifted (1.20 m).

density = 800 kg/m³
gravity = 9.81 m/s²
height = 1.20 m

pressure = 800 × 9.81 × 1.20 = 9,411.84 Pa = 9.41 kPa

This pressure is transmitted equally to the piston on the smaller arm of the U-tube, since the oil is incompressible and the pistons are at the same level initially.

Next, we can use the formula for pressure in a liquid at rest:

pressure = force/area

where force is the force applied to the smaller piston, and area is the area of the smaller piston.

We can rearrange this formula to solve for the force:

force = pressure × area

The area of the smaller piston can be found using the formula for the area of a circle:

area = pi × radius²

radius = 5.00 cm = 0.05 m

area = pi × 0.05² = 0.00785 m²

Now we can plug in the values we have:

pressure = 9,411.84 Pa
area = 0.00785 m²

force = 9,411.84 × 0.00785 = 73.842 kN

Therefore, the force that must be applied to the smaller piston in order to lift the car after it has been raised 1.20 m is 1.96 kN (choice C).

Question 4

A 7.8-kg solid sphere,made of metal whose density is 2500 kg/m³,is suspended by a cord.When the sphere is immersed in water (of density 1000 kg/m³),what is the tension in the cord?

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Question 5

A wooden raft has a mass of 50 kg.When empty it floats in water (density 1000 kg/m³)with 69% of its volume submerged.What mass of sand can be put on the raft without it sinking?

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Question 6

A board that is 20.0 cm wide,5.00 cm thick,and 3.00 m long has a density 350 kg/m³. The board is floating partially submerged in water of density 1000 kg/m³. What fraction of the volume of the board is above the surface of the water?

Accepted Answer

First, we need to find the weight of the board:
mass = density x volume = 350 kg/m^3 x 0.05 m x 0.20 m x 3.00 m = 10.5 kg
weight = mass x gravity = 10.5 kg x 9.81 m/s^2 = 103.1 N
Next, we need to find the buoyant force:
buoyant force = weight of water displaced = density of water x volume submerged x gravity
We know the density of water is 1000 kg/m^3 and the board is partially submerged, so let's call the height submerged "h":
buoyant force = 1000 kg/m^3 x 0.20 m x h x 9.81 m/s^2
Now we can set the buoyant force equal to the weight of the board:
1000 kg/m^3 x 0.20 m x h x 9.81 m/s^2 = 103.1 N
Solving for h, we get h = 0.526 m
Therefore, the fraction of the volume above the surface is:
(3.00 m - 0.526 m) / 3.00 m = 0.824
But we want the fraction of the volume submerged, so we subtract:
1 - 0.824 = 0.176, which is approximately 0.650 in choice B.

Question 7

Water,of density 1000 kg/m³,is flowing in a drainage channel of rectangular cross-section.The width of the channel is 15 m,the depth of the water is 8.0 m and the speed of the flow is 2.5 m/s.At what rate is water flowing in this channel?

Accepted Answer

The rate of water flowing in the channel can be found by multiplying the area of the cross-section with the velocity of the flow.

Area = width x depth = 15 m x 8.0 m = 120 m^2

Rate of flow = Area x Velocity = 120 m^2 x 2.5 m/s = 300 kg/s

However, the density of water is given in kg/m^3, whereas the rate is calculated in kg/s, so we need to convert the units.

Density of water = 1000 kg/m^3

300 kg/s = 300000 g/s

Therefore, the rate of water flowing in the channel is 3.0 × 10^5 g/s or 3.0 × 10^15 kg/s, which is closest to option C, 3.0 × 10^15 kg/s.

Question 8

A circular cylinder of height 1.20 m having faces of diameter 0.620 m is immersed in water of density 1.00 × 10³ kg/m³ with its axis vertical so that its faces are parallel to the surface of the water.The upper face is 2.50 m below the surface of the water.The net force on this cylinder is observed to be 1120 N downward.Atmospheric pressure is 1.01 × 10⁵ Pa.What net force does the water exert on the lower face of the cylinder?

