der Rücken
back
The rear part of the human or animal body between the neck and the buttocks.
Ich habe vom langen Sitzen Schmerzen im Rücken.
I have back pain from sitting for so long.

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back
The rear part of the human or animal body between the neck and the buttocks.
Ich habe vom langen Sitzen Schmerzen im Rücken.
I have back pain from sitting for so long.

The word 'Rücken' (back) is the basis for many other words that have to do with 'hinten' (behind/back) or 'zurück' (back), for example 'zurück' (back), 'Rückseite' (back side), 'Rucksack' (backpack), or 'Rückgrat' (backbone).
Für die Wanderung packe ich meinen Rucksack.
I'm packing my backpack for the hike.
In connection with the body part, fixed phrases like 'Schmerzen im Rücken haben' (to have back pain), 'auf dem Rücken liegen' (to lie on one's back), or colloquially 'Rücken haben' (to have back pain) are very common.
Ich schlafe am liebsten auf dem Rücken.
I prefer to sleep on my back.
The idiom 'jemandem in den Rücken fallen' means to betray someone. The person who is betrayed is always in the dative.
Er hat mir versprochen zu helfen, aber dann ist er mir in den Rücken gefallen.
He promised to help me, but then he stabbed me in the back.
The idiom 'den Rücken frei haben' means that you can focus on something without worries or distractions. You have the support you need.
Dank der Hilfe meiner Eltern habe ich den Rücken frei, um mich auf mein Studium zu konzentrieren.
Thanks to my parents' help, I have my back free to focus on my studies.
In a figurative sense, 'Rücken' (back) stands for support. A common phrase is 'jemandem den Rücken stärken' (to strengthen someone's back), which means to support someone morally.
In der schwierigen Situation hat mir meine Familie immer den Rücken gestärkt.
In the difficult situation, my family always had my back.
support
'die Unterstützung' and 'der Rücken' can both have to do with help and backing. 'Unterstützung' is the normal word for practical, moral, or other kinds of support. 'Rücken' is used in this meaning more figuratively, for example when someone 'stärkt jemandem den Rücken' (backs someone up) or gives them support. It sounds more personal and often more emotional. For clear help in everyday life, work, or money matters, 'Unterstützung' is more natural. 'Rücken' fits better in fixed or figurative expressions.

Vor dem schwierigen Gespräch stärkt das Team ihr den Rücken.
Before the difficult conversation, the team backs her up.
Bei der Bewerbung braucht Mehmet die Unterstützung seiner Lehrerin.
Mehmet needs his teacher's support with the application.