Hi

The first equation gives Fr = f(alpha)

The second equation gives N = g(alpha)

Substitute into the third equation

f(alpha) cos(beta) - g(alpha) sin(beta) = 2 Io alpha / L

This leads to alpha = 3g sin(beta) / L

Substitute into the first equation Fr = 3mg sin(beta) cos(beta) / 2

and into the second equation