Cylinder piston surface area is a big deal in hydraulics; it influences the system pressure, the cylinder force and its speed. This puzzle provides several cap-end cylinder calculation challenges for you to sharpen your skills on.

- For the sake of these calculations round 𝛑 to 3.14.
- All calculations will refer to the cap end chamber of the cylinder(s).

A lumber stacking system uses 2 cylinders of the same diameter to lift a completed stack to the conveyor belt above. The system pressure gauge reads 51 psi when the load weighs 1000 lbs.

The 2 vertical cylinders that power a shop lift each have a diameter of 2.25 in . The empty lift arms weigh 50 lbs in total.

Want to see how we suggest solving these problems? Read on for a detailed walkthrough of each, or skip directly to the end of the lesson.

The question tells us that there is a single 3 in diameter cylinder, and that the load is 1413 lb.

- The formula to find the surface area of a circle.

𝛑 × radius^{2}

- The FPA triangle, solving for pressure.

To use the FPA equation, we need the piston surface area. To find that, we need to start by finding the radius from the diameter.

- Radius = Diameter ÷ 2
- Radius = 3 ÷ 2
- Radius =
**1.5 in**

Now that we have the radius, we can use it to find the surface area of the piston.

- Surface Area = 𝛑 × radius
^{2} - Surface Area = 3.14 ×
**1.5**^{2} - Surface Area =
**7.069 in**^{2}

We can now use the surface area in the FPA equation to find the pressure.

- Pressure = Force ÷ Surface Area
- Pressure = 1413 ÷
**7.069** - Pressure =
**199.887 psi**

(Rounds to 200 psi)

We've been told in the question that the single cylinder has a 3.5 in diameter piston, and that it can apply 24 040 lbs of pressure.

- The formula to find the surface area of a circle.

- The FPA triangle, solving for pressure.

First, we'll need to find the surface area the force is being exerted against. In the case of our press, it's the piston.

Find the radius from the diameter.

- Radius = Diameter ÷ 2
- Radius = 3.5 ÷ 2
- Radius =
**1.75 in**

Use the radius to calculate the surface area.

- Surface Area = 𝛑 × radius
^{2} - Surface Area = 3.14 ×
**1.75**^{2} - Surface Area =
**9.616**in^{2}

Next, divide the force by that surface area.

- Pressure = Force ÷ Surface Area
- Pressure = 24 040 ÷
**9.616** - Pressure =
**2500 psi**

We've been told in the question that there are 2 cylinders, and that the system registers 51 psi when lifting a load of 1000 lbs.

- The formula to find the surface area of a circle.

Permutation to find the radius

- The FPA triangle, solving for pressure.

To find the combined surface area of the cylinder pistons, we'll need to use the FPA equation.

- Surface Area = Force ÷ Pressure
- Surface Area = 1000 ÷ 51
- Surface Area =
**19.608 in**^{2}

In the last question we calculated the total surface area for both pistons. Let's start by dividing it in half, to get the surface area for a single piston.

- SA
_{1}= SA_{total}÷ 2 - SA
_{1}=**19.608**÷ 2 - SA
_{1}=**9.804 in**^{2}

We'll need to re-arrange the 𝛑r^{2} equation to get the radius, and then the diameter.

- Radius = √(Surface Area ÷ 𝛑)
- Radius = √(
**9.804**÷ 3.14) - Radius =
**1.767 in**

Now that we have the radius, we can use it to calculate the diameter.

- Diameter = Radius × 2
- Diameter =
**1.767**× 2 - Diameter =
**3.534 in**

(Rounds to 3.5 in)

We've been told in the question that there are 2 cylinders, each with a diameter of 2.25 in, and that the empty lift arms weight 50 lbs total.

- The formula to find the surface area of a circle.

Since we're working with 2 matched cylinders, the surface area will be doubled.

(𝛑 × radius- The FPA triangle, solving for pressure.

Force ÷ Surface Area = Pressure

First, we'll need to find the surface area the force is being exerted against. In the case of our press, it's the piston.

Find the radius from the diameter.

- Radius = Diameter ÷ 2
- Radius = 2.25 ÷ 2
- Radius =
**1.125 in**

Use the radius to calculate the surface area.

- Surface Area = 𝛑 × radius
^{2} - Surface Area = 3.14 ×
**1.125 in**^{2} - Surface Area of a single cylinder =
**3.974**sq. in

Double the surface area to include both pistons.

*SA*_{total}= SA_{1}+ SA_{2}- SA
_{total}=**3.974**+**3.974** - SA
_{total}=**7.948 in**^{2}

We know that the empty arms weigh 50 lbs, and that they are lifting a 50 lb weight. Added together, we are working with a force of **100** lbs.

Divide the force by the total surface area to get the system pressure.

- Pressure = Force ÷ Surface Area
- Pressure =
**100**÷**7.948** - Pressure =
**12.582**

(Rounds to 13 psi)

We need to determine whether lifting 10 050 lbs (the 10 000 lbs weight and the 50 lbs added by the arms) will induce a system pressure near or over 1600 psi.

Recall that we've already calculated the total surface area: **7.948** in^{2}, so all that's left to do is to solve for pressure.

- Pressure = Force ÷ Surface Area
- Pressure = 10 050 ÷
**7.948** - Pressure =
**1264.469 psi**

**1264 psi** is less than 1600 psi, so this system should be able to lift the 10 000 lb weight.

You've solved all of the cap-end puzzles. Are you ready to go on to some rod-end puzzles?

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