`S_(2)` is a fixed rough sphere and `S_(1)` is a uniform solid sphere `S_(1)` is given a negligible velocity, So that at starts moving on the sphere `S_(2)`, It rolls without slipping. Find the angle formed with the vertical, by the line joining the centres of `S_(1)` and `S_(2)` when `S_(1)` leaves `S_(2)`

Answer

`theta= cos^(-1)((10)/(17))`

Explanation

When sphere leaves constant `N = 0`
`Mg cos alpha - N = (mv^(2))/(R )`
`rArr mg cos alpha = (mv^(2))/(R )` ….(i)
By energy conservation
`K_(i) + U = K + U_(f)`
`0 + mgR (1 - cos alpha) = (1)/(2) mv^(2) + (1)/(2)t omega^(2)`
`rArr MgR (1 - cos alpha) = (7)/(10)mv^(2)`
`( :. T + 2//5 mR^(2) "and" v = omega)`
`rArr mgR = (17)/(10mg R cos alpha`
`rArr cos alpha = (10)/(17) rArr alpha = cos^(-1) ((10)/(17))`