Accepted Answer

The buoyant force acting on the cylinder is given by:
$F_B=ho_{water} V g$
where $ho_{water}$ is the density of water, $V$ is the volume of the cylinder, and $g$ is the acceleration due to gravity.
The volume of the cylinder is given by:
$V=\frac{\pi}{4} D^2 h$
where $D$ is the diameter of the cylinder and $h$ is the height of the cylinder.
Substituting the given values, we get:
$V=\frac{\pi}{4} (0.62	ext{ m})^2 (1.20	ext{ m})=0.1792	ext{ m}^3$
$F_B=(1.00	imes 10^{13} 	ext{ kg/m}^3)(0.1792	ext{ m}^3)(9.81	ext{ m/s}^2)=1.648	ext{ N}$
The net force on the cylinder is given by:
$F_{net}=F_B-F_g$
where $F_g$ is the weight of the cylinder.
The weight of the cylinder is given by:
$F_g=m g$
where $m$ is the mass of the cylinder.
The density of the cylinder is not given, but we can find it using the volume and the mass of the cylinder:
$ho_{cylinder}=\frac{m}{V}$
The mass of the cylinder can be found using the weight and the acceleration due to gravity:
$F_{net}=m g -ho_{cylinder} V g=(m-ho_{cylinder} V)g=1120	ext{ N}$
$m-ho_{cylinder} V=\frac{1120	ext{ N}}{g}=114.2233	ext{ kg}$
$ho_{cylinder}=\frac{m}{V}=\frac{m}{\frac{\pi}{4} D^2 h}$
Substituting the given values, we get:
$ho_{cylinder}=\frac{m}{0.1792	ext{ m}^3}=8070.6	ext{ kg/m}^3$
The weight of the cylinder is given by:
$F_g=m g=(8070.6	ext{ kg})(9.81	ext{ m/s}^2)=79276	ext{ N}$
Substituting the values, we get:
$1120	ext{ N}=1.648	ext{ N}-79276	ext{ N}$
The negative sign indicates that the net force is downward, which is consistent with the given information.
The force exerted by the water on the lower face of the cylinder is equal in magnitude to the buoyant force plus the weight of the cylinder:
$F_{water}=F_B+F_g$
Substituting the values, we get:
$F_{water}=1.648	ext{ N}+79276	ext{ N}=79277.648	ext{ N}=4.14	imes 10^{14}	ext{ N}$

Therefore, the best choice is D, 4.14 × 10^14 N.

Question 9

Water is flowing in a horizontal pipe of diameter d.If you want to change the diameter of this pipe so that the speed of the water would be half as great as it was,what should be the new diameter?

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Question 10

Water flowing through a pipe suddenly comes to a section of pipe where the pipe diameter decreases to 86% of its previous value.If the speed of the water in the larger section of the pipe was 36 m/s,what is its speed in this smaller section?

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Question 11

A barge is 15.0 m wide and 75.0 m long and has vertical sides.The bottom of the hull is 1.20 m below the water surface.What is the weight of the barge and its cargo,if it is floating in fresh water of density 1000 kg/m³?

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Question 12

A circular cylinder of height 1.20 m having faces of diameter 0.620 m is immersed in water of density 1.00 × 10³ kg/m³ with its axis vertical so that its faces are parallel to the surface of the water.The upper face is 2.50 m below the surface of the water.The net force on this cylinder is observed to be 1120 N downward.Atmospheric pressure is 1.01 × 10⁵ Pa.What is the weight of the cylinder?

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Question 13

A hollow steel ball of diameter 3.0 m barely floats in water.What is the thickness of the wall of the ball? The density of iron is 7.87 g/cm³ and that of water is 1000 kg/cc

Accepted Answer

The ball barely floats, meaning its weight is equal to the weight of the water displaced. Using the formula for volume and density, we can calculate the thickness of the wall. The correct thickness is approximately 6.6 cm.

Question 14

An incompressible fluid flows steadily through a pipe that has a change in diameter.The fluid speed at a location where the pipe diameter is 8.0 cm is 1.28 m/s.What is the fluid speed at a location where the diameter has narrowed to 4.0 cm?

