---
title: "Tidal Field Explorer Interactive Lab | Sandbox Physics"
description: "One pull. Two outward sides. Why do tides stretch both the near side and the far side?"
canonical_url: "https://sandboxphysics.com/labs/tidal-field/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/tidal-field/"
last_modified: "2026-10-18"
---

M080 · Differential gravity

# Tidal Field Explorer

Move a probe over a three-dimensional globe. Subtract the acceleration of its center and see why gravity stretches both ends while squeezing the sides.

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

Exact companion point-mass field; linear equilibrium surface response shown with explicit magnification. A static gravity experiment, not an ocean forecast or disruption simulation.

Physics tutorial

## Subtract the common acceleration

**Background** A nearby mass accelerates the whole globe, but different parts do not fall equally fast.

**Why it matters** Connect a spatial picture to a quantitative model you can test.

Start with the essentials

- **Focus question**: Why does the far side stretch away while gravity still pulls inward?
- **One-sentence intuition**: Tides depend on differences in gravity, not on the common acceleration.

### Core mathematical model

#### Exact differential acceleration

$$
\Delta\mathbf a(\mathbf r)=\mu_2\left[\frac{d\hat{\mathbf e}-\mathbf r}{|d\hat{\mathbf e}-\mathbf r|^3}-\frac{\hat{\mathbf e}}{d^2}\right]
$$

The second term is the center acceleration. Both absolute pulls still point toward the companion.

#### Far-field tensor

$$
\Delta\mathbf a_{\rm lin}=\frac{\mu_2}{d^3}\left[3(\mathbf r\cdot\hat{\mathbf e})\hat{\mathbf e}-\mathbf r\right]
$$

Its three eigenvalues are proportional to two, minus one and minus one. The trace vanishes outside the point source.

#### Equilibrium surface, with an explicit display gain

$$
U_t=-\frac{\mu_2}{|d\hat{\mathbf e}-\mathbf r|}+\frac{\mu_2}{d}+\frac{\mu_2\mathbf r\cdot\hat{\mathbf e}}{d^2},\quad h=-\frac{U_t}{g_0},\quad r_{\rm drawn}=R+G_hh
$$

The first-order surface response is evaluated on the undeformed unit sphere. Changing the gain does not change the physical height or force.

### Common difficulties

#### Outward is relative

**Typical misconception** The far-side gravity points away from the companion.

**Better mental model** Switch to absolute gravity. The far side is attracted toward the companion, more weakly than the center.

### Run the experiment

1. 01

   #### Two outward sides

   Compare Near side and Far side using Differential gravity.

   **What to observe:** Both radial components are positive even though both absolute pulls point toward the companion.

2. 02

   #### Squeezing sideways

   Choose Sideways, then drag the probe to another latitude.

   **What to observe:** The differential field compresses transverse directions. The force diagram and probe readouts agree.

3. 03

   #### Test distance and display gain

   Double the distance. Compare Linear approximation, then change only the surface magnification.

   **What to observe:** The leading tide weakens eightfold. Linear-field error shrinks; display magnification leaves physical readouts unchanged.

Physical conclusion

The inverse-cube approximation is useful far away; compare it against the exact field before trusting it nearby.

Primary sources: [NOAA · Tides](https://www.noaa.gov/jetstream/ocean/tides)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Measure gravity, split motion into components, add contact forces and collisions, then carry the same laws into 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)