The middle point of a bridge, in the form of a circular arc on a canal of width `24m` at height `4m` from either end. Find the maximum speed at which a car can safely pass over the bridge.

Explanation

First find the radius of circular arc.Let `O` is center of circle and `R` is radius.
`Oc^(2)+BC^(2)=OB^(2)`
`(R-4)^(2)+(12)^(2)=R^(2)`
`R^(2)-8R+16+144=R^(2)`
`8R=160impliesR=20m`
`mg cos theta-N=(mv^(2))/(R )`
`N=mg cos theta-(mv^(2))/(R )`
If the car passes safely over the bridge, then
`Nge0`
`mg cos theta-(mv^(2))/(R )ge0`
`v^(2)legR cos theta`
Maximum value of `v^(2)=g R cos theta`
`v^(2)` is maximum when `cos theta=1`.
`v_(max)=sqrt(gR)=sqrt(10xx20)`
`=10sqrt(2) m//s`