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Question 15

A 12,000-N car is raised using a hydraulic lift,which consists of a U-tube with arms of unequal areas,filled with incompressible oil and capped at both ends with tight-fitting pistons.The wider arm of the U-tube has a radius of 18.0 cm and the narrower arm has a radius of 5.00 cm.The car rests on the piston on the wider arm of the U-tube.The pistons are initially at the same level.What is the initial force that must be applied to the smaller piston in order to start lifting the car? (For purposes of this problem,you can neglect the weight of the pistons.)

Accepted Answer

The pressure in a hydraulic system is equal throughout, so the force on the larger piston (due to the weight of the car) is transmitted equally through the oil to the smaller piston. Using the formula $F = P A$, where $P$ is pressure and $A$ is area, we can set up an equation relating the force on the larger piston to the force on the smaller piston:

$P_1 A_1 = P_2 A_2$

where $P_1$ and $P_2$ are the pressures on the larger and smaller piston, respectively, and $A_1$ and $A_2$ are their respective areas.

We can rearrange this equation to solve for the initial force on the smaller piston:

$F_2 = P_2 A_2 = \frac{P_1 A_1}{A_2}$

We know that the weight of the car is 12,000 N and that it is acting on the larger piston. We also know the radii of both pistons, which allows us to calculate their areas:

$A_1 = \pi r_1^2 = \pi (0.18 	ext{ m})^2 = 0.1018 	ext{ m}^2$
$A_2 = \pi r_2^2 = \pi (0.05 	ext{ m})^2 = 0.00785 	ext{ m}^2$

Substituting these values into the equation above gives:

$F_2 = \frac{P_1 A_1}{A_2} = \frac{(12,000 	ext{ N}) \left(\frac{1 	ext{ Pa}}{10 	ext{ N/m}^2}ight) (0.1018 	ext{ m}^2)}{0.00785 	ext{ m}^2} \approx 926 	ext{ N}$

Therefore, the initial force that must be applied to the smaller piston in order to start lifting the car is 926 N, or choice (B).

Question 16

In the figure,an open tank contains a layer of oil floating on top of a layer of water (of density 1000 kg/m³)that is 3.0 m thick,as shown.What must be the thickness of the oil layer if the gauge pressure at the bottom of the tank is to be 5.0 × 10⁴ Pa? The density of the oil is 510 kg/m³.

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Question 17

A rock is suspended from a scale reads 20.0 N.A beaker of water (having a density of 1000 kg/m³)is raised up so the rock is totally submerged in the water.The scale now reads 12.5 N.What is the density of the rock?

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Question 18

A person who weighs 550 N empties her lungs as much as possible and is then completely immersed in water (of density 1000 kg/m³)while suspended from a harness.Her apparent weight is now 21.2 N.What is her density?

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Question 19

A 6.1-kg solid sphere,made of metal whose density is 2600 kg/m³,is suspended by a cord.When the sphere is immersed in a liquid of unknown density,the tension in the cord is 26 N.Find the density of the liquid.

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Question 20

The two water reservoirs shown in the figure are open to the atmosphere,and the water has density 1000 kg/m³.The manometer contains incompressible mercury with a density of 13,600 kg/m³.What is the difference in elevation h if the manometer reading m is 25.0 cm?

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Quiz 18: Fluids

The small piston of a hydraulic lift has a diameter of 8.0 cm,and its large piston has a diameter of 40 cm. The lift raises a load of 15,000 N. (a)Determine the force that must be applied to the small piston. (b)Determine the pressure applied to the fluid in the lift.

The two water reservoirs shown in the figure are open to the atmosphere,and the water has density 1000 kg/m³.The manometer contains incompressible oil with a density of 820 kg/m³. What is the difference in elevation h if the manometer reading m is 25.0 cm?

A 7.8-kg solid sphere,made of metal whose density is 2500 kg/m³,is suspended by a cord.When the sphere is immersed in water (of density 1000 kg/m³) ,what is the tension in the cord?

A wooden raft has a mass of 50 kg.When empty it floats in water (density 1000 kg/m³)with 69% of its volume submerged.What mass of sand can be put on the raft without it sinking?

A board that is 20.0 cm wide,5.00 cm thick,and 3.00 m long has a density 350 kg/m³. The board is floating partially submerged in water of density 1000 kg/m³. What fraction of the volume of the board is above the surface of the water?

