This root was connected to Tolkien’s attempt to translate the phrase “try harder” into Quenya around 1966-67. √RIK “strive” first appeared in rough notes as a replacement for √NDEB which Tolkien thought was too close to “endeavor” (PE17/167). Tolkien then firmed up the meaning of √RIK in another note, giving it the gloss “put forth effort, strive, endeavour”, so that “try!” = ā rike in Quenya (PE17/93). Another note had √RIK meaning “try” in the sense “endeavour, make an effort, strive (to do something against an obstacle or opponent)”; this note indicated the root was intransitive and the preposition “against” was required before any obstacle (PE17/182). It seems Tolkien did not entirely abandon √NDEB, however, since a similar form √NDAB “endeavor, try, seek opportunity” appeared in Late Notes on Verb Structure (LVS) from 1969 (PE22/151); see that entry for discussion.
Note that √RIK “strive” is also identical in form to √RIK or ᴹ√RIK(H) “twist”, a root that had a long history in Tolkien’s writings. For purposes of Neo-Sindarin, I assume the two roots co-exist.
This root first appeared as unglossed ᴱ√RIQI or ᴱ√RIKI in the Qenya Lexicon of the 1910s with derivatives ᴱQ. riqi- “wrench, twist” and ᴱQ. marikta “wrist” (QL/80). This root also had derivatives in the contemporaneous Gnomish Lexicon such G. raig “awry, twisted, distorted, perverse, wrong, leering (of face)”, G. rig “a snarl, a sneer”, and G. rig- “twist, contort” (GL/64-65). These forms were also linked to G. grinn “ankle (talgrin), wrist (mabrin)” (GL/42) and G. arc “fierce, harsh, ill tempered; awkward, difficult” < ᴱ✶r̄k- (GL/20). The latter reappeared as ᴱN. arch “rough, fierce” >> “rough” in Early Noldorin Word-lists of the 1920s (GL/137), though in the somewhat later Early Noldorin Dictionary, ᴱN. arch “rough” was given a new derivation from ᴱ✶a-rak-wa (PE13/160), and thus was no longer tied to ᴱ√RIKI.
The root ᴹ√RIK reappeared in The Etymologies of the 1930s (EtyAC/RIK(H)) but it was revised to ᴹ√RIK(H) “jerk, sudden move, flirt” with derivatives like ᴹQ. rinke “flourish, quick stroke” and ᴹQ. rihta-/N. rhitha- “jerk, twitch” (Ety/RIK(H)). In notes from 1959-60 the root appeared again as √RIK “twist” with a derivative Q. raika “crooked” (VT39/7), a word that in The Etymologies of the 1930s was derived from ᴹ√RAYAK. This 1959-60 appearance seems to be harkening back to its meaning in the 1910s.