Substitute r2=L-r 1 into r 1/r2=m2/m 1 to get r 1=m2L/(m 1+m2).

Substitute r 1=m2L/(m 1+m2) into r 1/r2=m2/m 1 to get R2 = m1l/(m1+m2);

Then replace r 1=m2L/(m 1+m2) and r2=m 1L/(m 1+m2) with GM1m2/L2 = m1w2r respectively.

W = √ [g (m 1+m2)/l 3] can be solved separately;

Then t = 2π/w = 2π √ [l 3/g (m 1+m2)].

If you don't understand, you can ask.