The giant Irish-born paceman could only send down 8.2 overs before he limped from the Sydney cricket ground late in the opening day.
Rankin, 29, who replaced Tim Bresnan in one of three English changes from the side beaten by eight wickets in the fourth Melbourne Test, bowled one ball in his ninth over before he came off for treatment before tea.
He later returned to the field and bowled just one more delivery before again pulling up short. He trudged from the field, reducing the tourists to just three front-line fast bowlers and tyro leg-spinner Scott Borthwick.
Cook had early success after winning his first toss of the series, sending the home side into bat and reducing Australia to 97 for five before Brad Haddin and Steve Smith restored the innings with a 128-run sixth-wicket stand.