Pascal's Principle and Hydraulics Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 2

Question 21Question →

A maintenance crew is selecting a hydraulic jack to lift a heavy machinery component. The load must be lifted exactly $2 \text{ inches}$ . Different jacks have different input (pump) and output (lifting) cylinder areas.

Match each hydraulic jack's specifications to the distance its input piston must be pushed down to achieve the $2 \text{-inch}$ output lift.

Click a left item, then click its matching right item

Show answer & explanation

Answer

To find the input distance, calculate the mechanical advantage (Output Area / Input Area) and multiply it by the desired output distance (

2 \text{ inches}

). Jack Alpha requires

10 \text{ inches}

, Jack Beta requires

8 \text{ inches}

, Jack Gamma requires

12 \text{ inches}

, and Jack Delta requires

4 \text{ inches}

.

Each correct match is found by calculating the mechanical advantage (Output Area divided by Input Area) and multiplying that ratio by the desired

2\text{-inch}

output lift. This ensures the volume of fluid pushed by the pump equals the volume required to lift the load.

Step-by-Step Solution

1

Determine the mathematical relationship for distance in a closed hydraulic system.

\text{Input Area} \times \text{Input Distance} = \text{Output Area} \times \text{Output Distance}

Because liquids are essentially incompressible, the volume of fluid displaced by the input piston must equal the volume of fluid pushing up the output piston.

2

Rearrange the formula to solve for the required Input Distance.

\text{Input Distance} = \text{Output Distance} \times (\frac{\text{Output Area}}{\text{Input Area}})

This isolates the unknown variable, making it easier to plug in the given values for each jack.

3

Calculate the required input distance for Jack Alpha.

2 \text{ inches} \times (\frac{10}{2}) = 2 \times 5 = 10 \text{ inches}

.

Applying the specific area measurements of Jack Alpha to the derived formula.

4

Apply the same formula to the remaining jacks to find their respective input distances.

Beta:

2 \times (\frac{12}{3}) = 8 \text{ inches}

. Gamma:

2 \times (\frac{24}{4}) = 12 \text{ inches}

. Delta:

2 \times (\frac{10}{5}) = 4 \text{ inches}

.

Completing the required calculations to find all corresponding values.

Key Concept

In a hydraulic system, the distance an input piston must travel is proportional to the ratio of the output area to the input area. A larger mechanical advantage requires a proportionally larger input distance to conserve the total volume of displaced fluid.

Question 22Question →

A dentist adjusts a patient's hydraulic examination chair. The system's input piston has a cross-sectional area of $8 \text{ in}^2$ , and the lifting piston beneath the chair has an area of $64 \text{ in}^2$ . If the dentist applies $35 \text{ lbs}$ of force to the input piston, how much upward force is generated by the lifting piston?

Show answer & explanation

Answer: 280

Answer

280

The hydraulic system provides a mechanical advantage equal to the ratio of the two piston areas (

64 / 8 = 8

). Multiplying the input force of

35 \text{ lbs}

by this mechanical advantage yields a total lifting force of

280 \text{ lbs}

.

Step-by-Step Solution

1

Identify the given variables.

Input area

A_1 = 8 \text{ in}^2

, output area

A_2 = 64 \text{ in}^2

, and input force

F_1 = 35 \text{ lbs}

.

These variables are required to apply Pascal's Principle.

2

Calculate the mechanical advantage (area ratio) of the hydraulic system.

\frac{A_2}{A_1} = \frac{64}{8} = 8

.

The ratio of the output area to the input area determines how much the input force will be multiplied.

3

Calculate the generated output force.

F_2 = F_1 \times \text{Area Ratio} = 35 \text{ lbs} \times 8 = 280 \text{ lbs}

.

Pascal's Principle states that pressure is transmitted equally throughout an enclosed fluid, meaning force increases proportionally with area.

Key Concept

According to Pascal's Principle, pressure applied to an enclosed fluid is transmitted equally in all directions. In a hydraulic system, this means the ratio of output force to input force is equal to the ratio of output area to input area (

F_1/A_1 = F_2/A_2

).

Estimated Time:45s

Question 23Question →

A mechanic uses a simple hydraulic lift to raise a motorcycle. The system's input piston has a cross-sectional area of $5 \text{ in}^2$ , and the output (lifting) piston has a cross-sectional area of $50 \text{ in}^2$ . If the mechanic applies a downward force of $40 \text{ lbs}$ on the input piston, what is the upward force exerted by the output piston?

