Technically OSU kicked off the 50s by winning the 1950 Rose Bowl that concluded the 1949 season.
1950 was Wes Fesler's final season as HC. They finished 6-3, good for second in the Big Ten, and Vic Janowic won the Heisman. All three losses were by a TD or less, two were against top ten teams in SMU and Illinois, and the other was against the Wolverines. The Michigan loss was the death blow for the Fesler regime, as he struggled mightily in that series, going 0-3-1 in that series as HC. Fesler had been an All American player at OSU during the late 20s, leading the Buckeyes to two Michigan wins in 28 and 29.
Enter Woody Hayes. WWH's first three seasons were nothing to write home about, although he did beat the Wolverines in 52; OSU's lone home game in the series during that stretch. But then he finally earned his keep by winning the NC in 1954, going undefeated/untied and besting USC in the Rose Bowl. In 55 they didn't repeat, but they did beat Mich again, and Hopalong Cassidy won the Heisman. The 1957 team also won the NC, beating Oregon in the Rose Bowl after winning THE GAME. OSU also defeated the Wolverines in 58 and 59.