---
title: "Hydrostatics & Hydraulic Press Interactive Lab | Sandbox Physics"
description: "Does a hydraulic press give out more work as well as more force? No more work comes out than goes in."
canonical_url: "https://sandboxphysics.com/labs/hydrostatic-press/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/hydrostatic-press/"
last_modified: "2026-10-27"
---

M091 · Fluid mechanics

# Hydrostatics & Hydraulic Press

**Does a hydraulic press give out more work as well as more force?** No more work comes out than goes in. At the same pressure a bigger piston feels a bigger force, but it moves a correspondingly shorter distance. A piston with ten times the area pushes ten times as hard and moves a tenth as far, so the work is the same.

Move a pressure probe through a cylindrical or flared reservoir and work out the liquid density from the pressures you acquire. Then operate two connected pistons and see how force, stroke and the liquid's gravitational energy trade off.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The liquid is uniform and incompressible, and every state is in quasi-static balance. Vessel dimensions are real lengths in metres; fixtures are schematic. Sensor readings are simulated. Each piston stroke you set is a balanced state, not a lift in motion.

Physics tutorial

## Pressure gains force, not free work

**Background** The vessels hold a uniform liquid in constant gravity. All pressures are gauge pressures, measured relative to the surrounding air.

**Why it matters** Force balance and conservation of liquid volume predict how the liquid and pistons behave. The question is how much of that you can recover from simulated sensor readings.

Start with the essentials

- **Focus question**: Does a hydraulic press give out more work as well as more force?
- **One-sentence intuition**: Depth sets the pressure. Piston area sets the force, and the stroke scales inversely with area.

### Core mathematical model

#### Pressure grows with depth

$$
p(z)=p_s+\rho g(z_s-z)
$$

At a given depth the pressure is the same, whatever the shape of the vessel.

#### Both pistons sweep the same volume

$$
A_1s_1=A_2s_2
$$

The volume of liquid pushed down by the input piston raises the output piston by the same volume.

#### Output force when the pistons sit at different heights

$$
F_2=A_2\!\left[\frac{F_1}{A_1}+\rho g(z_1-z_2)\right]
$$

When the piston faces are at different heights, the force gain needs a correction for the liquid column between them.

#### The missing work raises the liquid

$$
W_1-W_2=\frac{\rho g A_1s_1^2}{2}\!\left(1+\frac{A_1}{A_2}\right)
$$

The two liquid columns start level. As the faces move apart in height, the liquid gains gravitational potential energy, and that is where the work difference goes.

#### Density from acquired pressure

$$
p_i=\alpha d_i+\beta,\qquad\widehat\rho=\widehat\alpha/g
$$

With known depths and gravity, the fitted slope of pressure against depth gives the density. The intercept takes up the surface pressure.

### Common difficulties

#### Acquire before inferring

**Typical misconception** The estimate just copies the value on a reference slider.

**Better mental model** Clear the observations and the estimate disappears. Add noise or change the reference parameter, then acquire again.

#### Know the approximation

**Typical misconception** If the apparatus looks smooth, every state it shows must be physically reliable.

**Better mental model** Check the domain status. Extrapolated states stay visible, but you cannot acquire data from them.

### Run the experiment

1. 01

   #### Change the vessel

   Compare cylindrical and flared presets at equal height.

   **What to observe:** The bottom pressure stays the same; the force from the walls balances the extra weight of liquid.

2. 02

   #### Move and acquire

   Drag the gold probe track, then acquire a depth profile.

   **What to observe:** The slope of the fitted pressure profile gives the density.

3. 03

   #### Operate the press

   Select the press preset and increase input stroke.

   **What to observe:** The output piston moves less, and part of the input work goes into raising the liquid.

4. 04

   #### Test the head correction

   Select the Elevation matters preset and look at the work budget residual.

   **What to observe:** As the piston heights move apart, the force predicted for equal heights drifts away from the actual force.

Physical conclusion

Acquired pressures and the model reference curve answer different questions. Use either one only where the approximation holds.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)