= Solution
\b[The dynamical method.] In <Lagrangian mechanics>, choose independent <generalized coordinates> $q_j$ consistent with the constraints, form the <Lagrangian> $\mathcal L=T-V$, and solve the <Euler-Lagrange equations>,
$$
\frac{d}{dt}\frac{\partial\mathcal L}{\partial\dot q_j}
-\frac{\partial\mathcal L}{\partial q_j}=0.
$$
This form applies to a Newtonian system with ideal constraints, so constraint forces do no allowed <virtual work>, and with the remaining forces derivable from the stated potential. Smooth <holonomic constraints> can be built into the coordinates. Additional nonconservative <generalized forces> would appear on the right. Prescribed shaft rotation is a driven constraint; its constant $\Omega$ is a parameter, not a conserved angular momentum.
\b[Energy of the two balls.] The diagram labels each ball $m/2$, so their combined mass is $m$. It also shows $\theta$ between an arm and the downward vertical, rather than the full angle between the arms. Each ball has horizontal radius $l\sin\theta$, vertical coordinate $-l\cos\theta$, and squared speed $l^2\dot\theta^2+l^2\Omega^2\sin^2\theta$. Thus
$$
T=\frac12ml^2\dot\theta^2+\frac12ml^2\Omega^2\sin^2\theta,\qquad
V=-mgl\cos\theta,
$$
and
$$
\boxed{\mathcal L=\frac12ml^2\dot\theta^2
+\frac12ml^2\Omega^2\sin^2\theta+mgl\cos\theta.}
$$
The massless connecting rods and sliding collar enforce the geometry but contribute no energy.
\b[Effective potential.] Define $\omega_0^2=g/l$. At fixed $\Omega$,
$$
\mathcal L=\frac12ml^2\dot\theta^2-V_{\rm eff}(\theta),\qquad
\boxed{\frac{V_{\rm eff}}{ml^2}
=-\frac12\Omega^2\sin^2\theta-\omega_0^2\cos\theta.}
$$
This <effective potential> determines the reduced $\theta$ dynamics. Its conserved effective energy is not the full lab-frame mechanical energy: the shaft drive can exchange work and angular momentum with the balls.
\b[The change of stability.] Near $\theta=0$,
$$
\frac{V_{\rm eff}(\theta)-V_{\rm eff}(0)}{ml^2}
=\frac12(\omega_0^2-\Omega^2)\theta^2
+\left(\frac{\Omega^2}{6}-\frac{\omega_0^2}{24}\right)\theta^4+O(\theta^6).
$$
For $|\Omega|<\omega_0$, the downward state is a minimum. At $|\Omega|=\omega_0$, its quadratic stiffness vanishes but the quartic coefficient is $+\omega_0^2/8$. Above that speed, the downward state becomes a maximum and stable nonzero openings appear. This is the <gravity-restored centrifugal governor bifurcation>.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2019/paper-1-c9-governor-potential.png]
{title=Original calculated effective potentials below, at and above the critical shaft speed; signed angles display the symmetric mathematical extension of the opening coordinate}
The physical opening is nonnegative; the two minima in the signed extension represent opposite transverse displacements. The stable opening increases continuously from zero toward a right angle as $|\Omega|$ grows. The apparatus is frictionless in the model, so a displaced system oscillates rather than automatically settling at the equilibrium shown by the minima.
\b[Equilibria.] Differentiation gives
$$
\frac{V_{\rm eff}'}{ml^2}=\sin\theta(\omega_0^2-\Omega^2\cos\theta).
$$
Hence the downward equilibrium is $\theta=0$, and the open equilibria satisfy
$$
\boxed{\cos\theta_{\rm eq}=\frac{\omega_0^2}{\Omega^2},\qquad
\theta_{\rm eq}=\arccos\frac{g}{l\Omega^2},
\quad |\Omega|\ge\sqrt{g/l}.}
$$
The signed extension includes the negative of this angle. Formally $\theta=\pi$ is also stationary, but it is an unstable inverted configuration and lies outside the usual opening range of the mechanism.
\b[Small oscillations.] Write $\theta=\theta_{\rm eq}+\eta$. The linear equation is $\ddot\eta+\nu^2\eta=0$, where
$$
\nu^2=\frac{V_{\rm eff}''(\theta_{\rm eq})}{ml^2}
=\omega_0^2\cos\theta_{\rm eq}-\Omega^2\cos2\theta_{\rm eq}.
$$
At the downward state and at a nonzero open state, respectively,
$$
\boxed{\nu_0=\sqrt{\omega_0^2-\Omega^2}\quad(|\Omega|<\omega_0),\qquad
\nu_{\rm open}=\sqrt{\Omega^2-\frac{\omega_0^4}{\Omega^2}}
=|\Omega|\sin\theta_{\rm eq}.}
$$
These are <angular frequencies> in $\mathrm{s^{-1}}$; ordinary frequencies are $\nu/(2\pi)$. Negative $\nu^2$ indicates exponential instability rather than an oscillation. At $\theta=\pi$, $\nu^2=-\omega_0^2-\Omega^2<0$. At the critical speed, the linear frequency is zero and the positive quartic potential gives amplitude-dependent nonlinear restoring motion.
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