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Zorluk: OrtaPascal's Principle and Hydraulics

A barber chair uses a closed hydraulic lift mechanism to raise customers. The barber operates a foot pedal connected to an input cylinder with a cross-sectional area of 4 in24 \text{ in}^2. The chair is mounted on a larger output cylinder with a cross-sectional area of 80 in280 \text{ in}^2. If the total weight of the chair and the customer is 320 lbs320 \text{ lbs}, how much downward force must the barber apply to the input cylinder to lift the load?

  1. A
    4 lbs4 \text{ lbs}
  2. 16 lbs16 \text{ lbs}Cevap
  3. C
    1280 lbs1280 \text{ lbs}
  4. D
    6400 lbs6400 \text{ lbs}

Cevap

The barber must apply 16 lbs16 \text{ lbs} of force.
The correct answer applies Pascal's Principle, which indicates that the mechanical advantage of a hydraulic system is determined by the ratio of the piston areas. The output cylinder is 2020 times larger than the input cylinder (80/4=2080 / 4 = 20). Therefore, the input force only needs to be 1/201/20 th of the output load. Dividing the 320 lbs320 \text{ lbs} load by 2020 yields an input force requirement of 16 lbs16 \text{ lbs}.

Adım Adım Çözüm

1
Identify the given values for the hydraulic system.
Input area (A1A_1) = 4 in24 \text{ in}^2, Output area (A2A_2) = 80 in280 \text{ in}^2, Output force/load (F2F_2) = 320 lbs320 \text{ lbs}.
Organizing known variables is the first step in solving a Pascal's Principle problem.
2
State Pascal's Principle equation relating force and area.
F1A1=F2A2\frac{F_1}{A_1} = \frac{F_2}{A_2}
Pascal's Principle dictates that pressure is transmitted equally throughout a confined incompressible fluid.
3
Rearrange the formula to solve for the unknown input force (F1F_1).
F1=F2×(A1A2)F_1 = F_2 \times \left(\frac{A_1}{A_2}\right)
Isolating F1F_1 prepares the equation for direct substitution.
4
Substitute the known values and calculate.
F1=320×(480)=320×(120)=16 lbsF_1 = 320 \times \left(\frac{4}{80}\right) = 320 \times \left(\frac{1}{20}\right) = 16 \text{ lbs}.
Performing the arithmetic reveals the exact force required at the pedal.

Anahtar Kavram

Pascal's Principle states that a change in pressure at any point in an enclosed fluid is transmitted equally throughout the fluid. In a hydraulic system, a small force applied over a small area can lift a large weight over a proportionally larger area.
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Konular

Pascal's Principle and Hydraulics

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