Domino Chain Simulator

The motion of a single domino rotating about its base is governed by the rotational version of Newton's second law. The angular acceleration depends on the net torque applied.

Here, $I = \frac{mh^2}{3}$ is the moment of inertia for a thin rod rotating about its end (a good model for a domino), $\tau_g = mg\frac{h}{2}\sin\theta$ is the torque due to gravity pulling on the center of mass, and $- c\dot{\theta}$ is a small damping torque that slows motion, representing air resistance and friction.

The chain reaction is powered by the transfer of angular momentum during the brief collision. When a falling domino strikes its neighbor, it delivers an impulse that provides the initial "kick" to start the next domino's rotation.

$\Delta L$ is the change in angular momentum of the struck domino. This impulse depends on the impact speed and the geometry—specifically, where on the domino the hit occurs. This is why the spacing between dominoes is a critical parameter you can experiment with.

Rube Goldberg Machines & Educational Demonstrations: Domino chains are classic examples of energy transfer and chain reactions. They are used in physics classrooms to visually demonstrate concepts of potential/kinetic energy, momentum transfer, and wave propagation in a tangible way.

Modeling Cascading Failures: The domino effect is a powerful metaphor and mathematical model for cascading failures in networks, such as electrical grid blackouts, financial market collapses, or the spread of information (and misinformation) in social networks.

Entertainment & Art: Large-scale domino topples are staged for world records and artistic performances. Planning these requires understanding the physics to ensure reliable propagation, choose correct spacing, and create visual patterns like waves and spirals.

Conceptual Engineering Design: While not a direct CAE tool, the simulator's underlying physics—rigid body dynamics, collision detection, and impulse resolution—are foundational for professional engineering software used in crash testing, robotics, and mechanism design.

First, you might think that "the narrower the spacing, the faster the chain reaction," but in fact, there is an optimal spacing. For example, if the spacing is extremely narrow relative to the tile's height (e.g., less than 10% of the height), a falling tile will hit the top of the adjacent tile, resulting more in a "pushing in" motion rather than a clean push-over. This fails to transmit effective torque (as $r_{\perp}$ becomes smaller) and can actually slow down or even stop the chain. As a guideline, a spacing of about 20-30% of the tile's height often allows for the most efficient transfer of angular momentum.

Next, even if you set the "coefficient of friction" to zero in the simulation, this is impossible in reality. If the friction with the table were completely zero, the tile's pivot point would slip as it falls, preventing a clean rotational motion. This tool assumes infinite friction with the table (no slipping), allowing you to observe pure rotational motion. The pitfall is that when reproducing this with physical objects, you need to use non-slip mats or similar to approximate this condition.

Finally, note that "increasing gravity g does not make the chain infinitely faster." While the angular acceleration of the fall is proportional to g, the collision moment cannot be ignored. If g is increased too much, the falling speed becomes so high that upon collision, tiles may bounce back or generate impact forces large enough to damage the next tile. Since the simulation does not model "destruction," you might observe unnatural bouncing. In practical chain design, it's crucial to identify the range of g that allows for stable propagation.