I have a m1.small instance in amazon with 8GB hard disk space on which my rails application runs. It runs smoothly for 2 weeks and after that it crashes saying the memory is ful
As Preston mentioned you don't have a memory problem (over 40% free), you have a disk full problem. du reports most of the storage is consumed in /root/data.
You could use find to identify very large files, eg, the following will show all files under that dir greater than 100MB in size.
sudo find /root/data -size +100M
If unicorn is still running, lsof (LiSt Open Files) can show what files are in use by your running programs or by a specific set of processes (-p PID), eg:
sudo lsof | awk '$5 ~/REG/ && $7 > 100000000 { print }'
will show you open files greater than 100MB in size