Solution
= Solution
The <Euler-Lagrange equation> from <9/a> is $2MR^2\ddot\theta+MgR\sin\theta=0$. Linearizing about $\theta=0$ gives
$$
\boxed{\omega_{\rm fixed}=\sqrt{\frac{g}{2R}},\qquad f_{\rm fixed}=\frac1{2\pi}\sqrt{\frac{g}{2R}},\qquad T_{\rm fixed}=2\pi\sqrt{\frac{2R}{g}}.}
$$
Both <angular frequency> and ordinary <frequency> are given to distinguish the two conventions. The stable position places the extra <mass> below the pivot.