In a specialized heavy-duty hydraulic system, the master cylinder has a piston diameter of… — Armed Services Vocational Aptitude Battery (ASVAB)

In a specialized heavy-duty hydraulic system, the master cylinder has a piston diameter of $2 \text{ cm}$ and the lifting cylinder has a piston diameter of $10 \text{ cm}$ . If the master cylinder piston is pushed through a total stroke distance of $75 \text{ cm}$ , how far, in cm, will the lifting cylinder piston move?

Answer: 3 cm

Answer

In a closed hydraulic system, the volume of fluid displaced by the input piston must equal the volume of fluid that moves the output piston. Because volume is the product of the piston's cross-sectional area and its stroke distance (

V = A \times d

), and area is proportional to the square of the diameter (

A \propto D^2

), the distance moved is inversely proportional to the square of the diameter. The lifting cylinder has a diameter

5

times larger than the master cylinder (

10 / 2 = 5

), which means its area is

25

times larger (

5^2 = 25

). Therefore, to conserve fluid volume, the lifting piston will move

1/25

of the distance of the master piston. Dividing the master piston's distance of

75 \text{ cm}

25

gives an output distance of

3 \text{ cm}

Step-by-Step Solution

State the principle of volume conservation for incompressible fluids.

V_{in} = V_{out}

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

Express volume in terms of piston diameter and stroke distance.

V \propto D^2 \times d

Volume equals cross-sectional area times distance, and area is proportional to the square of the diameter.

Set up the equation using the given values.

2^2 \times 75 = 10^2 \times d_{out}

Substituting

D_{in} = 2

d_{in} = 75

, and

D_{out} = 10

allows us to solve for the unknown output distance.

Simplify and solve for the output distance.

300 = 100 \times d_{out} \rightarrow d_{out} = 3

Basic algebra isolates the variable to find the final distance.

Key Concept

Hydraulic volumetric conservation and area-distance tradeoffs

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