A barber chair operates using a closed hydraulic lifting system to raise a customer. The i… — Armed Services Vocational Aptitude Battery (ASVAB)

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?

Answer: 0.5 inches

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

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.

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.

Calculate the total fluid volume displaced.

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

Multiplying area by distance gives the volume of fluid moved.

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).

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