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Zorluk: Çok zorPascal's Principle and Hydraulics

A heavy machinery hydraulic lift is being redesigned to alter its performance characteristics. Match each proposed modification to the hydraulic pistons with its resulting effect on the output force and output piston travel distance. (Assume the input force and input piston stroke distance remain constant in all scenarios).

  • Decreasing the input piston diameter to half its original sizeOutput force increases by a factor of 4; output distance decreases to one-fourth.
  • Doubling the output piston diameter and doubling the input piston cross-sectional areaOutput force increases by a factor of 2; output distance decreases to one-half.
  • Tripling the diameter of both the input and output pistons simultaneouslyBoth the output force and output distance remain unchanged.
  • Doubling the input piston diameter while keeping the output piston unchangedOutput force decreases to one-fourth; output distance increases by a factor of 4.

Cevap

Match each structural modification by first converting any diameter changes to area changes (by squaring the factor), then calculating the new mechanical advantage ratio, and applying the inverse factor to the stroke distance to conserve work.
Each modification changes the ratio of the output piston area to the input piston area, which dictates the mechanical advantage (MA). Because hydraulic systems conserve work, any factor that multiplies the output force must inversely scale the output distance. Crucially, while area scales directly, changes given as diameter must be squared to find the effect on area.

Adım Adım Çözüm

1
Analyze the relationship between diameter and cross-sectional area for circular pistons.
Area is proportional to the square of the diameter (A∝D2A \propto D^2). Doubling a diameter quadruples the area, and halving a diameter reduces the area to one-fourth.
To correctly determine how geometric modifications affect the actual area used in Pascal's principle.
2
Calculate the new mechanical advantage (MA) for the first modification (decreasing input diameter to half).
Input area becomes 1/41/4 of the original. MA=AoutAinMA = \frac{A_{out}}{A_{in}}, so the new MA is 11/4=4\frac{1}{1/4} = 4 times the original. Output force becomes 4×4\times.
To match the first modification to its force multiplication effect.
3
Apply the principle of conservation of work to determine the effect on output distance for the first modification.
Since Work In = Work Out (Fin×din=Fout×doutF_{in} \times d_{in} = F_{out} \times d_{out}), an increase in output force by a factor of 4 must result in a proportional decrease in output distance to 1/41/4.
Pascal's principle dictates an inverse relationship between force multiplication and distance traveled.
4
Evaluate the remaining modifications using the same area scaling and work principles.
Modification 2 yields 4/2=2×4/2 = 2\times MA. Modification 3 yields 9/9=1×9/9 = 1\times MA. Modification 4 yields 1/4=1/4×1/4 = 1/4\times MA. Output distances scale inversely for each.
To correctly pair all remaining items based on their calculated force and distance multipliers.

Anahtar Kavram

Pascal's Principle, Mechanical Advantage, and Work Conservation
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Konular

Pascal's Principle and Hydraulics

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