Getting into a rut and getting into a groove aren't entirely different things - they're similar things except the first has a negative spin and the second has a positive spin.
They're both situations where you get into a pattern of doing something and it's relatively easy to keep on doing the same thing - just sometimes it's a pattern that you want to stay in (in which case we call it getting into a groove), and sometimes it's a pattern you don't want to stay in (in which case we call it getting into a rut).
Which one's called 'in a rut' and which one's called 'in a groove' is maybe a bit arbitrary. Though - I think the physical sense of 'in a rut' also have negative connotations; I don't think there's a similar positive parallel with 'groove', but not a negative one either.
I think "getting into the groove" is a more recent expression, from record players, when the needle would be "in the groove" and the music was playing and everything was, well, groovy.
Dunno about "in a rut" I think it's gotta do with farming.
A rut is, specifically, a track caused by a wheel or "habitual passage", or a groove in which something runs. Ruts built specifically to ease transport are called "wagonways" and are the origin of modern railways.
I think ruts are deeper and harder to get out of. If you were 4 wheeling, a groove would make it easier to travel the road because someone's been there before, but a rut might be too deep and you could get hung up there. Although I bet the etymology of groove is more like ncp stated.
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They're both situations where you get into a pattern of doing something and it's relatively easy to keep on doing the same thing - just sometimes it's a pattern that you want to stay in (in which case we call it getting into a groove), and sometimes it's a pattern you don't want to stay in (in which case we call it getting into a rut).
Which one's called 'in a rut' and which one's called 'in a groove' is maybe a bit arbitrary. Though - I think the physical sense of 'in a rut' also have negative connotations; I don't think there's a similar positive parallel with 'groove', but not a negative one either.
At least, that's my take on the phrases.
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Dunno about "in a rut" I think it's gotta do with farming.
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But I have nothing intelligent to add.