---
title: "Center of Mass & Tipping Interactive Lab | Sandbox Physics"
description: "When an object could tip over any of several edges, which one goes first? The edge that needs the smallest lift of the mass center."
canonical_url: "https://sandboxphysics.com/labs/center-of-mass-tipping/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/center-of-mass-tipping/"
last_modified: "2026-10-27"
---

M022 · Rigid-body statics

# Center of Mass & Tipping

**When an object could tip over any of several edges, which one goes first?** The edge that needs the smallest lift of the mass center. Tipping about an edge raises the mass center until it sits directly above that edge, and only then does the body fall over. The edge with the smallest rise is the lowest barrier, so the body tips over it first.

Rearrange a rigid assembly of boxes and find its center of mass by hanging it from different points. Then compare how far the center sits inside the base with the energy barrier that gravity puts in the way of tipping.

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

The boxes are uniform, the floor is horizontal, nothing slides, and the rigid brackets have negligible mass. After release the body turns about one edge until it falls back onto its base or a box first touches the floor somewhere new; motion after that contact is not modeled. Hanging trials use a separate fixture whose bearing turns in one plane only.

Physics tutorial

## A mass center defines both stability and a barrier

**Background** A body built from parts can tip over even when every part on its own is symmetric.

**Why it matters** MIT's notes on the center of mass of extended objects and OpenStax's section on stability connect the way a body hangs with the way it rests on its base.

Start with the essentials

- **Focus question**: When an object could tip over any of several edges, which one goes first?
- **One-sentence intuition**: While the center of mass lies over the base, tilting the body outward first lifts the center of mass.

### Core mathematical model

#### The center of mass is a mass-weighted average

$$
\mathbf r_C=\frac{\sum_i m_i\mathbf r_i}{\sum_i m_i}
$$

The sum uses the center of every uniform box, the fixed base and mast included. The brackets have negligible mass, so they are left out.

#### Staying over the base

$$
|x_C|\leq\frac w2,\qquad |z_C|\leq\frac d2
$$

All four edges count. The Lab tips the body about the edge with the smallest signed margin.

#### Potential energy while tilting about an edge

$$
U(\phi)=Mg\left[h\cos\phi+\ell\sin\phi\right]
$$

Height is measured from the floor. The margin is positive when the center lies inside the chosen edge, and a positive angle tips the body outward.

#### The energy barrier to tipping

$$
\phi_* =\arctan\frac\ell h,\qquad\Delta U=Mg\left(\sqrt{h^2+\ell^2}-h\right)
$$

With a positive margin, the potential energy peaks when the center passes directly above the edge. With zero or negative margin there is no barrier to tipping outward.

#### Hanging angles locate the center

$$
(q_C-a_q)\cos\alpha+(h_C-a_h)\sin\alpha=0
$$

Each hanging trial, with a known eye and a measured angle, gives one line that the center must lie on. Trials in two vertical planes together fix all three coordinates by least squares.

### Common difficulties

#### Height changes how easily it tips

**Typical misconception** Two bodies with the same base and the same center position seen from above need the same energy to tip.

**Better mental model** With the same mass and margin, a higher center of mass lowers the energy barrier.

#### Hanging in one plane cannot find depth

**Typical misconception** Two plumb lines in the side-to-side plane are enough to find all three coordinates.

**Better mental model** To find depth, you need trials in a second vertical plane.

#### Floor contact changes the motion

**Typical misconception** The body keeps turning about the edge, right through the floor.

**Better mental model** The model stops when a box first touches the floor somewhere new and counts the energy absorbed there.

### Run the experiment

1. 01

   #### Move mass in two directions

   Change Payload lateral position and Payload depth position and watch the base footprint.

   **What to observe:** The nearest edge can change, even when the body looks stable from the side.

2. 02

   #### Measure the center of mass

   Hang the body from each eye, or press Measure all four eyes, and compare the fitted center with the model center.

   **What to observe:** The measured lines cross near the model center; noise leaves a residual you can measure.

3. 03

   #### Tip past the peak

   Choose Tall and vulnerable and release it from tilts below and above the barrier angle.

   **What to observe:** Released at rest below the peak, the body falls back onto its base; above the peak it tips outward.

4. 04

   #### Start already unstable

   Choose Lateral overhang or Depth overhang and release with zero tilt.

   **What to observe:** With a negative margin, gravity turns the body outward with no push needed to start it.

Physical conclusion

Keep three questions apart: how far the center sits inside the base, how much energy it takes to tip, and where the body first touches the floor again.

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)