A mechanic uses a hydraulic jack to lift a 2,000-pound engine. The output piston supportin… — Armed Services Vocational Aptitude Battery (ASVAB)

A mechanic uses a hydraulic jack to lift a 2,000-pound engine. The output piston supporting the engine has a diameter that is exactly 4 times larger than the diameter of the input piston. Assuming an ideal system, if the mechanic pushes the input piston down a total distance of 16 inches, what is the required input force, and how far will the engine be lifted?

125 pounds of force; the engine is lifted 1 inchCevap
B
500 pounds of force; the engine is lifted 4 inches
C
125 pounds of force; the engine is lifted 256 inches
D
32,000 pounds of force; the engine is lifted 1 inch

Cevap

125 pounds of force; the engine is lifted 1 inch

In a hydraulic system, the mechanical advantage is determined by the ratio of the piston areas. Since the output piston's diameter is 4 times the input piston's diameter, its area is

4^2

, or 16 times larger. This provides a mechanical advantage of 16. To lift a 2,000-pound engine, the required input force is

2,000 \div 16 = 125

pounds. Furthermore, because work is conserved (

F_{in} \times d_{in} = F_{out} \times d_{out}

), the input piston must move 16 times farther than the output piston. Therefore, pushing the input piston down 16 inches will lift the engine

16 \div 16 = 1

inch.

Adım Adım Çözüm

Determine the area ratio between the two pistons.

Area Ratio =

4^2 = 16

In hydraulics, mechanical advantage depends on the cross-sectional area of the pistons. Because area scales with the square of the diameter (

A = \pi r^2

), a piston with 4 times the diameter has

4^2

, or 16 times, the area.

Calculate the required input force using Pascal's Principle.

Input Force =

2,000 \div 16 = 125

pounds.

The hydraulic system provides a mechanical advantage of 16. Therefore, the input force required is

1/16

th of the output load.

Calculate the distance the output piston (and engine) will move.

Output Distance =

16 \text{ inches} \div 16 = 1 \text{ inch}

By the principle of conservation of work (Work In = Work Out), if force is multiplied by 16, the distance traveled must be divided by 16 to keep the total volume of fluid displaced equal.

Anahtar Kavram

Pascal's Principle and Mechanical Advantage

Alternatif Yöntem

You can verify your answers using the principle of conservation of energy (Work In = Work Out). The work done by the input piston is

125 \text{ lbs} \times 16 \text{ inches} = 2,000 \text{ in-lbs}

. The work done on the engine is

2,000 \text{ lbs} \times 1 \text{ inch} = 2,000 \text{ in-lbs}

. Since the work values match, the force and distance calculations are correct.

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