4 \text{ lbs}

40 \text{ lbs}

200 \text{ lbs}

400 \text{ lbs}

Show answer & explanation

Answer: $400 \text{ lbs}$

Answer

The upward force exerted by the output piston is

400 \text{ lbs}

.

According to Pascal's Principle, pressure is transmitted equally throughout a confined fluid. The pressure generated at the input piston is

40 \text{ lbs} / 5 \text{ in}^2 = 8 \text{ psi}

. This exact same pressure acts on the larger

50 \text{ in}^2

output piston, resulting in an upward force of

8 \text{ psi} \times 50 \text{ in}^2 = 400 \text{ lbs}

. Alternatively, using the mechanical advantage ratio:

50 / 5 = 10

, so the output force is

10 \times 40 = 400 \text{ lbs}

.

Step-by-Step Solution

1

Calculate the pressure exerted by the input piston.

P = \frac{F_{in}}{A_{in}} = \frac{40 \text{ lbs}}{5 \text{ in}^2} = 8 \text{ psi}

Pascal's Principle states that pressure applied to an enclosed fluid is transmitted equally in all directions.

2

Calculate the output force by applying the pressure to the output piston's area.

F_{out} = P \times A_{out} = 8 \text{ psi} \times 50 \text{ in}^2 = 400 \text{ lbs}

The output force is the product of the constant fluid pressure and the area of the output piston.

Key Concept

Pascal's Principle and Hydraulic Multiplication

Estimated Time:45s

Question 24Question →

A barber chair uses a closed hydraulic lift mechanism to raise customers. The barber operates a foot pedal connected to an input cylinder with a cross-sectional area of $4 \text{ in}^2$ . The chair is mounted on a larger output cylinder with a cross-sectional area of $80 \text{ in}^2$ . If the total weight of the chair and the customer is $320 \text{ lbs}$ , how much downward force must the barber apply to the input cylinder to lift the load?

4 \text{ lbs}

16 \text{ lbs}

1280 \text{ lbs}

6400 \text{ lbs}

Show answer & explanation

Answer: $16 \text{ lbs}$

Answer

The barber must apply

16 \text{ lbs}

of force.

The correct answer applies Pascal's Principle, which indicates that the mechanical advantage of a hydraulic system is determined by the ratio of the piston areas. The output cylinder is

20

times larger than the input cylinder (

80 / 4 = 20

). Therefore, the input force only needs to be

1/20

th of the output load. Dividing the

320 \text{ lbs}

load by

20

yields an input force requirement of

16 \text{ lbs}

.

Step-by-Step Solution

1

Identify the given values for the hydraulic system.

Input area (

A_1

) =

4 \text{ in}^2

, Output area (

A_2

) =

80 \text{ in}^2

, Output force/load (

F_2

) =

320 \text{ lbs}

.

Organizing known variables is the first step in solving a Pascal's Principle problem.

2

State Pascal's Principle equation relating force and area.

\frac{F_1}{A_1} = \frac{F_2}{A_2}

Pascal's Principle dictates that pressure is transmitted equally throughout a confined incompressible fluid.

3

Rearrange the formula to solve for the unknown input force (

F_1

).

F_1 = F_2 \times \left(\frac{A_1}{A_2}\right)

Isolating

F_1

prepares the equation for direct substitution.

4

Substitute the known values and calculate.

F_1 = 320 \times \left(\frac{4}{80}\right) = 320 \times \left(\frac{1}{20}\right) = 16 \text{ lbs}

.

Performing the arithmetic reveals the exact force required at the pedal.

Key Concept

Pascal's Principle states that a change in pressure at any point in an enclosed fluid is transmitted equally throughout the fluid. In a hydraulic system, a small force applied over a small area can lift a large weight over a proportionally larger area.

Question 25Question →

Match the following hydraulic system components and concepts with their correct mechanical effects or descriptions.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The small input piston matches with traveling a greater distance for less force. The large output piston matches with traveling a shorter distance for greater force. Hydraulic fluid matches with the incompressible medium transmitting pressure. Piston area ratio matches with determining mechanical advantage.

Each component in a basic hydraulic system has a distinct mechanical role. The small input piston trades distance for force, while the large output piston trades force for distance. The area ratio defines this mechanical advantage, and the incompressible fluid ensures the pressure is transmitted without loss.

Step-by-Step Solution

1

Analyze the mechanical relationship between piston size and movement distance.

