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

According to the question, the cyclist moves along the circular path and the centripetal force is provided by the frictional force. Thus from the equation `F_n=mw_n`
`f r=(mv^2)/(r)` or `kmg=(mv^2)/(l)`
or `k_0(1-r/R)g=v^2/r` or `v^2=k_0(r-r^2//R)g` (1)
For `v_(max)`, we should have `(d(r-r^2/R))/(dr)=0`
or, `1-(2r)/(R)=0`, so `r=R//2`
Hence `v_(max)=1/2sqrt(k_0gR)`