Hong Life CalculatorsUnderstanding the hydraulic press calculator — concept and principle
A hydraulic press multiplies force by connecting a small piston and a large piston through a confined liquid. The same principle powers car brakes, hydraulic jacks, excavators and lifts. Given the input force and the two piston diameters, this calculator returns the output force, the fluid pressure and the force ratio.
How it is calculated
- Pressure: p = F₁ ÷ A₁, where A₁ = π (d₁ ÷ 2)² is the small piston area. Diameters are entered in mm and converted to metres; the result in pascals is divided by 100,000 to give bar.
- Output force: F₂ = F₁ × A₂ ÷ A₁ = F₁ × (d₂ ÷ d₁)², in newtons (N).
- Force ratio: (d₂ ÷ d₁)². Because area scales with the square of diameter, a piston 10 times wider has 100 times the area.
The principle
Pascal's law states that a pressure change applied to a confined incompressible fluid is transmitted undiminished to every part of the fluid and to the walls of its container. The pressure under the small piston, F₁ ÷ A₁, is the same as under the large piston, so the large piston receives F₂ = p × A₂.
Energy is still conserved. The liquid that leaves the small cylinder fills the large one, so A₁ × s₁ = A₂ × s₂: the large piston moves only A₁ ÷ A₂ as far. Force times distance is the same on both sides (ignoring friction), so a hydraulic press trades distance for force, just like a lever.
Worked example
- Inputs: F₁ = 100 N, d₁ = 20 mm, d₂ = 200 mm.
- A₁ = π × (0.01 m)² ≈ 3.14159 × 10⁻⁴ m².
- p = 100 ÷ 3.14159 × 10⁻⁴ ≈ 318,310 Pa = 3.183098862 bar.
- Force ratio = (200 ÷ 20)² = 10² = 100.
- F₂ = 100 × 100 = 10,000 N.
| Small piston diameter (mm) | Output force (N) |
|---|---|
| 5 | 160,000 |
| 10 | 40,000 |
| 15 | 17,777.77778 |
| 20 | 10,000 |
| 25 | 6,400 |
| 30 | 4,444.444444 |
| 40 | 2,500 |
How to read the result
10,000 N is the weight of about 1,020 kg, lifted with a push of about 10 kgf. The price is travel: to raise the large piston by 1 mm the small piston must move 100 mm. A pressure of 3.18 bar is modest; industrial hydraulics often run at about 100–350 bar, which is why they can be compact. Compare the pressure output with the rated pressure of the cylinders, hoses and seals.
Common mistakes
- Using the diameter ratio instead of its square, which understates the force by a factor of 10 here.
- Entering radius where diameter is expected.
- Expecting more output work than input work; friction and seal losses make real output slightly lower.
Frequently asked questions
Why do car brakes use hydraulics?
Pedal force on a small master cylinder becomes larger force at the wheel cylinders, and the fluid transmits it evenly to every wheel.
Does the pipe length or shape between pistons matter?
Not for the static force: pressure is the same throughout a fluid at rest at one level. In motion, long or narrow lines add flow losses.
How do I convert bar to other units?
1 bar = 100 kPa ≈ 14.5 psi, so 3.18 bar is about 46.2 psi.
Sources: Blaise Pascal, Traité de l'équilibre des liqueurs (published 1663) · Halliday, Resnick and Walker, Fundamentals of Physics, chapter on fluids