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What is the difference between "unter" and "wegen" when used in a sentence? In my dictionary one can either use "unter" or "wegen" with "müde":
Ich war unter den Hausaufgaben sehr müde.
Ich war wegen der Hausaufgaben sehr müde.
Do "unter" and "wegen" have different meanings in this context?
Here "Unter" suggests something like "while doing". You wouldn't use it, when there is no time involved. You would most likely even say "ich war unter den Hausaufgaben sehr müde geworden" to stress the passing of time. "Unter" in this sense is uncommon in colloquial language.
Actually, neither unter nor wegen are connected with müde in a way that the adjective would require any of these prepositions. You can replace the adjective with any word, and the sentence is equally fine or not fine, respectively:
Ich war wegen der Hausaufgaben sehr langsam.
*Ich war unter den Hausaufgaben sehr langsam.
The reason for taking unter or wegen depends on something else. Let's take another example first:
Ich leide unter Müdigkeit(=Ich bin sehr oft müde und das macht mir zu schaffen).
I would translate this sentence as to suffer from tiredness. In theory, you could also say Ich leide wegen meiner Müdigkeit. There is, however, a subtle difference. I'd translate that as I suffer because of my tiredness. But strictly speaking, it's at least very uncommon if not wrong. I didn't find any Google hits. But I found a few results for Ich leider wegen meiner Größe. Nevertheless, it's still more common to say Ich leider unter meiner Größe.
Anyway, unter etwas leiden is just a fixed expression and the actual definiton of unter isn't really applied. What I mean: you cannot translate it literally. So, you simply choose unter because of it being an expression and not because of the word Müdigkeit.
Back to your example: as already mentioned wegen means because of. So, you're saying:
Because of my homework, I was tired.
First off, using unter would indicate a very different meaning. Unter means during, so while doing the homework. But the above sentence indicates that you're tired after you've finished your homework.
Whatever, unter doesn't work very well here. There's also a fixed expression unter etwas sein, as in unter Druck sein, or you can apply this literally, i.e. you are, for instance, below a surface.
But in your very example unter would be applied in the sense of during. When using unter in this meaning, it usually comes in collocation with Woche or Tag(-> unter Tags or tagsunter). Also take a look in Duden.
As you might realize, the phrase unter der Woche is quite generic. You don't refer to any specific time or action. In contrast, unter den Hausaufgaben would be very specific. So remember, you don't use unter when referring to a particular time frame or action. Again, this is how you would use unter in that sense:
Unter der Woche bin ich häufig müde.
This sentences only says that you are often tired but not when and why exactly. It's not even sure, that you're tired every day.
Your example is weird, but other sentences would work, e.g.:
Ich war unter der Therapie sehr müde.
Ich war wegen der Therapie sehr müde.
(I was so tired during the therapy.)
unter etwas is quite common in medical German.
