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Two hemispheres of negligible of radius R combined together by two minimum forces ‘F’ to form a sphere (completely filled with water) as shown in figure. A small pin hole of radius a at top of the sphere as shown in figure. If the system is in equilibrium at $\tan θ = \frac{x}{2}$ . Find the value of x. (gravity is present and no other support is given to hemispheres)

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answer is 3.

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Detailed Solution

$2 F \cos θ = m g F \cos θ = \frac{m g}{2} F \sin θ = (ρ g R) π R^{2} \tan θ = \frac{2 ρ g R^{3} π}{m g} \Rightarrow \frac{2 ρ g R^{3} π}{ρ \times \frac{4}{3} π R^{3} \times g} = \frac{3}{2}$

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