On March 9 1983, Sire/Warner released Madonna’s second single, Burning Up/Physical Attraction, in North America.
Unlike her first single, Burning Up/Physical Attraction was available only as a 12-inch single domestically, while many international markets later released it on 7-inch as well. A promotional 7-inch was issued to promote the release to radio in the U.S., however it oddly featured the b-side, Physical Attraction, on both sides.
The artwork for the single was created by Madonna’s close friend and roommate at the time, the late Martin Burgoyne.
Burning Up was written by Madonna and produced by Reggie Lucas. The song had previously appeared on the four-track demo tape that landed Madonna her recording contract with Sire Records.
Today In Madonna History 
Make no mistake about it, second Warner Rabbit hit Burning Up finds Rebel Madonna in raging full-on Rock mode with her diamond-black Gibson Les Paul guitar and Marshall amplifier cranked up and turned up to eleven-The Unapologetic Bitch with a Rebel Heart gets her first crack at rock & roll and succeeds admirably! Way to go La Ciccone Evita Madge!
It’s interesting that “Burning Up” wasn’t included on the promo sent to radio, nor was it released as a 7″ single. I would guess those are the reasons “Burning Up” didn’t chart on the Billboard Hot 100 (or its Bubbling Under chart). It was basically handicapped both in terms of sales and airplay. Without a 7″, there wasn’t anything for most music consumers to buy, and without a promo, radio stations wouldn’t have been playing it. Seems like Madonna was being marketed solely to a dance crowd with this release.