What is the origin of the idiom "let something rip"?

I would like to know the origin of the idiom "let something rip".

Does anyone know where this usage came from?

The OED’s first citation for ‘let her rip’ is dated 1840. In this sense, the dictionary describes it as colloquial and originally of US origin. It is defined as ‘an exhortation not to restrict the speed of something; (hence) an invitation to act without restraint or to pursue a reckless course’.

Let her rip (English-American, 1840 onwards). Let her go as she wants.

This phrase has a very striking history. "When rival river steam-boats were fully established on the Mississippi and other American rivers, the rival captains would put on every ounce of steam in order to keep ahead. Too frequently the boiler would burst, or 'rip', as emphatically it would when bursting. ' Let her rip ' came to be a common expression amongst these captains when more timid passengers or sensible sub-officers urged him to lower the steam pressure.

Oxford Reference (presumably taken from Green's Dictionary of Slang)

The idiomatic phrase 'let [something] rip', in the sense of "[a command or exhortation to] proceed unchecked", more likely derived from gunners' exhortations than the cries of steamboat captains. The earliest uses of the phrase that I could find refer to gunfire, rather than riverboats.

The earliest use that turned up was in the 29 Dec 1837 issue of The Times-Picayune mentioned in RaceYouAnytime's answer. The reference to cannonfire is explicit:

Amos calculates that when the said cannon are fired off, the mails will reach their destination with as great regularity and far greater velocity than by the ordinary conveyance. He may stuff ours into the cannon pointed South and "let her rip," as the Yankees say — we'll run all risks.

The next clear use of the phrase in the sense at hand was in an 1841 article from the 27 Apr Boston Post. Here the use specifically ascribes the phrase to "gunners":

The cover of a salt-box was hung upon a nail in the bake-house door, as the mark, and M'Donald "let her rip,", as gunners say, and sent the charge a little to the right of the mark, and clear through the door....

An earlier, 1840, use of the phrase, marked as peripheral attestation in OED although undoubtedly a pun on the sense at hand, appears to on its face to be literal:

"Let her rip!" as the noisy politician said when he tore his shirt hurraing for Harrison.

Similarly, a slightly later 1841 use than appears in the 27 Apr 1841 Boston Post shown above, is figurative, and so of little help in tracing the origins of the phrase:

If I found a girl fastened to my desires with the adhesive glue of love, and the dictates of prudence demanded a separation, I always laid hold with the firm grasp of resolution and "let her rip," unmindful of the consequences; ....

Wisconsin Express, Madison, Wisconsin, 01 Sep 1841

Effectively contemporaneous with OED's 31 Aug 1846 attestation of the phrase ("Daily Picayune (New Orleans) 31 Aug. 648/2 Why in the name of h-ll's eternal flints don't the engineer pitch in more pine knots and crack on more steam? Let her rip."), which does appear to refer to steamboat boilers, another use in the 12 Nov 1846 issue of The Perry County Democrat (Bloomfield, Pennsylvania) again refers to gunfire:

'No, sir. Don't be a coward: take aim — one, two, THREE, let 'er rip!'

An ad, also found in 1846, appears to allude to a boast, common since the late 1700s in ads for ship sales, that ships made of oak were of superior strength. The reference is a subtext; the ad is for a grocery store opening, and headlines this quote: "Let her rip — she's all oak!"

I found the phrase used figuratively with reference to starting a game of cards in 1848. In 1849, the phrase is again used jocularly in a pun ostensibly refering to clothes ripping (as in the 1840 use).

In 1850, the cries of "jolly tars" in a sailing regatta suggest another possible origin of the phrase:

Up went the gaff topsails, and up the flying jibs. The faint-hearted cried out for the reefing of the mainsails, but the jolly tars shook out the last canvas rag, and cried, "sink or swim, but let 'em rip!"

New Orleans Weekly Delta, 15 Apr 1850.

Despite the appealing possibility of an early origin in sails ripping, the preponderance of early uses of the phrase 'let [something] rip', where the reference is not obscured by humor or figures, refer to gunfire. Such include the earliest, 1837, the next in 1841, and another in 1846. I could not, however, find any uses refering to steamboats earlier than the 1846 attestation of the phrase in OED.