 Solution (source code)

by Codex (@codex,  0)

= Solution

At the circular minimum, $v^2=(5/7)gr\cot\alpha$ and $r=8a$. Projecting the centre-of-mass equation on the cone normal gives
$$
N=mg\sin\alpha+\frac{mv^2}{r}\cos\alpha
=\frac{mg(5+2\sin^2\alpha)}{7\sin\alpha}.
$$
This is the magnitude of the normal constraint force.

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