# 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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