Smaller pistons move farther with less force; larger pistons move less distance with more force.

This relationship is dictated by the conservation of work (Work = Force × Distance) in the hydraulic system.

2

Identify the primary role of the fluid in a hydraulic system.

The fluid serves as the incompressible medium that transmits pressure.

Pascal's Principle states that pressure applied to an enclosed fluid is transmitted equally in all directions.

3

Define what determines the force multiplication factor.

The ratio of the output piston area to the input piston area dictates the mechanical advantage.

Since pressure is force per unit area, applying equal pressure across different areas results in proportional forces.

Key Concept

Pascal's Principle and conservation of work in hydraulic systems

Question 26Question →

A hydraulic rescue spreader operates using a closed fluid system. The tool's spreading arms are driven by an output piston with a cross-sectional area of $20 \text{ in}^2$ . During an operation, the output piston must exert a constant force of $6000 \text{ lbs}$ and extend exactly $4 \text{ inches}$ . The hand-operated pump uses an input piston with a cross-sectional area of $2 \text{ in}^2$ .

To achieve this full $4 \text{ inch}$ extension of the output piston, what is the total distance, in inches, that the input piston must travel?

Show answer & explanation

Answer: 40

Answer

The input piston must travel a total distance of 40 inches.

In a closed hydraulic system, the volume of fluid displaced by the input piston must equal the volume received by the output piston (

A_1 \times d_1 = A_2 \times d_2

). Since the output piston requires

80 \text{ in}^3

of fluid (

20 \text{ in}^2 \times 4 \text{ in}

), the input piston, with an area of

2 \text{ in}^2

, must travel

40 \text{ inches}

(

80 / 2

) to displace that same volume. The force value (

6000 \text{ lbs}

) is not needed to find the distance.

Step-by-Step Solution

1

State the principle of conservation of volume for a closed hydraulic system.

V_{in} = V_{out}

or

A_1 \times d_1 = A_2 \times d_2

Because liquids are essentially incompressible, the volume of fluid pushed by the input piston equals the volume that moves the output piston.

2

Calculate the volume of fluid displaced at the output cylinder.

V_{out} = 20 \text{ in}^2 \times 4 \text{ inches} = 80 \text{ in}^3

This is the total volume of fluid required to extend the spreading arms by the specified distance.

3

Calculate the required travel distance for the input piston.

2 \text{ in}^2 \times d_1 = 80 \text{ in}^3 \rightarrow d_1 = 40 \text{ inches}

Dividing the total required volume by the input piston's area gives the total distance the input piston must be pushed.

Key Concept

Conservation of fluid volume and mechanical advantage in hydraulic systems.

Question 27Question →

A hydraulic press used for stamping metal parts has an input piston with a cross-sectional area of $2 \text{ in}^2$ and an output piston with a cross-sectional area of $6 \text{ in}^2$ . If a downward force of $30 \text{ lbs}$ is applied to the input piston, how much force is exerted by the output piston?

10 \text{ lbs}

30 \text{ lbs}

90 \text{ lbs}

180 \text{ lbs}

Show answer & explanation

Answer: $90 \text{ lbs}$

Answer

90 \text{ lbs}

According to Pascal's Principle, pressure is transmitted equally throughout a confined fluid. The pressure generated at the input is

P = \frac{F_1}{A_1} = \frac{30}{2} = 15 \text{ psi}

. This same pressure acts on the output piston. The output force is

F_2 = P \times A_2 = 15 \times 6 = 90 \text{ lbs}

. Alternatively, using the mechanical advantage, since the output area is

3

times larger than the input area, the output force is

3

times the input force.

Step-by-Step Solution

1

Identify the given values from the problem description.

A_1 = 2 \text{ in}^2

,

F_1 = 30 \text{ lbs}

,

A_2 = 6 \text{ in}^2

These variables are required to apply Pascal's Principle.

2

Determine the mechanical advantage by finding the ratio of the output area to the input area.

Ratio =

\frac{A_2}{A_1} = \frac{6}{2} = 3

The output force is multiplied by the ratio of the piston areas.

3

Calculate the output force (

F_2

).

F_2 = 30 \times 3 = 90 \text{ lbs}

Multiplying the input force by the mechanical advantage gives the total output force.

