Textbook says "not really acceptable": http://nualeargais.ie/gnag/modal.htm#brauchen
If I read the tips and notes correctly, then
"Tá uirthi rith"
rather expresses being forced to do something. If you want / need something, then you would say:
"Tá / teastaíonn uaithi rith."
note difference: "on her to run" (= must, being forced from outside) vs. "from her to run" (sort of coming from inside and of own will).
Then of course may be discussion in English, at what stage the verb "need" is more related to "want" and at what stage more to "must". Maybe this is again a case of subtle differences between the languages Irish and English.
I'm confused. What makes this (and similar others) a "must" sentence? (you must, she must, etc.)
As explained in the Tips and Notes section:
The basic meaning of this word is on. For example, Ritheann sé ar bhóthar means He runs on a road.
When used with the verb bí, it conveys the idea of obligation to do something. For example, Tá orm rith means I must run. (The literal translation of the phrase would be "It is on me to run".)
Tá ort snámh You must swim
Tá ar Phól éisteacht Paul must listen
Tá orthu siúl They must walk"
What is the difference between "Tá uirthi rith" agus "Caithfidh sí rith" I had never before heard of ar being used but had always used the verb caith to express compulsion or obligation.
What's the difference between "She has to run", "She must run", "She will have to run" or "She is obliged to run"? They all mean basically the same thing, but you might have a preference for one form over another in certain circumstances. It's much the same with Tá ar and Caithfidh (and is gá de and ní mór de and various other phrases). You can probably stick with Caithfidh for your own use, but you need to recognize the other forms when you encounter them in other people's Irish.
Aside from tá uirthi rith and is gá di rith, you could also say caithfidh sí rith, ní mór di rith, ní foláir di rith or tá aici le rith.
It's contained in uirthi. One of the ways to express "must" or "have to" in Irish involves the preposition ar (so, it's "on" someone) with the verbal noun. The person doing it is the object of the preposition, therefore it inflects.
No it's present. We haven't met the past tense yet, but you'd make it past tense by replacing the present-tense verb to be tá ("is") with the past-tense form of the verb to be bhí ("was"): bhí uirthi rith: she had to run.
This phrase is Tá ar X Y, where X is a person, and Y is a verb, meaning "X must Y". When you use a pronoun for the person, it gets combined with ar - X is "her", so uirthi.
There is a completely different phrase Tá Z ó X, where X is still a person, but Z is a noun, not a verb, meaning "X wants a Z". Note that the order of X and Z has changed, and the combination of ó with her is uaithi.