---
title: "Conservative Forces & Path Dependence Interactive Lab | Sandbox Physics"
description: "Does a force do the same work along any path between the same two points? Only if the force is conservative."
canonical_url: "https://sandboxphysics.com/labs/path-dependent-work/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/path-dependent-work/"
last_modified: "2026-10-27"
---

M043 · Work and energy

# Conservative Forces & Path Dependence

**Does a force do the same work along any path between the same two points?** Only if the force is conservative. In a swirling force field the work depends on the path, and running a loop backward flips its sign. Friction never flips: it takes energy whichever way you go. One loop that happens to give zero work does not prove a potential energy exists; that needs zero work around every loop.

Edit two paths that share the same endpoints. Separate the work done by the potential part of the force, the circulation of its rotational part and the loss to sliding. Then compare loop work computed from recorded force samples with the value predicted from the loop's geometry.

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

The paths are prescribed. Field work and sliding dissipation are computed separately. The drive and the contact dynamics are not modeled.

Physics tutorial

## Conservative Forces & Path Dependence

**Background** Two routes between the same points show the difference between work that depends only on the endpoints and work that depends on which way you go around a loop.

**Why it matters** Edit two paths between the same endpoints and pull apart three contributions: work from the potential, circulation of the rotational part of the field and loss to sliding. Then test recorded force samples against the loop work predicted from geometry.

Start with the essentials

- **Focus question**: Does the same destination require the same work?
- **One-sentence intuition**: Zero work around one loop does not prove that a potential exists everywhere.

### Core mathematical model

#### A force with conservative and rotational parts

$$
\begin{aligned}\mathbf F&=(-kx-\gamma y,\,-G-ky+\gamma x)\\U_c&=\tfrac12k(x^2+y^2)+Gy\end{aligned}
$$

Only the spring force and the uniform downward force come from this potential.

#### Work as an integral along the path

$$
W_C=\int_0^1\mathbf F(\mathbf c(q))\cdot\mathbf c^{\prime}(q)\,\mathrm dq
$$

Only the force component along the path's tangent contributes. The path parameter is not time; the apparatus is driven along the selected path.

#### Loop orientation and curl

$$
\begin{aligned}A_{\mathrm{or}}&=\tfrac12\oint(x\,\mathrm dy-y\,\mathrm dx)\\\oint\mathbf F\cdot\mathrm d\mathbf r&=2\gamma A_{\mathrm{or}}\end{aligned}
$$

Going forward along the first path and back along the second sets the orientation. Where the loop crosses itself, each part counts with the sign of its winding.

#### Dissipation does not reverse sign

$$
\begin{aligned}W_f&=-fL\\W_{f,\mathrm{loop}}&=-f(L_A+L_B)\end{aligned}
$$

Resistance depends on the direction of travel, so its loop work is not the curl integral of any field that depends only on position.

#### Work estimated from recorded forces

$$
\widehat W_C=\sum_i\frac{\mathbf F_i^{\mathrm{obs}}+\mathbf F_{i+1}^{\mathrm{obs}}}{2}\cdot(\mathbf r_{i+1}-\mathbf r_i)
$$

In this simulated acquisition the positions are exact, and the field-force components carry bounded deterministic noise.

### Common difficulties

#### Reversing a loop

**Typical misconception** If I go around the loop the other way, every work term flips sign.

**Better mental model** Work by a field that depends only on position does flip. Sliding resistance turns to oppose the new direction of motion, so it still takes energy out.

### Run the experiment

1. 01

   #### Keep the endpoints fixed

   Edit both internal points on each route.

   **What to observe:** The work of the potential part stays the same, even when the path length changes.

2. 02

   #### Add circulation to the field

   Choose Rotational field and change its sign.

   **What to observe:** The field work around the closed loop follows the oriented area and changes sign when the loop is reversed.

3. 03

   #### Add resistance

   Set a sliding resistance, then press Reverse both routes.

   **What to observe:** The field term changes sign, while the resistance term stays negative.

4. 04

   #### Measure the loop work

   Acquire the field forces, increase the scan intervals and compare with the model.

   **What to observe:** The measured loop work is a sum of force times displacement. The resistance is still calculated separately.

Physical conclusion

Reverse the loop and the field's circulation changes sign, but the loss to sliding does not.

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)