Key Concept

Pascal's Principle and Hydraulic Multiplication

Estimated Time:45s

Question 28Question →

A factory uses a closed hydraulic press system to stamp metal parts. The system features an input piston with a cross-sectional area of $3 \text{ in}^2$ and an output stamping piston with a cross-sectional area of $15 \text{ in}^2$ . If the output stamping piston must travel a distance of $4 \text{ inches}$ to complete the stamping operation, how far must the input piston be depressed?

20 \text{ inches}

60 \text{ inches}

5 \text{ inches}

0.8 \text{ inches}

Show answer & explanation

Answer: $20 \text{ inches}$

Answer

20 \text{ inches}

In a closed hydraulic system, the volume of fluid displaced by the input piston must equal the volume gained at the output piston (

V = A \times d

). Because the output piston has

5

times the area of the input piston (

15 / 3 = 5

), the input piston must travel

5

times as far to move the required volume of fluid. Therefore, multiplying the area ratio (

5

) by the output distance (

4 \text{ inches}

) gives an input travel distance of

20 \text{ inches}

.

Step-by-Step Solution

1

Calculate the mechanical advantage (area ratio) between the output and input pistons.

15 \text{ in}^2 / 3 \text{ in}^2 = 5

The area ratio dictates the proportional difference in how far each piston must travel.

2

Apply the principle of conservation of fluid volume (

A_1 \times d_1 = A_2 \times d_2

) to find the input distance.

d_{in} = 5 \times 4 \text{ inches} = 20 \text{ inches}

Because the input piston is 5 times smaller in area, it must travel 5 times the distance of the output piston to displace the same volume of fluid.

Key Concept

Conservation of fluid volume in hydraulics (

A_1 \times d_1 = A_2 \times d_2

)

Estimated Time:1m 0s

Question 29Question →

Match each modification made to a standard closed hydraulic lift system with its resulting effect on the output piston. Assume the applied input force remains constant unless otherwise specified.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Modifications changing the area ratio affect force and distance inversely; proportional area modifications maintain the status quo; and input distance modifications only affect output distance.

Each correct match applies Pascal's Principle (

F_2 = F_1 \times \frac{A_2}{A_1}

) and the principle of work conservation (

F_1 \times d_1 = F_2 \times d_2

). Changing the area ratio inversely affects force and distance. Changing only input distance affects fluid volume displaced, altering output distance without changing force.

Step-by-Step Solution

1

Analyze modifications changing a single piston area.

Doubling output area doubles force but halves distance; doubling input area halves force but doubles distance.

Pascal's principle states

F_2 = F_1 \times \frac{A_2}{A_1}

, and conservation of work states

d_2 = d_1 \times \frac{A_1}{A_2}

.

2

Analyze modifications changing both piston areas.

Doubling both areas keeps the ratio

A_2/A_1

constant, resulting in no change to force or distance.

The mechanical advantage depends strictly on the ratio of the areas, not their absolute sizes.

3

Analyze modifications changing only input distance.

Doubling input distance doubles output distance without affecting force.

Volume displaced (

V = A_1 \times d_1

) doubles, so output distance doubles. Force depends only on the area ratio, which is unchanged.

Key Concept

Pascal's Principle and Conservation of Energy in Hydraulics

Question 30Question →

A mechanic uses a hydraulic floor jack to lift the front axle of a delivery truck. The jack's output piston has a cross-sectional area of $25 \text{ in}^2$ and must support a load of $5,000 \text{ lbs}$ . If the input piston has a cross-sectional area of $0.5 \text{ in}^2$ , what is the minimum input force the mechanic must apply to lift the truck? (Assume a frictionless, ideal system)

100 \text{ lbs}

200 \text{ lbs}

10,000 \text{ lbs}

250,000 \text{ lbs}

Show answer & explanation

Answer: $100 \text{ lbs}$

Answer

100 \text{ lbs}

In a closed hydraulic system, pressure is uniform. The pressure created by the load on the output piston is its force divided by its area (

5,000 / 25 = 200 \text{ psi}

). To produce this same pressure at the input piston, you multiply the pressure by the input piston's area (

200 \times 0.5

), which equals an input force of

100 \text{ lbs}

.

Step-by-Step Solution

1

Calculate the fluid pressure required at the output piston using the formula P = F/A.

5,000 \text{ lbs} / 25 \text{ in}^2 = 200 \text{ psi}

Pascal's Principle states that pressure applied to a confined fluid is transmitted equally in all directions.

2

Apply the calculated pressure to the input piston to find the required input force using F = P × A.

