The Falling Stick Problem

Here's a little problem for physics teachers, mechanics enthusiasts, slo-mo videographers, science communicators, and other curious onlookers. Let a straight and rigid stick stand vertically on a flat horizontal surface. Then, without imparting any initial velocity, release the stick and let it fall from standstill. As it falls, the angle it makes with the vertical keeps increasing. Let's call this the fall angle (ϕ). At what fall angle does the bottom of the stick move away from where it was originally placed? In other words, what is the take-off angle (ϕm)?

In this IOP Bioinspiration and Biomimetics paper from 2015, we were looking at this problem as part of a modeling exercise on human walking, with a simplistic view considering the human as a 2D inverted pendulum, like the falling stick above. In that model, using some basic mechanics equations (no rocket science involved), we predict that the take-off angle is about 48°, cos^-1(2/3) to be exact. This result holds irrespective of the mass and length of the stick or even which planet you're doing this activity on, a heart-warming general result. The figure below shows the take-off angle for four different leg lengths (0.4m, 0.6m, 0.8m, 1.0m). The x-axis (vo) denotes the initial velocity of the bob of the inverted pendulum. Ping me if you are interested to read the paper but cannot access it in the above IOP link.

 Here is a snippet from the paper.

"As the inverted pendulum falls, the compression in the stance leg keeps reducing. At the instant the compression becomes zero, the ground will need to start pulling on the rod. This is not possible (unless the foot is stuck on the ground due to some adhesive or suction pads) and hence the inverted pendulum takes off." ~L Patnaik and L Umanand

Without slo-mo videography―and with more important fires to put out in order to finish a PhD―we never really verified the 48° claim experimentally. I would be happy to know if someone has, or can capture, slo-mo video footage of this mundane phenomenon, to help find the answer. The underlying message is to showcase the amazing fact that simple mathematical models can often make reasonably accurate predictions about the world around us.