Hydraulic lift

[Obi Wan]

I've been working on the Kaplan topical for fluids and solids, and this discrete showed up:

In a hydraulic lift, piston A has a cross sectional diameter of 2 m while piston B has a cross sectional diameter of 4 m. If the force exerted by Piston A on the liquid is doubled, the force lifting up Piston B is:

A) Decreased by a factor of 4
B) Increased by a factor of 2
C) Increased by a factor of 4
D) Increased by a factor of 8

I marked D), but the correct answer is B).

My reasoning:

F1 / A1 = F2 / A2

A1 = pi; A2 = 4pi

4F1 = F2; if F1 is doubled, then F2 should increase by 8.

The answer explanation states that the areas of the pistons don't change, so the Force on A is directly proportional to the Force on B.

What am I doing wrong? Shouldn't the equation agree with the solution?

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[Akarat]

I've been working on the Kaplan topical for fluids and solids, and this discrete showed up:

In a hydraulic lift, piston A has a cross sectional diameter of 2 m while piston B has a cross sectional diameter of 4 m. If the force exerted by Piston A on the liquid is doubled, the force lifting up Piston B is:

A) Decreased by a factor of 4
B) Increased by a factor of 2
C) Increased by a factor of 4
D) Increased by a factor of 8

I marked D), but the correct answer is B).

My reasoning:

F1 / A1 = F2 / A2

A1 = pi; A2 = 4pi

4F1 = F2; if F1 is doubled, then F2 should increase by 8.

The answer explanation states that the areas of the pistons don't change, so the Force on A is directly proportional to the Force on B.

What am I doing wrong? Shouldn't the equation agree with the solution?

Here's what I did:

F1 / A1 = F2 / A2

A1 = pi(r)^2
A2 = pi(2r)^2

Therefore, in order for the equation to still remain true, F1 must be multiplied by 1/4. So, now I have:

(1/4)F1 / A1 = F2 / (4)A2

Now, if I double the force exerted on piston A, I will have:

(1/2)F1 / A1 = F2 / (4)A2

In order to for the equation to remain true, F2 must be doubled in order to get 2/4, which will reduce to 1/2. I'll end up with:

(1/2)F1 / A1 = (1/2)F2 / A2

 

[Obi Wan]

That makes sense. Thanks Akarat!