A cyclist rides along the circular path of a circular horizontal plane of radius R, the firctional coefficient being dependent only on distance r from the centre O of the plane as `mu = mu_(0) (1 - (r )/(R ))`, where `mu_(0)` is a constant.
Find the radius of the circle such that the cyclist can ride with the maximum velocity.

Options

A. R
B. `(R )/(2)`
C. `(R )/(3)`
D. `(R )/(4)`

Answer

B

Explanation

`mu mg = (mv^(2))/(r )`
`mu_(0) (1 - (r )/(R )) = (mv^(2))/(r )` When v is max., `v^(2)` is also max.
So, `(d)/(dr) (v^(2)) = (mu_(0))/(m) (1 - (2r)/(R )) = 0`
`r = (R )/(2)`
`v = sqrt((mu_(0))/(m) (r - (r^(2))/(R ))) `
for `v_("max",) r = (R )/(2)`