---
title: "Buoyancy & Metacentric Stability Interactive Lab | Sandbox Physics"
description: "Can a floating hull still overturn? Yes. Floating only needs the weight to equal the buoyancy."
canonical_url: "https://sandboxphysics.com/labs/buoyancy-stability/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/buoyancy-stability/"
last_modified: "2026-10-27"
---

M092 · Fluid mechanics

# Buoyancy & Metacentric Stability

**Can a floating hull still overturn?** Yes. Floating only needs the weight to equal the buoyancy. Staying upright needs more: as the hull heels, buoyancy and weight must form a torque that pushes it back. Whether they do depends on the shape of the underwater volume and how high the ballast sits, and a hull that rights itself at small angles can have that torque reverse at large ones.

Move a ballast mass inside a sealed box hull or a flared prism hull. Compare the exact displaced geometry, the restoring curve at finite heel and the initial stability. Then infer the initial metacentric height from small-heel moments you record.

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

The hull is a watertight prism in still water. The heel angle is imposed, and the hull sits at the depth where its weight and buoyancy balance. Rolling motion, flooding and trim are not modeled, and nothing here certifies ship safety.

Physics tutorial

## Displacement, ballast and initial stability

**Background** This experiment tells you something only when the hull geometry, the forces on the hull and the recorded data agree.

**Why it matters** You link a floating hull to a quantitative task, finding its initial metacentric height, and then test where that approach works and where it fails.

Start with the essentials

- **Focus question**: Does floating mean stable?
- **One-sentence intuition**: Compare the full geometric model, its small-angle approximation and the separately recorded data.

### Core mathematical model

#### Vertical force balance

$$
\rho g\nabla=mg
$$

To float in equilibrium, the hull must displace enough water. A sealed hull that is overloaded cannot reach this balance.

#### Centre of the displaced volume

$$
\begin{aligned}\nabla&=LA_s\\\boldsymbol r_B&=\frac{1}{A_s}\int_{A_s}\boldsymbol r\,dA\end{aligned}
$$

Cut the tilted cross-section at the waterline, then integrate the submerged area and its centroid.

#### Restoring moment at any heel angle

$$
\begin{aligned}GZ&=x_B-x_G\\M_R&=mg\,GZ\end{aligned}
$$

For a floating hull, a positive moment resists a clockwise heel. A negative arm makes the heel worse.

#### Initial metacentric height

$$
\begin{aligned}BM&=\frac{I_{\mathrm{wp}}}{\nabla}\\GM&=KB+BM-KG\end{aligned}
$$

The second moment of the waterplane is taken about its centroid. This height describes only small angles and does not replace the curve for finite heel.

#### Initial slope measured from raw data

$$
\begin{aligned}\frac{M_i}{mg}&\simeq c+a\theta_i+b\theta_i^2+d\theta_i^3\\\widehat{GM}&=a\end{aligned}
$$

Angles are in radians. Fit a cubic with a free offset over four degrees, using the raw moments and the separately known mass.

### Common difficulties

#### A reference curve is not a measurement

**Typical misconception** The reference curve on screen is a measured result.

**Better mental model** Only the sensor channels you record count as observations, and changing the setup clears them. The perturbations are deterministic, so they give no statistical certification.

### Run the experiment

1. 01

   #### Move weight down

   Compare Low ballast with High ballast, unstable.

   **What to observe:** Both hulls float, but near upright their restoring curves slope in opposite directions.

2. 02

   #### Look beyond the initial slope

   Drag the gold heel selector out to large angles.

   **What to observe:** The curve from the full geometry moves away from the small-angle sine approximation.

3. 03

   #### Move the ballast sideways

   Select Off-centre ballast and look at zero heel.

   **What to observe:** The initial stability can be positive while the upright hull still feels a nonzero moment.

4. 04

   #### Measure first, then infer

   Acquire the small-heel sweep, add perturbations and export the CSV.

   **What to observe:** The slope estimate comes from the raw moments. Clear them and the estimate disappears.

Physical conclusion

Neither a small-angle approximation nor a polished scene can stand in for the finite-angle geometry, an independent reference check and raw measurements.

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)