That's not how you make ice cream

A sphere of negligible mass and radius \(r = 1 \text{ cm}\) is placed inside a smooth hollow cone with semi-vertical angle \(\alpha = \dfrac{\pi}{6}\).

Then, a liquid of density \(\rho = 1000 \text{ kg/m}^{3}\) rises into the cone until no more liquid can possibly enter.

If the normal reaction force per unit length \(N_\lambda\) between the sphere and the walls of the cone can be expressed in SI units as \(\frac{a\sqrt{b}}{c}\) where \(a\), \(b\) and \(c\) are natural numbers with \(b\) square free and \(a\) and \(c\) coprime, what is the value of \(a+b+c\)?

Details and Assumptions:

  • Assume that the sphere forms a tight seal with the walls of the cone and prevents the liquid from rising further

  • A small hole is present at the vertex of the cone to allow air to escape

  • Take \(g = 10 \text{m/s}^2 \).


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