c. 1300, "implement for throwing stones," from an unidentified continental Germanic source (such as Middle Low German slinge "a sling"); see sling (v.). The notion probably is of a sling being twisted and twirled before it is thrown. Sense of "loop for lifting or carrying heavy objects" first recorded early 14c. Meaning "piece of cloth tied around the neck to support an injured arm" is first attested 1720.
<a href="https://www.etymonline.com/word/slung">Etymology of slung by etymonline</a>
Harper, D. (n.d.). Etymology of slung. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/slung