= Solution
\b[Fixed-angular-momentum form.] Eliminating $\dot\phi$ using the conserved <angular momentum> $h$ gives
$$
\boxed{\ddot\theta=\frac{h^2}{m^2l^4}\frac{\cos\theta}{\sin^3\theta}-\frac gl\sin\theta.}
$$
Equivalently, the polar motion has <effective potential>
$$
V_{\mathrm{eff}}(\theta)=\frac{h^2}{2ml^2\sin^2\theta}-mgl\cos\theta,
\qquad ml^2\ddot\theta=-V_{\mathrm{eff}}'(\theta).
$$
For the steady <conical pendulum> at $\theta_0$, $\ddot\theta=0$, so
$$
\boxed{h^2=m^2gl^3\frac{\sin^4\theta_0}{\cos\theta_0},\qquad
h=\pm m\sqrt{gl^3}\frac{\sin^2\theta_0}{\sqrt{\cos\theta_0}}.}
$$
The sign of $h$ agrees with the sign of $\Omega$; substituting $h=ml^2\sin^2\theta_0\Omega$ verifies the constant-angle result in part (d).
\b[Period of small polar oscillations.] Put $\theta=\theta_0+\alpha(t)$, with $|\alpha|\ll1$ and $h$ held fixed. This is the appropriate <small nutation about a conical pendulum>; keeping $\dot\phi$ fixed instead would violate the <angular momentum> constraint during the oscillation. If
$$
f(\theta)=\frac{h^2}{m^2l^4}\frac{\cos\theta}{\sin^3\theta}-\frac gl\sin\theta,
$$
then $f(\theta_0)=0$ and
$$
f'(\theta_0)=-\frac{h^2}{m^2l^4}\left(\frac1{\sin^2\theta_0}+\frac{3\cos^2\theta_0}{\sin^4\theta_0}\right)-\frac gl\cos\theta_0
=-\frac gl\frac{1+3\cos^2\theta_0}{\cos\theta_0}.
$$
The <linearization> is $\ddot\alpha+\omega_\theta^2\alpha=0$, giving
$$
\boxed{\omega_\theta^2=\frac gl\frac{1+3\cos^2\theta_0}{\cos\theta_0}
=\Omega^2(1+3\cos^2\theta_0),\qquad
T_\theta=2\pi\sqrt{\frac{l\cos\theta_0}{g(1+3\cos^2\theta_0)}}.}
$$
Since $0<\theta_0<\pi/2$, the coefficient is positive and these small polar oscillations are stable. To first order, $\dot\phi=\Omega(1-2\alpha\cot\theta_0)$, making the compensating change in azimuthal <angular velocity> explicit. The limit $\omega_\theta/|\Omega|\to2$ as $\theta_0\to0$ concerns radial modulation of a small nonzero circular orbit; the exactly planar $h=0$ pendulum is a different, degenerate limit of these polar coordinates.
\Image[/past-exam-of-the-physics-1-paper-of-the-university-of-oxford/2015-resit-spherical-pendulum-nutation.png]
{title=Conical motion and the frequency of small polar oscillations at fixed vertical angular momentum}
{description=Original schematic of the <spherical pendulum> and a plot of $\omega_\theta/|\Omega|=\sqrt{1+3\cos^2\theta_0}$ on the nontrivial conical-orbit interval. Open endpoints denote limits, not additional constant-angle solutions.}
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