Solution
= Solution
The radial <effective potential> is $V_{\rm eff}=mgr\cot\alpha+5J^2/(14mr^2)$. Initially $r=8a$, $J=(7m/5)(8a)v_0$. Comparing the initial energy with $V_{\rm eff}(2a)$ gives
$$
\frac7{10}mv_0^2+8mga\cot\alpha\le \frac{56}{5}mv_0^2+2mga\cot\alpha,
$$
which is exactly $v_0^2\ge(4/7)ga\cot\alpha$. Larger speed raises the centrifugal barrier at the neck faster than it raises the initial <kinetic energy>.