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Difficulty: MediumPascal's Principle and Hydraulics

A hydraulic rescue tool, often used by emergency responders to pry open crushed vehicles, operates based on Pascal's Principle. The tool's pump applies 200 lbs200\text{ lbs} of force to an input piston that has a cross-sectional area of 0.5 in20.5\text{ in}^2. In order to generate 4,000 lbs4,000\text{ lbs} of spreading force at the output piston, what must be the cross-sectional area of the output piston?

  1. 10 in210\text{ in}^2Answer
  2. B
    20 in220\text{ in}^2
  3. C
    40 in240\text{ in}^2
  4. D
    0.025 in20.025\text{ in}^2

Answer

The cross-sectional area of the output piston must be 10 in210\text{ in}^2.
In a hydraulic system, pressure is transmitted equally in all directions based on Pascal's Principle. First, determine the system pressure generated by the input piston by dividing the applied force by the area: 200 lbs/0.5 in2=400 psi200\text{ lbs} / 0.5\text{ in}^2 = 400\text{ psi}. Since this same pressure acts on the output piston, the required area to produce 4,000 lbs4,000\text{ lbs} of lifting force is found by dividing the output force by the pressure: 4,000 lbs/400 psi=10 in24,000\text{ lbs} / 400\text{ psi} = 10\text{ in}^2. Alternatively, you can identify that the output force is 20 times the input force, meaning the output area must be 20 times the input area: 0.5 in2×20=10 in20.5\text{ in}^2 \times 20 = 10\text{ in}^2.

Step-by-Step Solution

1
Calculate the fluid pressure generated by the input piston using the formula Pressure = Force / Area.
P=200 lbs0.5 in2=400 psiP = \frac{200\text{ lbs}}{0.5\text{ in}^2} = 400\text{ psi}
Pascal's Principle states that pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid, so the output piston experiences this same pressure.
2
Use the calculated pressure to find the required area of the output piston using the rearranged formula Area = Force / Pressure.
A2=4,000 lbs400 psi=10 in2A_2 = \frac{4,000\text{ lbs}}{400\text{ psi}} = 10\text{ in}^2
Dividing the desired output force by the uniform system pressure yields the necessary cross-sectional area to achieve that force.

Key Concept

Pascal's Principle (Hydraulic Multiplication)
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Topics

Pascal's Principle and Hydraulics

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