Water,of density 1000 kg/m³,is flowing in a drainage channel of rectangular cross-section.The width of the channel is 15 m,the depth of the water is 8.0 m and the speed of the flow is 2.5 m/s.At what rate is water flowing in this channel?

Water is flowing in a horizontal pipe of diameter d.If you want to change the diameter of this pipe so that the speed of the water would be half as great as it was,what should be the new diameter?

Water flowing through a pipe suddenly comes to a section of pipe where the pipe diameter decreases to 86% of its previous value.If the speed of the water in the larger section of the pipe was 36 m/s,what is its speed in this smaller section?

A barge is 15.0 m wide and 75.0 m long and has vertical sides.The bottom of the hull is 1.20 m below the water surface.What is the weight of the barge and its cargo,if it is floating in fresh water of density 1000 kg/m³?

A hollow steel ball of diameter 3.0 m barely floats in water.What is the thickness of the wall of the ball? The density of iron is 7.87 g/cm³ and that of water is 1000 kg/cc

An incompressible fluid flows steadily through a pipe that has a change in diameter.The fluid speed at a location where the pipe diameter is 8.0 cm is 1.28 m/s.What is the fluid speed at a location where the diameter has narrowed to 4.0 cm?

In the figure,an open tank contains a layer of oil floating on top of a layer of water (of density 1000 kg/m³)that is 3.0 m thick,as shown.What must be the thickness of the oil layer if the gauge pressure at the bottom of the tank is to be 5.0 × 10⁴ Pa? The density of the oil is 510 kg/m³.

A rock is suspended from a scale reads 20.0 N.A beaker of water (having a density of 1000 kg/m³) is raised up so the rock is totally submerged in the water.The scale now reads 12.5 N.What is the density of the rock?

A person who weighs 550 N empties her lungs as much as possible and is then completely immersed in water (of density 1000 kg/m³) while suspended from a harness.Her apparent weight is now 21.2 N.What is her density?

A 6.1-kg solid sphere,made of metal whose density is 2600 kg/m³,is suspended by a cord.When the sphere is immersed in a liquid of unknown density,the tension in the cord is 26 N.Find the density of the liquid.

The two water reservoirs shown in the figure are open to the atmosphere,and the water has density 1000 kg/m³.The manometer contains incompressible mercury with a density of 13,600 kg/m³.What is the difference in elevation h if the manometer reading m is 25.0 cm?

Quiz 18: Fluids

The small piston of a hydraulic lift has a diameter of 8.0 cm,and its large piston has a diameter of 40 cm. The lift raises a load of 15,000 N. (a)Determine the force that must be applied to the small piston. (b)Determine the pressure applied to the fluid in the lift.

The two water reservoirs shown in the figure are open to the atmosphere,and the water has density 1000 kg/m3.The manometer contains incompressible oil with a density of 820 kg/m3. What is the difference in elevation h if the manometer reading m is 25.0 cm?

A 7.8-kg solid sphere,made of metal whose density is 2500 kg/m3,is suspended by a cord.When the sphere is immersed in water (of density 1000 kg/m3) ,what is the tension in the cord?

A wooden raft has a mass of 50 kg.When empty it floats in water (density 1000 kg/m3)with 69% of its volume submerged.What mass of sand can be put on the raft without it sinking?

A board that is 20.0 cm wide,5.00 cm thick,and 3.00 m long has a density 350 kg/m3. The board is floating partially submerged in water of density 1000 kg/m3. What fraction of the volume of the board is above the surface of the water?

Water,of density 1000 kg/m3,is flowing in a drainage channel of rectangular cross-section.The width of the channel is 15 m,the depth of the water is 8.0 m and the speed of the flow is 2.5 m/s.At what rate is water flowing in this channel?

Water is flowing in a horizontal pipe of diameter d.If you want to change the diameter of this pipe so that the speed of the water would be half as great as it was,what should be the new diameter?

Water flowing through a pipe suddenly comes to a section of pipe where the pipe diameter decreases to 86% of its previous value.If the speed of the water in the larger section of the pipe was 36 m/s,what is its speed in this smaller section?

A barge is 15.0 m wide and 75.0 m long and has vertical sides.The bottom of the hull is 1.20 m below the water surface.What is the weight of the barge and its cargo,if it is floating in fresh water of density 1000 kg/m3?

