I came across this copy of page about simple pendulums some time ago. (If anyone knows the original source for this image/page please let me know and I will add the reference.) From the artistry I think it might be from an older text.
What I remember about seeing this image for the first time was that I immediately focused on the denominator of the equation for Fig. 32. The denominator could have a zero or negative value. (There were no limits/conditions mentioned on the page.) If the numerator is negative then “bw” is greater than “AW” and in the real world the pendulum would actually invert, so that if the pendulum assembly were left free hanging, “bw” would be below and “AW” above the pivot. For the case where the denominator approaches zero, the equivalent length “l” and the period of oscillation “t” approach infinity, which eventually results in a balanced pendulum, one that wouldn’t oscillate.
It is fairly simple but I am still always pleased when you see something that might be a bit odd mathematically but can still be reconciled physically.