Ok, I'm wondering . If Uchiha and Uzumaki have a child together, there are possibility that the child inherited Sharingan right? Since Uzumaki have ties with Senju then, can the child awaken Rinnegan?
No, just an average/general Uchiha and Uzumaki marriage cannot give birth to a child who is capable of activating Rinnegan. In order to activate Rinnegan one requires chakra of both Indra and Ashura which enables Hagoromo's chakra. Please refer the below url. http://naruto.wikia.com/wiki/Rinnegan
The correct answer is IT DEPENDS.
Rinnegan, as multiple users have pointed out, can be obtained by combining chakra from Indra and Asura. In each generation, one descendant of the brothers inherits their Chakra.
Thus iff the "chosen" descendents of both Indra and Asura have a child together, their offspring would inherit both Chakras and would have a Rinnegan naturally. Since both Uchihas and Uzumakis have the potential to inherit the Chakra from Indra and Asura, it MAY be possible for an offspring to inherit Rinnegan naturally.
Disclaimer: The above scenario is hypothetical and is intended to point out that Natural Rinnegan may be a possibility. However, its highly likely that all "chosen" successors are male from what we've seen.
Madara did say you need senju and uchiha powers but it's actually the chakra of Indra and ashura combined that could awaken the rinnegan. As for the sharingan its probably very possible the child will get it, sasukes daughter (who is half an uchiha and half of some random clan with no special abilities) has the sharingan and because the uzumaki and uchiha are related (by blood but this is slight as the clans expanding caused the blood ties to become slimmer and slimmer) there is a high chance of the child getting it. It would probably be a very powerful kid if they have 2 powerful clan's blood running through them!
All of you are wrong about this. All Uchiha have Indra's chakra and All Senju(Uzumaki related) have Ashura's chakra. Now Madara/Sasuke and Hashirama/Naruto are special because Indra's and Ashura's reincarnated into them. To answer the question, yes the child will be able to obtain rinnegan. He will be most likely like the sage of the six paths before he obtained sharingan and rinnegan.
The sage wasn't born with rinnegan. He started with nothing, his eyes were almost like a byakugan. When he lost his lover he obtained the sharingan. Afterward, learned how to control sage power and went fight his mother. His mother made him kill his brother (he healed him a few moments after Ain't got time to explain all) which allowed him to awaken the EMS. He had the EMS and sage powers and he ended up with the rinnegan.
Uchiha possess the sharingan Senju possess the sage chakra
Of course if they have a child he will be able to obtain the rinnegan. Although he will have to overcome hardships like losing a friend (sharingan) a lover (MS) and take sibling eyes (EMS) and since he already have sage chakra from Senju parent, the rinnegan will be awakened