A cyclist rides along the circumference of a circular horizontal plane of radius `R`, with the friction coefficient `mu=mu_(0)(1-(r )/(R ))`, where `mu_(0)` is constant and `r` is distance from centre of plane `O`. Find the radius of the circle along which the cyclist can ride with the maximum velocity, what is this valocity?

Explanation

`f=(mv^(2))/(r )`
`muN=mu_(0)(1-(r )/(R ))mg=(mv^(2))/(r )`
`v^(2)=mu_(0)(r-(r^(2))/(R ))g`
For `v` to be maximum
`(dv)/(dr)` or `(d(v^(2)))/(dr)=0`
`mu_(0)g(1-(2r)/(R ))=0impliesr=(R )/(2)`
`v_(max)=sqrt(mu_(0)[(R )/(2)-((R//2)^(2))/(R )]g)=(1)/(2)sqrt(mu_(0)gR)`