A heavy equipment mechanic is operating a closed hydraulic lifting system. The system's ma… — Armed Services Vocational Aptitude Battery (ASVAB)

A heavy equipment mechanic is operating a closed hydraulic lifting system. The system's master cylinder has a piston diameter of $2 \text{ inches}$ , while the lifting cylinder has a piston diameter of $10 \text{ inches}$ . If the mechanic applies an input force of $40 \text{ lbs}$ and pushes the master piston down by a distance of $25 \text{ inches}$ , what is the resulting output force and how far does the lifting piston move?

Output force of $1000 \text{ lbs}$ and a distance of $1 \text{ inch}$ Answer
B
Output force of $200 \text{ lbs}$ and a distance of $5 \text{ inches}$
C
Output force of $1000 \text{ lbs}$ and a distance of $5 \text{ inches}$
D
Output force of $200 \text{ lbs}$ and a distance of $125 \text{ inches}$

Answer

Output force of

1000 \text{ lbs}

and a distance of

1 \text{ inch}

The correct answer accurately applies Pascal's Principle by first recognizing that mechanical advantage depends on the cross-sectional areas of the pistons. Since the output diameter is 5 times larger than the input diameter, the output area is

5^2 = 25

times larger. This means the force is multiplied by 25 (

40 \times 25 = 1000 \text{ lbs}

). Because work is conserved, the output piston only moves

1/25

th of the distance the input piston moves (

25 / 25 = 1 \text{ inch}

Step-by-Step Solution

Calculate the ratio of the piston diameters.

Diameter ratio =

10 \text{ inches} / 2 \text{ inches} = 5

The diameter ratio is needed to find the area ratio, which dictates the mechanical advantage.

Calculate the ratio of the piston areas.

Area ratio =

5^2 = 25

According to Pascal's Principle, the mechanical advantage of a hydraulic system is equal to the ratio of the piston areas, not just their diameters. Area is proportional to the square of the diameter.

Calculate the output force.

Output force =

40 \text{ lbs} \times 25 = 1000 \text{ lbs}

The output force is the input force multiplied by the mechanical advantage (area ratio).

Calculate the output distance.

Output distance =

25 \text{ inches} / 25 = 1 \text{ inch}

To conserve the volume of the incompressible fluid, the distance the output piston moves must be inversely proportional to the area ratio. Therefore, the input distance is divided by the mechanical advantage.

Key Concept

Pascal's Principle and Mechanical Advantage in Hydraulics

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