Question

Pistons are fitted to two cylindrical chambers connected through a horizontal tube to form a hydraulic system. The piston chambers and the connecting tube are filled with an incompressible fluid. The cross-sectional areas of piston 1 and piston 2 are A1 and A2, respectively. A force F1 is exerted on piston 1. Rank the resultant force F2 on piston 2 that results from the combinations of F1, A1, and A2 given from greatest to smallest. If any of the combinations yield the same force, give them the same ranking. (Use only ">" or "=" symbols. Do not include any parentheses around the letters or symbols.)

F1 = 4.0 N; A1 = 0.9 m2; and A2 = 1.8 m2
F1 = 2.0 N; A1 = 0.9 m2; and A2 = 0.45 m2
F1 = 2.0 N; A1 = 1.8 m2; and A2 = 3.6 m2
F1 = 4.0 N; A1 = 0.45 m2; and A2 = 1.8 m2
F1 = 4.0 N; A1 = 0.45 m2; and A2 = 0.9 m2
F1 = 2.0 N; A1 = 1.8 m2; and A2 = 0.9 m2

Answer 1

Answer:

order   d> a = e> c> b = f

Explanation:

Pascal's law states that a change in pressure is transmitted by a liquid, all points are transmitted regardless of the form

      P₁ = P₂

Using the definition of pressure

      F₁ / A₁ = F₂ / A₂

      F₂ = A₂ /A₁   F₁

Now we can examine the results

a) F1 = 4.0 N A1 = 0.9 m2 A2 = 1.8 m2

     F₂ = 1.8 / 0.9 4

     F₂a = 8 N

b) F1 = 2.0 N A1 = 0.9 m2 A2 = 0.45 m2

    F₂b = 0.45 / 0.9 2

    F₂b = 1 N

c) F1 2.0 N A1 = 1.8 m2 A2 = 3.6 m2

    F₂c = 3.6 / 1.8 2

    F₂c = 4 N

d) F1 = 4.0N A1 = 0.45 m2 A2 = 1.8 m2

    F₂d = 1.8 / 0.45 4.0

    F₂d = 16 m2

e) F1 = 4.0 N A1 = 0.45 m2 A2 = 0.9 m2

   F₂e = 0.9 / 0.45 4

   F₂e = 8 N

f) F1 = 2.0N A1 = 1.8 m2 A2 = 0.9 m2

   F₂f = 0.9 / 1.8 2.0

   F₂f = 1 N

Let's classify the structure from highest to lowest

F₂d> F₂a = F₂e> F₂c> F₂b = F₂f

I mean the combinations are

 d> a = e> c> b = f

Answer 2

The resultant force on piston 2 is ranking in the following order,

F₂(d) > F₂(a) = F₂(e) > F₂(c) > F₂(b) = F₂(f)

Pressure transmitted in the fluid

The pressure transmitted at every point in the fluid is determined by applying pascal principle.

P = F/A

$\frac{F_1}{A_1} = \frac{F_2}{A_2}\\\\ F_2 = \frac{F_1A_2}{A_1}$

(a) When, F1 = 4.0 N; A1 = 0.9 m²; and A2 = 1.8 m²

$F_2 = \frac{4(1.8)}{0.9} \\\\F_2 = 8.0 \ N$

(b) When, F1 = 2.0 N; A1 = 0.9 m²; and A2 = 0.45 m²

$F_2 = \frac{2(0.45)}{0.9} \\\\F_2 = 1.0 \ N$

(c) When, F1 = 2.0 N; A1 = 1.8 m²; and A2 = 3.6 m²

$F_2 = \frac{2(3.6)}{1.8} \\\\F_2 = 4.0 \ N$

(d) When, F1 = 4.0 N; A1 = 0.45 m²; and A2 = 1.8 m²

$F_2 = \frac{4(1.8)}{0.45} \\\\F_2 = 16 \ N$

(e) When, F1 = 4.0 N; A1 = 0.45 m²; and A2 = 0.9 m²

$F_2 = \frac{4(0.9)}{0.45} \\\\F_2 = 8 \ N$

(f) When, F1 = 2.0 N; A1 = 1.8 m²; and A2 = 0.9 m²

$F_2 = \frac{2(0.9)}{1.8} \\\\F_2 = 1.0 \ N$

Ranking of the resultant force on piston 2

F₂(d) > F₂(a) = F₂(e) > F₂(c) > F₂(b) = F₂(f)

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