If words in file2 are unique then you can reverse the order of the files, build an array from file2 and apply it to file1:
$ awk 'NR==FNR{a[$1]=$2;next}{print $1,a[$2]}' file2 file1
word_1 root_of_word_1
word_2 root_of_word_2
word_3 root_of_word_3
word_4 root_of_word_1
(Obviously you can redirect the output to another file from within the awk, but redirection part of your question is not clear to me).