Hi Nick,
I did what I should have done at the start: if everything else fails, read the instructions
http://www.surfboard.com/wp-content/uploads/2014/07/ARRIS_SURFboard_SB6141_User_Guide.pdf
And clearly the computer does plug into the modem! There's nothing at all in the user manual about how to add a router so that anyone with more than one device in the house can build a LAN. (It does happen!)
I found some advice from DLink about the recommended procedure in general for powering up a cable modem and router, but I expect you do that anyway
http://www.dlink.com/uk/en/support/...-router-to-work-with-a-cable-modem-connection
And in relation to Motorola modems I found this
http://ccm.net/faq/4853-connect-a-wireless-router-to-a-motorola-modem
And, finally, I came across this topic - cable modem and router not talking to each other - which might make sense to you, though I'd advise you not to do anything until you've read it through completely and even then only if you're absolutely sure
http://forums.anandtech.com/showthread.php?t=2321608
If none of the above helps, I'd recommend phoning your ISP and the Motorola tech help number in the user manual. Please Tell us how you get on.
Martin
EDIT. Just seen the posting that went up 3 mins ago. Pleased you sorted it and that it was so simple.