This root was connected to Elvish words for strength and physical ability for most of Tolkien’s life. It first appeared as ᴱ√POLO “have stength” in the Qenya Lexicon of the 1910s with derivatives like ᴱQ. poldor “physical strength” and ᴱQ. polka “pig”, though the latter was marked by Tolkien with a “?” (QL/75). There were also derivatives in the contemporaneous Gnomish Lexicon such as G. polm “strength (physical)” and G. polod “power, might, authority” (GL/64). The root ᴹ√POL reappeared in The Etymologies of the 1930s with the gloss “physically strong”, extended form ᴹ√POLOD and derivatives like ᴹQ. polda “strong, burly” and ᴹQ. poldore as an element in the name ᴹQ. Poldórea (Ety/POL); this name was variously glossed “Strong One” (SM/79) or “Valiant” (LR/206), though in the later sense it was eventually replaced by Q. Astaldo (S/28).
√POL appeared in a list of roots from around 1959-60 with a minor shift in meaning:
> √pol can, have physical power and ability [as in] “I can jump that”. polin quete means I can speak (because mouth and tongue are free)” (VT41/6).
In another note around this time but probably later (and thus maybe in the early 1960s), Tolkien wrote:
> √POL. This cannot refer to strength. (Too obvious a reminiscence of [Latin] pollens); also it does not account for poli- “meal”, grist. √POL- should have senses “pound up”, break up small, reduced to powder etc. Poldórea, as adjective applied to Tulkas, should be derived from the Elvish nickname of Tulkas (not being derived from Valarin), Poldor, Poldorno: “breaker up of the hard/tough”, √DOR- “hard, tough” (PE17/181).
The mention of poli- “meal” seems to be a reference to a different early root ᴱ√POL-I from the Qenya Lexicon of the 1910s with derivatives like ᴱQ. pole “oats, grain; flour” and ᴱQ. polu “kernel” (QL/75), whereas in The Etymologies of the 1930s the word for “flour, meal” was ᴹQ. pore derived from ᴹ√POR (Ety/POR). This shift in sense to “pound up” seems to be motivated by the similarity of √POL “strength” to Latin “pollens” (able, strong).
However, in another later-still note Tolkien wrote the phrase Q. á rike empollie that seems to mean “try harder” (PE17/167), likely a rough contemporary of other notes from around 1967 exploring the same phrase (PE17/94). This later use of empollie seems to be connected to physical effort, and thus is in line with the 1959-60 note with √POL “can, have physical power and ability” mentioned above.
Finally in a note from around 1968, Tolkien wrote:
> Q. pol, large, big (strong). polda, big. DELETE pole “meal”! Make it mule (PE17/115).
This is explanation appears in a set of notes having to do with “large and small”, probably from around 1968, the date given for an apparently related root-list with similar information (VT47/26 note #26). It seems to firmly discard the connection of √POL to “pound, meal” and restored its connection to “strength” along with a new connection to “largeness”.
Assuming this analysis is correct, the conceptual evolution seems to be:
1910s: ᴱ√POLO “have stength”; ᴱ√POL-I “*grain, flour”.
1930s: ᴹ√POL “physically strong”; ᴹ√POR “*flour, meal”.
1959-60: √POL “can, have physical power and ability”.
early 1960s: √POL “pound up” > ✶poli “meal”.
late 1960s: √POL “large, big (strong)” vs. Q. mulë “meal”.
Neo-Eldarin: For purposes of Neo-Eldarin I think it best to assume √POL has to do with “strength” and “physical ability”. I think it better to assume the connection of the root to √POL “meal, flour” was abandoned, and use √MUL and possibly ᴹ√POR* for that purpose instead.