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It was really cool to see this gif because we explored this exact scenario in my classical mechanics class! In the gif, when the book was spun around the first two axes it was stable, but the last one it was wobbling. In that class, we showed that for small angle deviations from the rotation axis, in the last scenario, the small perturbations would grow exponentially and thus it wouldn't be stable! When spinning around the other two axes the small perturbations went to zero and didn't impact the spinning.