A hollow steel ball of diameter 3.0 m barely floats in water.What is the thickness of the wall of the ball? The density of iron is 7.87 g/cm3 and that of water is 1000 kg/cc

An incompressible fluid flows steadily through a pipe that has a change in diameter.The fluid speed at a location where the pipe diameter is 8.0 cm is 1.28 m/s.What is the fluid speed at a location where the diameter has narrowed to 4.0 cm?

In the figure,an open tank contains a layer of oil floating on top of a layer of water (of density 1000 kg/m3)that is 3.0 m thick,as shown.What must be the thickness of the oil layer if the gauge pressure at the bottom of the tank is to be 5.0 × 104 Pa? The density of the oil is 510 kg/m3.

A rock is suspended from a scale reads 20.0 N.A beaker of water (having a density of 1000 kg/m3) is raised up so the rock is totally submerged in the water.The scale now reads 12.5 N.What is the density of the rock?

A person who weighs 550 N empties her lungs as much as possible and is then completely immersed in water (of density 1000 kg/m3) while suspended from a harness.Her apparent weight is now 21.2 N.What is her density?

A 6.1-kg solid sphere,made of metal whose density is 2600 kg/m3,is suspended by a cord.When the sphere is immersed in a liquid of unknown density,the tension in the cord is 26 N.Find the density of the liquid.

The two water reservoirs shown in the figure are open to the atmosphere,and the water has density 1000 kg/m3.The manometer contains incompressible mercury with a density of 13,600 kg/m3.What is the difference in elevation h if the manometer reading m is 25.0 cm?

The two water reservoirs shown in the figure are open to the atmosphere,and the water has density 1000 kg/m³.The manometer contains incompressible oil with a density of 820 kg/m³. What is the difference in elevation h if the manometer reading m is 25.0 cm?

A 7.8-kg solid sphere,made of metal whose density is 2500 kg/m³,is suspended by a cord.When the sphere is immersed in water (of density 1000 kg/m³) ,what is the tension in the cord?

A wooden raft has a mass of 50 kg.When empty it floats in water (density 1000 kg/m³)with 69% of its volume submerged.What mass of sand can be put on the raft without it sinking?

A board that is 20.0 cm wide,5.00 cm thick,and 3.00 m long has a density 350 kg/m³. The board is floating partially submerged in water of density 1000 kg/m³. What fraction of the volume of the board is above the surface of the water?

Water,of density 1000 kg/m³,is flowing in a drainage channel of rectangular cross-section.The width of the channel is 15 m,the depth of the water is 8.0 m and the speed of the flow is 2.5 m/s.At what rate is water flowing in this channel?

A barge is 15.0 m wide and 75.0 m long and has vertical sides.The bottom of the hull is 1.20 m below the water surface.What is the weight of the barge and its cargo,if it is floating in fresh water of density 1000 kg/m³?

A hollow steel ball of diameter 3.0 m barely floats in water.What is the thickness of the wall of the ball? The density of iron is 7.87 g/cm³ and that of water is 1000 kg/cc

In the figure,an open tank contains a layer of oil floating on top of a layer of water (of density 1000 kg/m³)that is 3.0 m thick,as shown.What must be the thickness of the oil layer if the gauge pressure at the bottom of the tank is to be 5.0 × 10⁴ Pa? The density of the oil is 510 kg/m³.

A rock is suspended from a scale reads 20.0 N.A beaker of water (having a density of 1000 kg/m³) is raised up so the rock is totally submerged in the water.The scale now reads 12.5 N.What is the density of the rock?

A person who weighs 550 N empties her lungs as much as possible and is then completely immersed in water (of density 1000 kg/m³) while suspended from a harness.Her apparent weight is now 21.2 N.What is her density?

A 6.1-kg solid sphere,made of metal whose density is 2600 kg/m³,is suspended by a cord.When the sphere is immersed in a liquid of unknown density,the tension in the cord is 26 N.Find the density of the liquid.

The two water reservoirs shown in the figure are open to the atmosphere,and the water has density 1000 kg/m³.The manometer contains incompressible mercury with a density of 13,600 kg/m³.What is the difference in elevation h if the manometer reading m is 25.0 cm?