Ball, Don't Fall! - Part 2

A solid spherical ball of radius \(r=2 \ \text{m}\) is carefully placed on top of a fixed hemisphere of radius \(R=10 \ \text{m}\) as shown. It is then pushed very slightly.

If the ball doesn't slip till it makes an angle \(\theta=30^{\text{o}}\) with the vertical, then the minimum required value of coefficient of friction between the ball and the hemisphere is \[\dfrac{2}{a\sqrt{b}-c}\]

where \(a,b\) and \(c\) are co-prime positive integers. Find \(a+b+c\)

Details And Assumptions

• Take \(g=9.8 \ \text{m}\text{s}^{-2}\) in the downward direction if needed.