A problem or difficulty that prevents something from progressing normally, or a psychological or emotional problem that troubles someone.
From 'hang up' (phrasal verb, early 1900s) meaning to disconnect a telephone or to suspend something; combined into a noun form to describe obstacles or psychological issues.
During the early phone era, 'hanging up' literally meant ending a call—but by mid-20th century, therapists borrowed the term metaphorically for emotional obstacles, and it stuck so well we now talk about 'communication hangups' everywhere from workplaces to relationships.
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