I needed a quick exam question related to equilibrium - so I made up the following:
A 10 kg metal bar of length 1 meter is suspended in a horizontal position by two mass-less cables attached to its ends. Cable 1 makes a 30 degree angle with the axis of the rod and cable 2 makes a 60 degree angle with the axis of the rod. What are the tensions in the two cables? (I didn't really write the question out this way - I drew a diagram that was equivalent - can't draw a diagram on the list.)
I ran a quick analysis using the sum of the forces in the x and y directions, checked my results, and they appeared to work. Many of my best students agreed with my answers. However, two of my best students got a different answer for the tensions - and they agreed with each other. I checked their results and their arithmetic was fine. However, they solved the problem using torques about the two ends of the rod.
It took me a quite few minutes to figure out what was wrong - and I was quite embarrassed. The moral of the story is don't make up quickie problems for an exam.