"cut off," 1510s, from Middle English loppe (n.) "small branches and twigs trimmed from trees" (early 15c.), of unknown origin. Related: Lopped (mid-15c.); lopping. Place name Loppedthorn is attested from 1287.
"droop, hang loosely," 1570s, probably a variant of lob or of lap (v.); cf. lopsided (1711), originally lapsided, first used of ships. Lop-eared attested from 1680s. Related: Lopped; lopping.