200 \text{ psi} \times 0.5 \text{ in}^2 = 100 \text{ lbs}

The input piston must generate the identical system pressure to balance and lift the load.

Key Concept

Pascal's Principle (Hydraulic Mechanical Advantage)

Question 31Question →

An industrial cardboard baler utilizes a closed hydraulic system to compress recyclable materials. The system features an input pump piston with a cross-sectional area of $3 \text{ in}^2$ and a large output driving piston with a cross-sectional area of $45 \text{ in}^2$ . If the input piston is pushed downward for a single stroke distance of $15 \text{ inches}$ , how many inches will the output piston move? (Assume no fluid is lost and the fluid is incompressible.)

Show answer & explanation

Answer: 1

Answer

1

In a hydraulic system, the volume of fluid displaced must be conserved. The output piston has 15 times the cross-sectional area of the input piston (

45 \div 3 = 15

). Because it is 15 times larger, it will only travel 1/15th of the distance to displace the identical volume of fluid. Therefore, an input distance of 15 inches results in an output distance of exactly 1 inch.

Step-by-Step Solution

1

Identify the relationship for fluid displacement in a closed hydraulic system.

A_1 \times d_1 = A_2 \times d_2

Because liquids are incompressible, the volume of fluid pushed out by the input piston (

A_1 \times d_1

) must equal the volume of fluid pushing up the output piston (

A_2 \times d_2

).

2

Substitute the given values for the input piston (

A_1 = 3

,

d_1 = 15

) and output piston (

A_2 = 45

).

3 \times 15 = 45 \times d_2

This sets up the equation with all known variables to solve for the unknown distance (

d_2

).

3

Calculate the total volume displaced by the input piston.

45 = 45 \times d_2

Multiplying the input area by its travel distance gives a total displaced volume of 45 cubic inches.

4

Solve for the output distance (

d_2

) by dividing the total volume by the output piston's area.

d_2 = 1

Dividing 45 cubic inches by 45 square inches yields a travel distance of 1 inch.

Key Concept

Conservation of fluid volume and the distance-force tradeoff in hydraulics

Question 32Question →

A hydraulic dump truck bed lift system operates using a small input pump cylinder and a larger output lifting cylinder. The pump cylinder has a cross-sectional area of $2 \text{ in}^2$ and the lifting cylinder has a cross-sectional area of $20 \text{ in}^2$ . To raise the truck bed, the lifting cylinder must extend a distance of $8 \text{ inches}$ . Assuming an ideal closed hydraulic system, what is the total distance the pump cylinder must travel to achieve this lift?

80 \text{ inches}

0.8 \text{ inches}

160 \text{ inches}

10 \text{ inches}

Show answer & explanation

Answer: $80 \text{ inches}$

Answer

The input pump cylinder must travel a total distance of

80 \text{ inches}

.

Because liquids are essentially incompressible, the volume of fluid pushed by the input cylinder must equal the volume received by the output cylinder. Volume is calculated as Area multiplied by Distance (

V = A \times d

). Setting the volumes equal gives

2 \times d_1 = 20 \times 8

. Solving this equation shows that the input cylinder must travel

80 \text{ inches}

to move the larger cylinder

8 \text{ inches}

.

Step-by-Step Solution

1

Identify the given values for the hydraulic system.

A_1 = 2 \text{ in}^2

,

A_2 = 20 \text{ in}^2

, and

d_2 = 8 \text{ inches}

.

These values are needed to apply the principle of volume conservation in an incompressible fluid system.

2

Apply the volume conservation formula for hydraulic systems.

A_1 \times d_1 = A_2 \times d_2

In a closed hydraulic system, the volume of fluid displaced by the input piston must equal the volume of fluid displacing the output piston.

3

Substitute the known values into the equation and solve for the input distance (

d_1

).

2 \times d_1 = 20 \times 8 \Rightarrow 2 \times d_1 = 160 \Rightarrow d_1 = 80 \text{ inches}

.

Isolating

d_1

provides the total travel distance required by the input pump to move the lifting cylinder by the required amount.

Key Concept

In a hydraulic system, work and fluid volume are conserved. A smaller input piston must travel a proportionally greater distance to move a larger output piston a shorter distance.

Question 33Question →

Match each closed hydraulic system scenario to its calculated unknown value. Assume the fluid is incompressible, there is no friction, and conditions are ideal.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches are based on applying Pascal's Principle and the conservation of work. Output Force is 250 lbs, Output Area is 15 in², Input Force is 50 lbs, and Output Distance is 1 in.

