A car is moving with uniform speed over a circular bridge of radius `R` which subtends an angle of `90^(@)` at its center. Find the minimum posible speed so that the car can cross the bridge without losing the contact any where

Explanation

Let the car losses the contact at angle `theta` with the vertical

` mg cos theta - N = (mv^(2))/(R)`
`N = mg cos theta - (mv^(2))/(R )` …(i)
For losing the contact `N = 0`
`rArr v = sqrt(Rg cos theta)` [from Eq(i)]
For minimum speed ` cos theta ` should be minimum so that the `theta` should be maximum
`theta_(max) = 45^(@) rArr cos 45^(@) = (1) /(sqrt(2)`
`v_(min) = v_(min) = ((Rg)/(sqrt(2)))^(1//2)`
so that if car cannot lose the cvontact at inital or point , car cannot be lose the contact anywhere