Using volumetric pressure to counter a force

[TaylorTexas]

Homework Statement

There are two opposing cylinders, C₁ and C₂. Each cylinder is sealed. Each cylinder has a movable piston at one end. The pistons of each cylinder face each other. The pistons are connected to each other by a straight Shaft. C₁ is connected to an air supply with an initial air pressure of 150 psi. Over time, the air pressure in the air supply, and thus C₁, is increased to 157 psi.

The dimensions of C₁ is .5" radius and 5" length. the initial air pressure in C₂ is 150 psi. What are the dimensions of C₂ (Length and Radius), to allow the Shaft to move 4" in the direction of C₂ as the pressure increases 7 psi in C₁, with a minimum of volume?

Homework Equations

I have used Boyle's law for two cylinders: P₁V₁=P₂V₂

The Attempt at a Solution

As the pressure increases in C₁, the shaft begins to move right, decreasing the volume, and increasing the pressure in C₂. When the shaft moves 4", the pressure in C₁ and C2 are equalized and the shaft movement stops.

P₁V₁=P₂V₂
Inserting V=π * r₁² * h₁, I get
P1(π * r₁² * h₁)=P₂(π * r₂² * h₂)
Where P₁=157, r₁=.5", h₁=5, P₂=150, r₂=.4, I solve for h₂.
Thus, h₂=8.1.
So, I would need C₂ to be 8.1" long and .4" radius to allow 4" of shaft movement with a change of 7 psi in C₁?

 

[paisiello2]

I think you have mistakenly applied Boyle's law to two different gases in two different cylinders?

 

[PeterDonis]

I have used Boyle's law for two cylinders: P1V1=P2V2

As paisiello2 has pointed out, this does not seem correct. Boyle's law relates the pressure and volume of a fluid in a single cylinder (or other confined fluid) before and after some process takes place, provided that the temperature of the fluid does not change. It does not relate the pressure and volume in two different cylinders (or other separate quantities of fluid).

 

[PeterDonis]

Over time, the air pressure in the air supply, and thus C1, is increased to 157 psi.

A key assumption has been left out of the problem statement: that the temperature of everything stays the same. I'm assuming that was the intent; otherwise everything gets a lot more complicated.

What are the dimensions of C2 (Length and Radius), to allow the Shaft to move 4" in the direction of C2 as the pressure increases 7 psi in C1, with a minimum of volume?

That last qualifier, "with a minimum of volume", is important. Is that an exact quote? Is the intent that there should be a minimum change in volume in cylinder C2, or that cylinder C2 should have the minimum possible total volume at the end, consistent with the other quantities in the problem?

As the pressure increases in C1, the shaft begins to move right, decreasing the volume, and increasing the pressure in C2.

Yes, this looks correct.

When the shaft moves 4", the pressure in C1 and C2 are equalized and the shaft movement stops.

Yes, this looks ok so far. But the next step is key: if the shaft moved 4 inches, what does that tell you? Specifically, what does it tell you about cylinder C1?