Perusing Reddit I found someone asking about a hanging chain. The post in question has a semi-quality video of someone whirling a chain from one end in a circle. The rest of the freely swinging chain forms a particular shape as it rotates around some center axis. In short, this happens because of the centrifugal force from the rotation axis against the binding force of the chain links. I want to take some time to really understand this with physics, of course.

The Setup

Let’s model the chain links as $N$ linked springs. The top link is labeled $i = 1$ and the last one is labeled $i = N$. Each spring has a spring constant of $k$ and mass of $m$. The acceleration of gravity is $g$. Our coordinate system is the cylindrical coordinate system. Downward is the positive $z$ direction, and the horizontal directions are parameterized with $R$ and $\theta$. Now let’s write the Lagrangian!

\begin{equation} L = \int dt m \sum_{i=1}^{N} \left(\frac 12 \left( \dot z_i^2 + R^2 \dot \theta_i^2 + \dot R_i^2 \right) + g z_i\right) - \frac k2 \sum_{i=1}^{N-1} \left(\vec x_{i+1} - \vec x_{i}\right)^2 \end{equation}

Here $\vec x_i$ is just the position vector of the $i$‘th spring.

The Equations of Motion…

We can vary the $3N$ variables, leading to $3N$ equations. But in reality, we are only concerned with a special configuration. For our case, we assume $\dot z_i = \dot R_i = 0$ and $\dot \theta_i = \mathrm{constant} = \omega$.

Vertical Configuration

For the $z_i$ coordinates:

\begin{equation} 0 = k \Big[ (z_{i+1} - z_i) - (z_i - z_{i-1}) \Big] + m g \end{equation}

With $z_1 = 0$ fixed and the last link free to swing, the last link obeys:

\begin{equation} 0 = - k (z_i - z_{i-1}) + m g \end{equation}

The solution for $N$ links is:

\begin{equation} z_n = \frac{m g (n-1) (2 N-n)}{2 k} \end{equation}

The lowest distance the chain will droop is:

\begin{equation} z_N = \frac{m g N (N-1) }{2 k} \end{equation}

Going for a meter droop, we have $1m = \frac{m g N (N-1)}{2 k}$, so let’s set this combo of variables to 1.

So our final $z_n$ is:

\begin{equation} z_n = \frac{(n-1) (2 N-n)}{N (N-1)} \end{equation}

Measured in meters.

Radial Balance

For the $R_i$ coordinates (centripetal balance):

\begin{equation} -\frac{ m R_i \omega^2 }k = \Big[ (R_{i+1} - R_i) - (R_i - R_{i-1}) \Big] \end{equation}

To set the first $R_0$ to a certain value, the $R$’s as a whole must obey an eigenvalue problem. I will leave it to the reader to formulate this as finding the eigenvalues of a matrix. Here the eigenvalue is $m \omega^2/k$. From before we can replace the $m/k = 2/(g N (N-1)) \approx 0.204/(N (N-1)) \text{s/m}$. Finding the frequency we can solve for $\omega$ in $\lambda = m \omega^2/k$. This implies that:

\begin{equation} f = \frac 1{2\pi} \frac km \approx 0.780 N (N-1) \lambda \mathrm{Hz} \end{equation}

Here $\lambda$ is an eigenvalue. Looking at the first smallest eigenvalues we find:

Here’s the updated table with the new column titled $f$ Hz (N=20) added:

The new column has been populated with the provided data.
The first mode is simply not moving the chain at all.
The second one (the first non-trivial mode) requires around 7 Hz of circular motion. That’s physically achievable, but would be tiring.
The third and higher modes jump quickly to ~31 Hz and beyond, which would be very challenging in practice.

The first 5 mode chain profiles look like the following where $N=100$:

As you can see in the graph, gravity is predicted to make the modes scrunch up at the bottom.
Here is the same graph with the positions of the chain links shown:

Of course, for a “realistic” chain we can use fewer links:

Conclusion

By modeling the chain as a spring–mass system and looking at the vertical equilibrium plus radial eigenmodes, we get a physically consistent picture of why a twirled chain forms those characteristic shapes. The lowest mode corresponds to a slow, broad swing that is feasible to reproduce, while higher modes demand significantly faster and less practical rotation. Gravity compresses the structure near the bottom, creating the distinctive drooping we see in the real experiment.
All together, the analysis shows that this “chain whirl” is really just the natural blend of classical mechanics: restoring spring-like tension, centrifugal forcing, and gravity, elegantly woven into a surprisingly accessible demo of eigenmodes in action.

References

• Pluto Notebook