rip-saw n.

"tear apart, cut open or off," c. 1400, rippen, "pull out sutures," probably from a North Sea Germanic language (compare Flemish rippen "strip off roughly," Frisian rippe "to tear, rip;" also Middle Dutch reppen, rippen "to rip") or else from a Scandinavian source (compare Swedish reppa, Danish rippe "to tear, rip"). Likely most or all of them are from a Proto-Germanic *rupjan- (from PIE root *reup-, *reub- "to snatch"). "Of somewhat obscure origin and history; it is not quite certain that all the senses really belong to the same word" [OED].

The meaning "to slash with a sharp instrument" is from 1570s. Intransitive sense of "be torn or split open" is by 1840. Related: Ripped; ripping. In old U.S. slang, "to utter strong language" (1772), often with out; hence "break forth with sudden violence." The meaning "to move with slashing force" (1798) is the sense in let her rip "allow something to go or continue unrestrained," an American English colloquial phrase attested by 1846.

At another time, when a charge was ordered one of the officers could not think of the word, and he shouted—'Let 'er rip!'—when the whole line burst out with a yell—'Let 'er rip!' and dashed in among the Mexicans, laughing and shouting this new battle cry. [from an account of Illinois volunteers in the Mexican-American War, in the Pensacola Gazette, March 29, 1851] 

  In garments we rip along the line at which they were sewed ; we tear the texture of the cloth; we say, "It is not torn; it is only ripped." More broadly, rip, especially with up, stands for a cutting open or apart with a quick, deep strike: as, to rip up a body or a sack of meal. Rend implies great force or violence. [Century Dictionary]

[toothed cutting tool] Middle English saue, from Old English sagu, from Proto-Germanic *sago "a cutting tool" (source also of Old English seax "knife," Old Norse sög, Norwegian sag, Danish sav, Swedish såg, Middle Dutch saghe, Dutch zaag, Old High German saga, German Säge "saw"), from PIE root *sek- "to cut" (source also of Latin secare "to cut").

In reference to its use as a musical instrument, by 1905. Saw-grass, the long, toothed grass found in the Southern U.S., is attested by 1822. The saw-fly (1773), destructive to plants, is so called for the construction of the insect's egg-depositing organ.