Question 13 of 13: Self-Righting Stability of a Falling Playing Card

Question 13: Self-Righting Stability of a Falling Playing Card (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A thin flat card falling broadside-on (large face roughly horizontal, i.e. perpendicular to its fall velocity), released either perfectly level or with a small tilt angle.

Find. How the streamline pattern changes with tilt, and how that change in pressure produces a self-righting (stabilizing) moment.

Level card. Falling broadside-on and perfectly level, the relative airflow approaches the underside symmetrically from below. The streamlines are deflected outward in a symmetric fan, and a broad region of elevated (near-stagnation) pressure builds up uniformly across the entire underside. Because this high-pressure region is symmetric about the card's centre, it produces a net UPWARD force (contributing to the drag that limits terminal fall speed) but NO net moment — there is nothing to rotate the card one way or the other, so a perfectly level card, once falling steadily, has no tendency to tip.

Tilted card. The instant the card tips slightly, one edge (the leading, lower edge in the direction of tilt) drops closer to being square-on to the oncoming relative flow, while the opposite (trailing) edge rotates further away from it. On the LOW edge, the streamlines are compressed into a smaller effective gap and the local flow is more strongly decelerated (closer to a true stagnation condition), so the pressure there RISES above the level-card value. On the HIGH edge, the card face is now more nearly aligned WITH the flow, the streamlines pass more freely underneath with less deflection, and the local pressure there FALLS below the level-card value.

Resulting stability. This asymmetric pressure distribution — high on the low (leading) edge, low on the high (trailing) edge — produces a net moment about the card's centre that pushes the LOW edge back UP and lets the HIGH edge drop back down: precisely a RESTORING moment, opposing the tilt and driving the card back toward level. This is the fluid-dynamic analogue of a weathervane or a dart's tail fins: any small angular disturbance is met by a pressure-generated moment that opposes (rather than amplifies) it, which is exactly the mechanism absent when the SAME card is dropped edge-first (vertically) — in that orientation there is no broad stagnating face to generate this restoring pressure asymmetry, so the smallest disturbance grows instead of being damped, producing the irregular tumbling and side-veering described in the question.