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Difficulty: MediumPascal's Principle and Hydraulics

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 in22 \text{ in}^2 and the lifting cylinder has a cross-sectional area of 20 in220 \text{ in}^2. To raise the truck bed, the lifting cylinder must extend a distance of 8 inches8 \text{ inches}. Assuming an ideal closed hydraulic system, what is the total distance the pump cylinder must travel to achieve this lift?

  1. 80 inches80 \text{ inches}Answer
  2. B
    0.8 inches0.8 \text{ inches}
  3. C
    160 inches160 \text{ inches}
  4. D
    10 inches10 \text{ inches}

Answer

The input pump cylinder must travel a total distance of 80 inches80 \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×dV = A \times d). Setting the volumes equal gives 2×d1=20×82 \times d_1 = 20 \times 8. Solving this equation shows that the input cylinder must travel 80 inches80 \text{ inches} to move the larger cylinder 8 inches8 \text{ inches}.

Step-by-Step Solution

1
Identify the given values for the hydraulic system.
A1=2 in2A_1 = 2 \text{ in}^2, A2=20 in2A_2 = 20 \text{ in}^2, and d2=8 inchesd_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.
A1×d1=A2×d2A_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 (d1d_1).
2×d1=20×8⇒2×d1=160⇒d1=80 inches2 \times d_1 = 20 \times 8 \Rightarrow 2 \times d_1 = 160 \Rightarrow d_1 = 80 \text{ inches}.
Isolating d1d_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.
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Topics

Pascal's Principle and Hydraulics

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