Each matching pair applies either Pascal's Principle (

P = \frac{F_1}{A_1} = \frac{F_2}{A_2}

) or the conservation of work (

F_1 \cdot d_1 = F_2 \cdot d_2

) to solve for the missing variable in the hydraulic system.

Step-by-Step Solution

1

Calculate the Output Force for the first scenario.

250 lbs

Using Pascal's Principle (

F_1/A_1 = F_2/A_2

), the system pressure is 50 lbs / 2 in² = 25 psi. The output force is 25 psi * 10 in² = 250 lbs.

2

Calculate the Output Area for the second scenario.

15 in²

The system pressure is 30 lbs / 3 in² = 10 psi. Rearranging the formula for area (

A = F/P

), the output area is 150 lbs / 10 psi = 15 in².

3

Calculate the Input Force for the third scenario.

50 lbs

System pressure is determined by the output side: 100 lbs / 20 in² = 5 psi. The required input force is 5 psi * 10 in² = 50 lbs.

4

Calculate the Output Distance for the fourth scenario.

1 in

By conservation of work (

F_1 \times d_1 = F_2 \times d_2

), the input work is 40 lbs * 5 in = 200 in-lbs. Thus, the output distance is 200 in-lbs / 200 lbs = 1 in.

Key Concept

Pascal's Principle states that pressure applied to an enclosed fluid is transmitted undiminished. The work done on the input piston equals the work done by the output piston.

Question 34Question →

A hydraulic lifting mechanism on a heavy-duty forklift contains an input cylinder and an output cylinder. The output cylinder has $5$ times the cross-sectional area of the input cylinder. If the operator applies an input force of $50 \text{ lbs}$ to move the input piston down by $15 \text{ inches}$ , what is the resulting lifting force and how far does the output piston move?

250 \text{ lbs}

of force and

3 \text{ inches}

of movement

250 \text{ lbs}

of force and

75 \text{ inches}

of movement

10 \text{ lbs}

of force and

3 \text{ inches}

of movement

10 \text{ lbs}

of force and

75 \text{ inches}

of movement

Show answer & explanation

Answer: $250 \text{ lbs}$ of force and $3 \text{ inches}$ of movement

Answer

The output cylinder produces

250 \text{ lbs}

of lifting force and moves

3 \text{ inches}

.

The hydraulic system provides a mechanical advantage equal to the ratio of the piston areas. Since the output cylinder has

5

times the area, the output force is

5

times the input force (

50 \times 5 = 250 \text{ lbs}

). However, to conserve energy and fluid volume, the output piston only moves

1/5

of the distance of the input piston (

15 / 5 = 3 \text{ inches}

).

Step-by-Step Solution

1

Determine the mechanical advantage of the hydraulic system.

Mechanical Advantage =

5

.

The output cylinder has

5

times the cross-sectional area of the input cylinder.

2

Calculate the output lifting force.

F_2 = 50 \times 5 = 250 \text{ lbs}

.

According to Pascal's Principle, force is directly proportional to the area. The larger piston exerts

5

times the force.

3

Calculate the distance the output piston moves.

d_2 = 15 / 5 = 3 \text{ inches}

.

Because the volume of incompressible fluid displaced remains constant, the distance moved is inversely proportional to the area.

Key Concept

Pascal's Principle and Conservation of Energy in Hydraulics

Estimated Time:1m 0s

Question 35Question →

A barber chair operates using a closed hydraulic lifting system to raise a customer. The input pump piston has a cross-sectional area of $4 \text{ in}^2$ , and the output lifting piston has a cross-sectional area of $60 \text{ in}^2$ . If the barber pushes the input piston down a distance of $7.5 \text{ inches}$ , how many inches does the output piston move upward?

Show answer & explanation

Answer: 0.5

Answer

0.5

In a hydraulic system, the mechanical advantage that multiplies the output force must correspondingly divide the output distance. The output piston has an area 15 times larger than the input piston (

60 / 4 = 15

). Therefore, while it can lift 15 times the force, it will only travel

1/15\text{th}

of the input distance (

7.5 / 15 = 0.5 \text{ inches}

).

Step-by-Step Solution

1

Identify the relevant principle for hydraulic piston travel distances.

The volume of fluid displaced by the input piston equals the volume gained by the output piston:

A_1 \times d_1 = A_2 \times d_2

.

Because hydraulic fluid is essentially incompressible, the total volume in the closed system remains constant.

2

Set up the equation using the given values.

4 \times 7.5 = 60 \times d_2

Input area (

A_1

) is 4, input distance (

d_1

) is 7.5, and output area (

A_2

) is 60.

3

Calculate the total fluid volume displaced.

4 \times 7.5 = 30 \text{ in}^3

Multiplying area by distance gives the volume of fluid moved.

4

Solve for the output piston's upward distance (

d_2

).

30 / 60 = 0.5 \text{ inches}

Dividing the displaced volume by the output piston's larger area yields the smaller travel distance.

Key Concept

Conservation of volume in hydraulic systems dictates that distance and area are inversely proportional (Mechanical Advantage trades distance for force).

Question 36Question →

In an aircraft's hydraulic braking system, the master cylinder (input) connects to a brake caliper cylinder (output). The cross-sectional area of the caliper cylinder is $5$ times the area of the master cylinder. If the master cylinder piston is pushed with $40 \text{ lbs}$ of force and travels $5 \text{ inches}$ , what is the resulting force and travel distance of the caliper cylinder piston?

200 \text{ lbs}

of force and

1 \text{ inch}

of travel

200 \text{ lbs}

of force and

25 \text{ inches}

of travel

8 \text{ lbs}

of force and

25 \text{ inches}

of travel

8 \text{ lbs}

of force and

1 \text{ inch}

of travel

Show answer & explanation

Answer: $200 \text{ lbs}$ of force and $1 \text{ inch}$ of travel

Answer

The caliper cylinder piston will output

200 \text{ lbs}

of force and move

1 \text{ inch}

.

Based on Pascal's principle and the conservation of energy, the larger output cylinder multiplies the input force by the area ratio (

40 \times 5 = 200 \text{ lbs}

). Correspondingly, the output distance must be divided by that same ratio (

5 \div 5 = 1 \text{ inch}

) to ensure the work input equals the work output.

Step-by-Step Solution

1

Determine the area ratio between the input and output cylinders.

Area Ratio = 5.

The problem states the caliper cylinder has an area 5 times larger than the master cylinder.

2

Calculate the resulting force at the output cylinder.

Output Force = Input Force x Area Ratio =

40 \text{ lbs} \times 5 = 200 \text{ lbs}

.

According to Pascal's principle, pressure is transmitted equally, so the force is multiplied by the ratio of the piston areas.

3

Calculate the travel distance of the output cylinder.

Output Distance = Input Distance / Area Ratio =

5 \text{ inches} / 5 = 1 \text{ inch}

.

Because fluid is incompressible, the volume displaced by the input piston equals the volume displaced by the output piston. A larger area requires proportionally less travel distance.

Key Concept

In a closed hydraulic system, moving from a smaller to a larger cylinder multiplies the force by the area ratio but decreases the travel distance by that same ratio, conserving the total work done.

Question 37Question →

Match each hydraulic jack scenario to its correctly calculated parameter based on Pascal's Principle.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Jack A matches to 160 lbs output force, Jack B matches to 3 inches output distance, Jack C matches to 4 in² input area, and Jack D matches to 30 psi fluid pressure.

Each hydraulic system operates on Pascal's Principle and the conservation of volume. By applying the formulas for pressure (

P = F/A

) and mechanical advantage (

MA = A_{out}/A_{in}

), the missing parameter for each jack can be calculated and uniquely matched.

Step-by-Step Solution

1

Calculate the output force for Jack A.

F_{out} = 160 \text{ lbs}

.

Using Pascal's principle, mechanical advantage is

12/3 = 4

.

F_{out} = 40 \times 4

.

2

Calculate the output distance for Jack B.

d_{out} = 3 \text{ inches}

.

Due to volume conservation,

d_{in} \times A_{in} = d_{out} \times A_{out}

.

15 \times 2 = d_{out} \times 10

.

3

Calculate the input area for Jack C.

A_{in} = 4 \text{ in}^2

.

Since pressure is equal across the system,

P = 180 / 24 = 7.5 \text{ psi}

.

A_{in} = 30 / 7.5

.

4

Calculate the fluid pressure for Jack D.

P = 30 \text{ psi}

.

Pressure equals force over area,

P = 45 / 1.5

.

Key Concept

Pascal's Principle, Fluid Volume Conservation, and Mechanical Advantage

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